To my son Philippe for his unbounded energy and optimism.
On the classification of Floer-type theories.
Nadya Shirokova.
Abstract
In this paper we outline a program for the classification of Floer-type theories, (or defining
invariants of finite type for families). We consider Khovanov complexes as a local system on the
space of knots introduced by V. Vassiliev and construct the wall-crossing morphism. We extend
this system to the singular locus by the cone of this morphism and introduce the definition of
the local system of finite type. This program can be further generalized to the manifolds of
dimension 3 and 4 [S2], [S3].
http://arxiv.org/abs/0704.1330v1
Contents
1. Introduction.
2. Vassiliev’s and Hatcher’s theories.
2.1. The space of knots, coorientation.
2.2. Vassiliev derivative.
2.3. The topology of the chambers of the space of knots.
3. Khovanov homology.
3.1. Jones polynomial as Euler characterictics. Skein relation.
3.2. Reidemeister and Jacobsson moves.
3.3. Wall-crossing morphisms.
3.4. The local system of Khovanov complexes on the space of knots.
4. Main definition, invariants of finite type for families.
4.1. Some homological algebra.
4.2. Space of knots and the classifying space of the category.
4.3. Vassiliev derivative as a cone of the wall-crossing morphism.
4.4. The definition of a theory of finite type.
5. Theories of finite type. Further directions.
5.1. Examples of combinatorially defined theories.
5.2. Generalizations to dimension 3 and 4.
5.3. Further directions.
6. Bibliography.
1. Introduction.
Lately there has been a lot of interest in various categorifications of classical scalar invariants,
i.e. homological theories, Euler characteristics of which are scalar invariants. Such examples
include the original instanton Floer homology, Euler characteristic of which, as it was proved by
C.Taubes [T], is Casson’s invariant. Ozsvath-Szabo [OS] 3-manifold theory categorifies Turaev’s
torsion, the Euler characteristic of their knot homologies [OS] is the Alexander polynomial. The
theory of M. Khovanov categorifies the Jones polynomial [Kh] and Khovanov-Rozhansky theory
categorifies the sl(n) invariants [KR] .
The theory that we are constructing will bring together theories of V. Vassiliev, A. Hatcher
and M. Khovanov, and while describing their results we will specify which parts of their con-
structions will be important to us.
The resulting theory can be considered as a ”categorification of Vassiliev theory” or a clas-
sification of categorifications of knot invariants. We introduce the definition of a theory of
finite type n and show that Khovanov homology theory in a categorical sense decomposes into
a ”Taylor series” of theories of finite type.
The Khovanov functor is just the first example of a theory satisfying our axioms and we
believe, that all theories mentioned above will fit into our template.
Our main strategy is to consider a knot homology theory as a local system, or a constructible
sheaf on the space of all objects (knots, including singular ones), extend this local system to
the singular locus and introduce the analogue of the ”Vassiliev derivative” for categorifications.
By studying spaces of embedded manifolds we implicitly study their diffeomorphism groups
and invariants of finite type. In his seminal paper [V] Vassiliev introduced finite type invariants
by considering the space of all immersions of S1 into R3 and relating the topology of the singular
locus to the topology of its complement via Alexander duality. He resolved and cooriented the
discriminant of the space and introduced a spectral sequence with a filtration, which suggested
the simple geometrical and combinatorial definition of an invariant of finite type, which was
later interpreted by Birman and Lin as a ”Vassiliev derivative” and led to the following skein
relation.
If λ be an arbitrary invariant of oriented knots in oriented space with values in some Abelian
group A. Extend λ to be an invariant of 1-singular knots (knots that may have a single
singularity that locally looks like a double point  ), using the formula
λ( ) = λ(!) − λ(")
Further extend λ to the set of n-singular knots (knots with n double points) by repeatedly
using the skein relation.
Definition We say that λ is of type n if its extension to (n + 1)-singular knots vanishes
identically. We say that λ is of finite type if it is of type n for some n.
Given the above formula, the definition of an invariant of finite type n becomes similar to
that of a polynomial: its (n+1)st Vassiliev derivative is zero.
It was shown that all known invariants are either of finite type, or are infinite linear combi-
nations of those, e.g. in [BN1] it was shown that the nth coeffitient of the Conway polynomial
is a Vassiliev invariant of order ≤ n.
In this paper we are working with Khovanov homology, which will be our main example,
however the latest progress in finding the combinatorial formula for the differential of the
Ozsvath-Szabo knot complex [MOS], makes us hopeful that more and more examples will be
coming.
For the construction of the local system it is important to understand the topological type
of the base. The topology of the connected components of the complement to the discriminant
in the space of knots, called chambers, was studied by A. Hatcher and R.Budney [H], [B].
They introduced simple homotopical models for such spaces. Recall that the local system is
well-defined on a homotopy model of the base, so Hatcher’s model is exactly what is needed to
construct the local system of Khovanov complexes.
Throughout the paper the following observation is the main guideline for our constructions:
local systems of the classifying space of the category are functors from this category to the
triangulated category of complexes.
It would be very interesting to understand the relation between the Vassiliev space of knots
is the classifying space of the category, whose objects are knots and whose morphisms are knot
cobordisms.
Our construction provides a Khovanov functor from the category of knots into the triangu-
lated category of complexes.
This allows us to translate all topological properties of the space of knots and Khovanov
local system on it into the language of homological algebra and then use the methods of
triangulated categories and homological algebra to assign algebraic objects to topological ones
(singular knots and links).
Recall that in his paper [Kh] M.Khovanov categorified the Jones polynomial, i.e. he found
a homology theory, the Euler characteristics of which equals the Jones polynomial. He starts
with a diagram of the knot and constructs a bigraded complex, associated to this diagram,
using two resolutions of the knot crossing:
0-resolution 1-resolution
The Khovanov complex then becomes the sum of the tensor products of the vector space
V, where the homological degree is given by the number of 1’s in the complete resolution of
the knot. The local system of Khovanov homologies on the Vassiliev’s space of knots can be
considered as invariants of families of knots.
The discriminant of Vassiliev’s space corresponds to knots with transversal self-intersection,
i.e. moving from one chamber to another we change overcrossing to undercrossing by passing
through a knot with a single double point. We study how the Khovanov complex changes under
such modification and find the corresponding morphism.
After defining a wall-crossing morphism we can extend the invariant to the singular locus
by the cone of a morphism which is our ”categorification of the Vassiliev derivative”. Then we
introduce the definition of a local system of finite type: the local system is of finite type n if for
any selfintersection of the discriminant of codimension n, its n’th cone is an acyclic complex.
The categorification of the Vassiliev derivative allows us to define the filtration on the Floer
- type theories for manifolds of any dimension.
In [S4] we prove the first finiteness result:
Theorem [S4]. Restricted to the subcategory of knots with at most n crossing, Khovanov
local system is of finite type n, n ≥ 3 and of type zero n = 0, 1, 2.
This definition can be generalized to the categorifications of the invariants of manifolds of
any dimension: we construct spaces of 3 and 4-manifolds by a version of a Pontryagin-Thom
construction, consider homological invariants of 3 and 4-manifolds as local systems on these
spaces and extend them to the discriminant.
In subsequent papers our main example will be the Heegaard Floer homology [OS], the Euler
characteristic of which is Turaev’s torsion. We show that local systems of such homological
theories on the space of 3 - manifolds [S1] will carry information about invariants of finite type
for families and information about the diffeomorphism group. We also have a construction [S2]
for the refined Seiberg-Witten invariants on the space of parallelizable 4-manifolds.
Acknowledgements. My deepest thanks go to Yasha Eliashberg for many valuable discus-
sions, for inspiration and for his constant encouragement and support. I want to thank Maxim
Kontsevich who suggested that I work on this project, for his attention to my work during my
visit to the IHES and for many important suggestions.
I want to thank graduate students Eric Schoenfeld and Isidora Milin for reading the paper
and making useful comments.
This paper was written during my visits to the IAS, IHES, MPIM and Stanford and I am
grateful to these institutions for their exceptional hospitality. This work was partially supported
by the NSF grant DMS9729992.
2. Vassiliev theory, invariants of finite type.
2.1. The space of knots, coorientation
Vassiliev considered the space of all maps E = f : S1 → R3. This space is a space of
functions, so it is an infinite-dimensional Euclidean space. It is linear, contractible, and consists
of singular (D) and nonsingular(E - D) knots. The discriminant D forms a singular hypersurface
in E and subdivides into chambers, corresponding to different isotopy types of knots. To move
from one chamber to another one has to change one overcrossing to undercrossing, passing
through a singular knot with one double point.
The discriminant of the space of knots is a real hypersurface, stratified by the number of the
double points, which subdivides the infinite-dimensional space into chambers, corresponding
to different isotopy types of knots.
Vassiliev resolved and cooriented the discriminant, so we can assume that all points of
selfintersection are transversal, with 2n chambers adjacent to a point of selfintersection of the
discriminant of codimension n.
To study the topology of the complement to the discriminant, Vassiliev wrote a spectral
sequence, calculating the homology of the discriminant and then related it to the homology of
its complement via Alexander duality. His spectral sequence had a filtration, which suggested
the simple geometrical and combinatorial definition of an invariant of finite type: an invariant
is of type n if for any selfintersection of the discriminant of codimension (n+1) its alternated
sum over the 2n+1 chambers adjacent to a point of selfintersection is zero.
For our constructions it will be very important to have a coorientation of the discriminant,
which was introduced by Vassiliev.
Definition. A hypersurface in a real manifold is said to be coorientable if it has a non-zero
section of its normal bundle, i.e. if there exists a continuous vector field which is not tangent
to the hypersurface at any point and doesn’t vanish anywhere.
So there are two sides of the hypersurface : one where this vector field is pointing to and
the other is where it is pointing from. And there are two choices of such vector field. The
coorientation of a coorientable hypersurface is the choice of one of two possibilities.
For example, Mobius band in R3 is not coorientable.
Vassiliev shows [V] that the discriminant of the space of knots has a coorientation, the
conistent choice of normal directions.
Recall that the nonsingular point ψ ∈ D of the discriminant is a map S1 → R3, gluing to-
gether 2 distinct points t1, t2 of S
1, s.t. derivatives of the map ψ at those points are transversal.
Coorientation of the discriminant. Fix the orientation of R3 and choose positively
oriented local coordinates near the point ψ(t1) = ψ(t2). For any point ψ1 ∈ D close to ψ define
the number r(ψ1) as the determinant:
|t2, ψ1(t1)− ψ1(t2))
with respect to these coordinates. This determinant depends only of the pair of points t1, t2,
not on their order. A vector in the space of functions at the point ψ ∈ D, which is transversal
to the discriminant, is said to be positive, if the derivative of the function r along this vector
is positive and negative, if this derivative is negative.
This rule gives the coorientation of the hypersurface D at all its nonsingular points and also
of any nonsingular locally irreducible component of D at the points of selfintersection of D.
The consistent choice of the normal directions of the walls of the discriminant will give the
”directions” of the cobordisms (which are embedded into E× I) between knots of the space E.
Note. It is interesting to compare this construction with the result of E.Ghys [Gh], who
introduced a metric on the space of knots and 3-manifolds.)
2.3. The topology of the chambers of the space of knots.
The study of the topology of the chambers of the space of knots was started by A. Hatcher
[H], who found a simple homotopy models for these spaces.
The main result is based on an earlier theorem regarding the topology of the classifying
space of diffeomorphisms of an irreducible 3-manifold with nonempty boundary.
In the following theorem A. Hatcher and D. McCullough answered the question posed by M.
Kontsevich [K], regarding the finiteness of the homotopy type of the classifying space of the
group of diffeomorphisms [HaM]:
Theorem [HaM]. Let M be an irreducible compact connected orientable 3-manifold with
nonempty boundary. Then BDiff(M, rel∂) has the homotopy type of a finite aspherical CW-
complex.
The proof of this theorem uses the JSJ-decomposition of a 3-manifold.
When applied to knot complements, the JSJ-decomposition defines a fundamental class
of links in S3, the ”knot generating links” (KGL). A KGL is any (n + 1)-component link
L = (L0, L1, · · · , Ln) whose complement is either Seifert fibred or atoroidal, such that the
n-component sub-link (L1, L2, · · · , Ln) is the unlink. If the complement of a knot f contains
an incompressible torus, then f can be represented as a ‘spliced knot’ f = J�L in unique way,
where L is an (n + 1)-component KGL, and J = (J1, · · · , Jn) is an n-tuple of non-trivial long
knots.
The spliced knot J�L is obtained from L0 by a generalized satellite construction. For any
knot there is a representation of a knot as an iterated splice knot of atoroidal and hyperbolic
KGLs. The order of splicing determines the ”companionship tree” of f , Gf , and is a complete
isotopy invariant of long knots.
Given a knot f ∈ K, denote the path-component of K containing f by Kf . The topology
of the chambers Kf was further studied by R. Budney The main result of his paper [Bu] is
the computation of the homotopy type of Kf if f is a hyperbolically-spliced knot ie: f = J�L
where L is a hyperbolic KGL.
The combined results can be summarized in the following theorem:
Theorem [Bu, H].
If f = J�L where L is an (n+ 1)-component hyperbolic KGL, then
Kf ⋍ S
SO2 ×Af
Af is the maximal subgroup of BL such that induced action of Af on K
n preserves
i=1KLi .
The restriction map Af → Diff(S
3, L0) → Diff(L0) is faithful, giving an embedding Af →
SO2, and this is the action of Af on SO2.
This result completes the computation of the homotopy-type of K since we have the prior
results:
H1 If f is the unknot, then Kf is contractible.
H2 If f is a torus knot, then Kf ≃ S
H3 If f is a hyperbolic knot, then Kf ⋍ S
1 × S1
H4 If a knot f is a cabling of a knot g then Kf ⋍ S
1 ×Kg.
B5 If the knot f is a connected sum of n ≥ 2 prime knots f1, f2, · · · , fn then Kf ⋍
((C2(n)×
i=1Kfi) /Σf . Here Σf ⊂ Sn is a Young subgroup of Sn, acting on C2(n)
by permutation of the labellings of the cubes, and similarly by permuting the factors
of the product
i=1Kfi . The definition of Σf ⊂ Sn is that it is the subgroup of Sn
that preserves a partition of {1, 2, · · · , n}, the partition being given by the equivalence
relation i ∼ j ⇐⇒ Kfi = Kfj .
B6 If a knot has a non-trivial companionship tree, then it is either a cable, in which case
H4 applies, a connect-sum, in which case B5 applies or is hyperbolically spliced. If
a knot has a trivial companionship tree, it is either the unknot, in which case H1
applies, or a torus knot in which case H2 applies, or a hyperbolic knot, in which case
H3 applies. Moreover, every time one applies one of the above theorems, one reduces
the problem of computing the homotopy-type of Kf to computing the homotopy-type
of knot spaces for knots with shorter companionship trees, thus the process terminates
after finitely-many iterations.
For constructing a local system we need only the homotopy type of the chamber. The
theorem of Hatcher and Budney provides us with a complete classification of homotopy types
of chambers, corresponding to all possible knot types.
3. Khovanov’s categorification of Jones polynomial.
3.1. Jones polynomial as Euler characterictics. Skein relation.
In his paper [Kh] M. Khovanov constructs a homology theory, with Euler characteristics
equal to the Jones polynomial.
He associated to any diagram D of an oriented link with n crossing points a chain complex
CKh(D) of abelian groups of homological length (n+1), and proved that for any two diagrams
of the same link the corresponding complexes are chain homotopy equivalent. Hence, the
homology groups Kh(D) are link invariants up to isomorphism.
His construction is as follows: given any double point of the link projection D, he allows two
smoothings:
0-resolution 1-resolution
If the the diagram has n double points, there are 2n possible resolutions. The result of each
complete smoothing is the set of circles in the plane, labled by n-tuples of 1’s and 0’s:
CKh(©, ...,©
︸ ︷︷ ︸
ntimes
) = V ⊗n
The cobordisms between links, i.e., surfaces embedded in R3 × [0, 1], should provide maps
between the associated groups. A surface embedded in the 4-space can be visualized as a
sequence of plane projections of its 3-dimensional sections (see [CS]). Given such a presentation
J of a compact oriented surface S properly embedded in R3 × [0, 1] with the boundary of S
being the union of two links L0 ⊂ R
3 × {0} and L1 ⊂ R
3 × {1}, , Khovanov associates to J a
map of cohomology groups
θJ : Kh
i,j(D0) → Kh
i,j+χ(S)(D1), i, j ∈ Z
The differential of the Khovanov complex is defined using two linear maps m : V ⊗ V → V
and ∆ : V → V ⊗ V given by formulas :
V ⊗ V
v+ ⊗ v− 7→ v− v+ ⊗ v+ 7→ v+
v− ⊗ v+ 7→ v− v− ⊗ v− 7→ 0
→ V ⊗ V
v+ 7→ v+ ⊗ v− + v− ⊗ v+
v− 7→ v− ⊗ v−
The differential in Khovanov complex can be informally described as ”all the ways of changing
0-crossing to 1-crossing”.
Homological degree of the Khovanov complex in the number of 1’s in the plane diagram
resolution. The sum of ”quantum” components of the same homological degree i gives the ith
component of the Khovanov complex.
One can see that the i-th differential di is the sum over ”quantum” components, it will
map one of the quantum components in homological degree i to perhaps several quantum
components of homological degree i+1.
Khovanov theory can be considered as a (1+1) dimensional TQFT. The cubes, that are
used in it’s definition come from the TQFT corresponding to the Frobenius algebra defined
by V,m,∆. As we will see later, our constructions will give the interpretation of Khovanov
local system as a topological D-brane and will suggest to study the structure of the category
of topological D-branes as a triangulated category.
We prove the following important property of the Khovanov’s complex:
Theorem 1. Let k denote the kth crossing point of the knot projection D, then for any k
the Khovanov’s complex C decomposes into a sum of two subcomplexes C = Ck0 ⊕ C
1 with
matrix differential of the form
d0 d0,1
Proof. Let Ck0 denote the subcomplex of C, consisting of vector spaces, which correspond
to the complete resolutions of D, having 0 on the kth place. The differential d0 obtained by
restricting d only to the arrows between components of Ck0 . We define C
1 the same way, by
restricting to the complete resolutions of D, having 1 on the kth place.
The only components of the differential, which are not yet used in our decomposition, are
the ones which change 0-resolution on the kth place of Ck0 to 1 on the kth place in C
1 , we
denote them d0,1.
One can easily see from the definition of the Khovanov’s differential (which can be intuitivly
described as ”all the ways to change 0-resolution in the ith component of the complex to the
1-resolution in the (i+1)st component”), that there is no differential mapping ith component
of Ck1 to the (i+1)st component of C
Mirror images and adjoints. Taking the mirror image of the knot will dualize Khovanov
complex. So if we want to invert the cobordism between two knots, we should consider the
”dual” cobordism between mirror images of these knots.
3.2. Reidemeister and Jacobsson moves.
A cobordism (a surface S embedded into R3 × [0, 1]) between knots K0 and K1 provide a
morphism between the corresponding cohomology:
FS : Kh
i,j(D0) → Kh
i,j+χ(S)(D1)
where D0 and D1 are diagrams of the knots K0 and K1 and χ(S) is the Euler characteristic
of the surface.
We will distinguish between two types of cobordisms - first, corresponding to the wall crossing
(and changing the type of the knot). And second, corresponding to nontrivial loops in chambers
which will reflect the dependence of Khovanov homologies on the selfdiffeomorphisms of the
knot, similar to the Reidemeister moves. In this paragraph we will discuss the second type of
cobordisms.
By a surface S in R4 we mean an oriented, compact surface S, possibly with boundary,
properly embedded in R3 × [0, 1]. The boundary of S is then a disjoint union
∂S = ∂0S ⊔ −∂1S
of the intersections of S with two boundary components of R3 × [0, 1]:
∂0S = (S ∩R
3 × {0})
−∂1S = (S ∩R
3 × {1})
Note that ∂0S and ∂1S are oriented links in R
The surface S can be represented by a sequence J of plane diagrams of oriented links where
every two consecutive diagrams in J are related either by one of the four Reidemeister moves
or by one of the four moves birth, death, fusion described by Carter-Saito [CS].
To each Reidemeister move between diagrams D0 and D1 Khovanov [Kh] associates a quasi-
isomorphism map of complexes C(D0) → C(D1).
Given a representation J of a surface S by a sequence of diagrams, we can associate to J a
map of complexes
ϕJ : C(J0) → C(J1)
Any link cobordism can be described as a one-parameter family Dt, t ∈ [0, 1] of planar dia-
grams, called a movie. The Dt are link diagrams, except at finitely many singular points which
correspond to either a Reidemeister move or a Morse modification. Away from these points
the diagrams for various t are locally isotopic . Khovanov explained how local moves induce
chain maps between complexes, hence homomorphisms between homology groups. The same is
true for planar isotopies. Hence, the composition of these chain maps defines a homomorphism
between the homology groups of the diagrams of links.
In his paper [Ja] Jacobsson shows that there are knots, s.t. a movie as above will give a
nontrivial morphism of Khovanov homology:
Theorem [Ja] For oriented links L0 and L1, presented by diagrams D0 and D1, an oriented
link cobordism Σ from L0 to L1, defines a homomorphism H(D0) → H(D1), invariant up
to multiplication by -1 under ambient isotopy of Σ leaving ∂Σ setwise fixed. Moreover, this
invariant is non-trivial.
Jacobsson constructs a family of derived invariants of link cobordisms with the same source
and target, which are analogous to the classical Lefschetz numbers of endomorphisms of man-
ifolds.
The Jones polynomial appears as the Lefschetz polynomial of the identity cobordism.
From our perspective the Jacobsson’s theorem shows that the Khovanov local system will
have nontrivial monodromies on the chambers of the space of knots.
3.3. Wall-crossing morphisms.
In 3.2 we described what kind of modifications can occur in the cobordism, when we consider
the ”movie” consisting only of manifolds of the same topological type. These modifications
implied corresponding monodromies of the Khovanov complex.
However, morphisms that are the most important for Vassiliev-type theories are the ”wall-
crossing” morphisms. We will define them now (locally).
Consider two complexes A• and B• adjacent to the generic wall of the discriminant. Recall,
that the discriminant is cooriented ( 2.2). If B• is ”right” via coorientation (or ”further in the
Ghys metric form the unknot) of A•, then we shift B•’s grading up by one and consider B•[1]:
A•|B•[1]
Note. In general, and this will be very important for us in subsequent chapters, if the complex
K• is n steps (via the coorientation) away from the unknot, we shift its grading up by n.
Thus adjacent complexes will have difference in grading by one (as above), defined by the
coorientation.
Now we want to understand what happens to the Khovanov complex when we change the
kth over-crossing (in the knot diagram D) to an under-crossing.
We will illustrate these changes on one of the Bar-Natan’s trademark diagrams (with his
permission)[BN1].
By ”I” we mark the arrow , connecting components of the complex which will exchange
places under wall-crossing morphisms when we change over-crossing to under-crossing for the
self-intersection point 1. By ”II” when we do it for point 2 and ”III” when we do it for 3:
2 V {1}
V ⊗2{2}
==zzzzzzzzzzzzzzzzzz
V {1}
<<yyyyyyyyyyyyyyyyyy
V ⊗2{2}
V ⊗3{3}
V {1}
wwwwwwww
;;wwwwwwww
V ⊗2{2}
;;vvvvvvvvvvvvvvvvv
d0 // J&K1
d1 // J&K2
d2 // J&K3
(0.1)
Now recall the theorem proved in (3.1): for any k, where k is the number of crossings of the
diagram D, the Khovanov complex can be split into the sum of two subcomplexes with the
uppertriangular differential.
Notice from the diagram above that when we change kth overcrossing to an undercrossing,
0 and 1-resolutions are exchanged , so A• = A•0 ⊕ A
•[1] = B•0 [1] ⊕ B
1 [1], thus for every k
we can define the wall-crossing morphism ω as follows:
Theorem 2. The map defined as the identity on A•0 and as a trivial map on A
ω : A•0
Id // B•0 [1]
ω : A•1
∅ // B•1 [1]
is the morphism of complexes.
Proof. From the Theorem 1 we know that for any crossing k the Khovanov complex can be
decomposed as a direct sum with uppertriangular differential:
d0 d0,1
It is an easy check that the wall-crossing morphism defined as above is indeed a morphism
of complexes (i.e. it commutes with the differential):
ω // B•
ω // B•
Since we defined the morphism as 0 on A•1, the diagram above becomes the following com-
mutative diagram:
Id // B•0 [1]
Id // B•0 [1]
3.4. The local system of Khovanov complexes on the space of knots.
In this paragraph we introduce the Khovanov local system on the space of knots.
Definition. A local system on the locally connected topological space M is a fiber bundle
over M, the sections of which are abelian groups. The fiber of the bundle depend continuously
on the point of the base (such that the group structure on the set of fibers can be extended
over small domains in the base).
Any local system on M with fiber A defines a representation π1(M) → Aut(A). To any loop
there corresponds a morphism of the fibers of the bundle over the starting point of the loop.
The set of isomorphism classes of local systems with fiber A are in one-to-one corresondence
with the set of such representations up to conjugation. For example any representation of an
arbitrary group π in Aut(A) uniquely (up to isomorphism) defines a local system on the space
K(π, 1) [GM].
Morphisms of local systems are morphisms of fiber bundles, preserving group structure in
the fibers. Thus introducing the continuation functions (maps between fibers) over paths in
the base will define a local system over the manifold M.
Next we set up the Khovanov complexes as a local system on the space of knots. If we
were doing it ”in coordinates”, we would introduce charts on the chambers of the space of
knots and define our local system via transition maps, starting with some ”initial” point . This
would be a very interesting and realistic approach, since the homotopy models for chambers
are understood [H], [B], e.g. we would have just one chart for the chamber, containing the
unknot (since that chamber is contractible), two for a torus knot, four for a hyperbolic one,
etc. Then monodromies of the Khovanov local system along nontrivial loops in the chamber
will be given via Jacobsson movies.
It would be also very interesting to find a unique special point in every chamber of the space
E and study monodromies of the local system with respect to this point.
The candidate for such point is introduced in the works of J. O’Hara, who studied the minima
of the electrostatical energy function of the knot [O’H]:
E(K) =
|(x− y)|−2dxdy
It was shown that under some assumptions and for perturbation of the above functional, its
critical points on the space of knots will provide a ’distinguished” point in the chamber.
The first natural question for this setup is: which nontrivial loops in the chamber EK ,
corresponding to the knot K are distinguised by Khovanov homologies and which are not?
However, assuming Khovanov’s theorem [Kh] (that his homology groups are invariants of
the knot, independent on the choices made) and assuming also the results of Jacobsson [J] , it
is enough for us to introduce the continuation maps, along any path γ in the chamber of the
space of knots.
These methods were developed by several authors (see [Hu]):
Let K1 and K2 be two knots in the same chamber of the space E, let Ki be generic, and let
γ = {Kt | t ∈ [0, 1]} be any path of equivalent objects in E from K1 to K2. Then a generic
path γ
induces a chain map
F (γ) : CKh∗(K1)−→CKh∗(K2)
called the “continuation” map, which has the following properties:
• 1)Homotopy A generic homotopy rel endpoints between two paths γ1 and γ2 with
associated chain maps F1 and F2 induces a chain homotopy
H : HKh∗(K1)−→HKh∗+1(K2)
∂H +H∂ = F1 − F2
• 2)Concatenation If the final endpoint of γ1 is the initial endpoint of γ2, then F (γ2γ1)
is chain homotopic to F (γ2)F (γ1).
• 3)Constant If γ is a constant path then F (γ) is the identity on chains.
These three properties imply that ifK1 andK2 are equivalent, thenHKh∗(K1) ≃ HKh∗(K2).
(Khovanov’s theorem).
This isomorphism is generally not canonical, because different homotopy classes of paths
may induce different continuation isomorphisms on Khovanov homology (Jacobsson moves).
However, since the loop is contractible, we do know that HKh∗(K) depends only on K, so we
denote this from now on by HKh∗(K).
We now define the restriction of the Khovanov local system to finite-dimensional subspaces
of the space of knots.
Note that in the original setting our complexes may have had different length. For example,
the complex corresponding to the standard projection of the unknot will have length 1, however,
we can consider very complicated ”twisted” projections of the unknot with an arbitrary large
number of crossing points. The corresponding complexes will be quasiisomorphic to the original
This construction resembles the definition of Khovanov homology introduced in [CK], [W].
They define Khovanov homology as a relative theory, where homology groups are calculated
relative to the twisted unknots.
When considering the restrictions of the Khovanov local system to the subcategories of knots
with at most n crossings, we would like all complexes to be of length n+ 1.
This can be achieved by ”undoing” the local system, starting with the knots of maximal
crossing number n and then using the wall-crossing morphisms, define complexes of length
(n + 1), quasiisomorphic to the original ones, in all adjacent chambers. We continue this
process till it ends, when we reach the chamber containing unknot.
Recall that Khovanov homology is defined for the knot projection (though is independent
of it by Khovanov’s theorem). So we will consider a ramification of Vassiliev space, a pair, the
embedding of the circle into R3 and its projection on (x,y)-plane. Then each chamber will be
subdivided into ”subchambers” corresponding to nonsingular knot projections and the ”sub-
discriminant” will consist of singular projections of the given knot. The local system, defined
on such ramification will live on the universal cover of the base, the original Hatcher chamber
corresponding to knot K and morphisms of the local system between ”subchambers” are given
by Reidemaister moves. The composition of such moves may constitute the Jacobsson’s movie
and will give nontrivial monodromies of the local system within the original chamber.
Note. As we will see later, if one assines cones of Reidemeister morphisms to the walls of
the ”subdiscriminant”, all such cones will be acyclic complexes. This statement in a different
form was proved in the original Khovanov [Kh] paper.
4. The main definition, invariants of finite type for families.
4.1. Some homological algebra.
We describe results and main definitions from the category theory and homological algebra
which will be used in subsequent chapters. The standard references on this subject are [GM],
[Th].
By constructing the local system of (3.4) we introduced the derived category of Khovanov
complexes. The properties of the derived category are summarized in the axiomatics of the
triangulated category, which we will discuss in this chapter.
Definition. An additive category is a category A such that
• Each set of morphisms Hom(A,B) forms an abelian group.
• Composition of morphisms distributes over the addition of morphisms given by the
abelian group structure, i.e. f ◦ (g + h) = f ◦ g + f ◦ h and (f + g) ◦ h = f ◦ h+ g ◦ h.
• There exist products (direct sums) A×B of any two objects A,B satisfying the usual
universal properties.
• There exists a zero object 0 such that Hom(0, 0) is the zero group (i.e. just the identity
morphism). ThusHom(0, A) = 0 = Hom(A, 0) for all A, and the unique zero morphism
between any two objects is the one that factors through the zero object.
So in an abelian category we can talk about exact sequences and chain complexes, and
cohomology of complexes. Additive functors between abelian categories are exact (respectively
left or right exact) if they preserve exact sequences (respectively short exact sequences 0 →
A→ B → C or A→ B → C → 0).
Definition. The bounded derived category Db(A) of an abelian category A has as objects
bounded (i.e. finite length) A-chain complexes, and morphisms given by chain maps with quasi-
isomorphisms inverted as follows. We introduce morphisms f for every chain map between
complexes f : Xf → Yf , and g
−1 : Yg → Xg for every quasi-isomorphism g : Xg
→ Yg. Then
form all products of these morphisms such that the range of one is the domain of the next.
Finally identify any combination f1f2 with the composition f1 ◦ f2, and gg
−1 and g−1g with
the relevant identity maps idYg and idXg .
Recall that a triangulated category C is an additive category equipped with the additional
data:
Definition. Triangulated category is an additive category with a functor T : X → X[1]
(where Xi[1] = Xi+1) and a set of distinguished triangles satisfying a list of axioms.
The triangles include, for all objects X of the category:
1) Identity morphism
X → X → 0 → X[1],
2) Any morphism f : X → Y can be completed to a distinguished triangle
X → Y → C → X[1],
3) There is also a derived analogue of the 5-lemma, and a compatibility of triangles known
as the octahedral lemma, which can be understood as follows:
If we naively interprete property 1) as the difference X −X = 0, property 2) as C = X −Y ,
then the octahedron lemma says:
(X − Y )− Z = C − Z = X − (Y − Z)
When topological spaces considered up to homotopy there is no notion of kernel or cokernel.
The cylinder construction shows that any map f : X → Y is homotopic to an inclusion
X → cyl (f) = Y ⊔ (X × [0, 1])/f(x) ∼ (x, 1), while the path space construction shows it is also
homotopic to a fibration.
The cone Cf on a map f : X → Y is the space formed from Y ⊔ (X × [0, 1]) by identifying
X × {1} with its image f(X) ⊂ Y , and collapsing X × {0} to a point.
It can be considered as a cokernel, i.e. if f : X → Y
is an inclusion, then Cf is homotopy equivalent to Y/X.
Taking the ith cohomologyHi of each term, and using the suspension isomorphismHi(ΣX) ∼=
Hi−1(X) gives a sequence
Hi(X) → Hi(Y ) → Hi(Y,X) → Hi−1(X) → Hi−1(Y ) → . . .
which is just the long exact sequence associated to the pair X ⊂ Y .
Up to homotopy we can make this into a sequence of simplicial maps, so that taking the
associated chain complexes we get a lifting of the long exact sequence of homology to the level
of complexes. It exists for all maps f , not just inclusions, with Y/X replaced by Cf .
If f is a fibration, Cf can act as the “kernel” or fibre of the map. If f : X →point, then
Cf = ΣX, the suspension of the fibre X.
Thus Cf acts as a combination of both cokernel and kernel, and if f : X → Y is a map
inducing an isomorphism of homology groups of simply connected spaces then the sequence
Hi(X) → Hi(Y ) → Hi(Cf ) → Hi−1(X) → Hi−1(Y ) → . . .
implies H∗(Cf )=0. Then Cf homotopy equivalent to a point. Thus we can give the following
definition.
Definition. If X and Y are simplicial complexes, then a simplicial map f : X → Y ,
defines (up to isomorphism) an object in triangulated category, called the cone of morphism
f, denoted Cf .
C•X ⊕ C
Y [1] with differential dCf =
0 dY [1]
where [n ] means shift a complex n places up.
Thus we can define the cone Cf on any map of chain complexes f : A
• → B• in an abelian
category A by the above formula, replacing C•X by A
• and C•Y by B
•. If A• = A and B• = B
are chain complexes concentrated in degree zero then Cf is the complex {A
→ B}. This has
zeroth cohomology h0(Cf ) = ker f , and h
1(Cf ) = coker f , so combines the two (in different
degrees). In general it is just the total complex of A• → B•.
So what we get in a derived category is not kernels or cokernels, but “exact triangles”
A• → B• → C• → A• [ 1 ].
Thus we have long exact sequences instead of short exact ones; taking ith cohomology hi of
the above gives the standard long exact sequence
hi(A•) → hi(B•) → hi(C•) → hi+1(A•) → . . .
The cone will fit into a triangle:
u // B
The “[1]” denotes that the map w increases the grade of any object by one.
4.2. Space of knots as a classifying space of the category.
In this paragraph we will construct the Khovanov functor from the category of knots into
the triangulated category of Khovanov complexes.
Definition. The category of knots K is the category, the objects of which are knots,
S1 → S3, morphisms are cobordisms, i.e. surfaces Σ properly embedded in R3 × [0, 1] with the
boundary of Σ being the union of two knots K1 ⊂ R
3 × {0} and K2 ⊂ R
3 × {1}.
We denote Kn the subcategory of knots with at most n crossings.
(Recall that a knot’s crossing number is the lowest number of crossings of any diagram of
the knot. )
Note that our cobordisms (morphisms in the category of knots) are directed via the coori-
entation of the discriminant of the space of knots.
Note that to reverse cobordism, we can consider the same cobordism between mirror images
of the knots.
Definition. The nerveN (C) of a category C is a simplicial set constructed from the objects
and morphisms of C, i.e. points of N (C) are objects of C, 1-simplices are morphisms of C,
2-simplices are commutative triangles, 3-simplices are commutative tetrahedrons of C, etc.
N (C) = (limN i(C))
The geometric realization of a simplicial set N (C) is a topological space, called the classi-
fying space of the category C, denoted B(C).
The following observation is the main guideline for our constructions: sheaves on the classi-
fying space of the category are functors on that category [Wi].
Once we prove that the Vassiliev space of knots is a classifying space of the category K, our
local system will provide a representation of the Khovanov functor.
Let C be a category and let Set be the category of sets. For each object A of C let Hom(A,)
be the hom functor which maps objects X to the set Hom(A,X).
Recall that a functor F : C → Set is said to be representable if it is naturally isomorphic
to Hom(A, ) for some object A of C. A representation of F is a pair (A,Ψ) where
Ψ : Hom(A, ) → F
is a natural isomorphism.
If E - the space of knots, denote KE the category of knots, whose objects are points in E
and morphisms Mor(x, y) = {γ : [0, 1] → X; s.t.γ(0) = x, γ(1) = y} and KK - subcategory
corresponding to knots of the same isotopy type K.
Proposition. The chamber EK of the space of long knots forK - unknot, torus of hyperbolic
knot is the classifying space of the category KK .
Proof. By Hatcher’s theorem [H] the chambers of the space of knots EK , corresponding to
unknot, torus or hyperbolic knot are K(π, 1).
By definition the space of long knots is E = {f : R1 → R3}, nonsingular maps which are
standard outside the ball of large radius. If f1, f2 are vector equations giving knots K1,K2,
then tf1 + (1 − t)f2 is a path in the mapping space, defining a knot for each value of t. The
cobordism between two embeddings is given by equations in R3× I. All higher cobordisms can
be contracted, since there are no higher homotopy groups in EK . So both the classifying space
of the category and the chamber of the space of knots are K(π, 1) with the same π. They are
the same as simplicial complexes.
Note, that in the case of hyperbolic knots one can choose the distinguished point in the
chamber - corresponding to the hyperbolic metric on the complement to the knot.
4.3. Vassiliev derivative as a cone of the wall-crossing morphism.
To be able to construct a categorification of Vassiliev theory, we have to extend the local
system, which we defined on chambers, to the discriminant of the space of knots.
Recall that according to the axiomatics of the triangulated category, described in (4.1), we
assign an new object to every morphism in the category:
for a complex X = (Xi, dix) define a complex
X[1] by
(X[1])i = Xi+1, dX[1] = −dX
For a morphism of complexes f : X → Y let f [1] : X[1] → Y [1] coincide with f component-
wise.
Let f : X → Y be a wall-crossing morphism. The cone of f is the following complex C(f):
X → Y → Z = C(f) → X[1]
C(f)i = X[1]i ⊕ Y i, dC(f)(x
i+1, yi) = (−dXx
i+1, f(xi+1)− dY y
Recall, that we set up the local system on the space of knots (3.4) s.t. if the complex X• is n
steps (via the coorientation) away from the unknot, we shift its grading up by n. So complexes
in adjacent chambers will have difference in grading by one, defined by the coorientation.
Thus, given a bigraded complex, associated to the generic wall of the discriminant, we get
two natural specialization maps into the neighbourhoods, containing X• and Y •:
So with any morphism f we associate the triangle:
// Y •
aaBBBBBBBB
With any commutative cube
u // •
u // •
(in the space of knots the above picture corresponds to the cobordism around the self-
intersection of the discriminant of codimension two), we associate the map between cones,
corresponding to the vertical and horisontal walls, and assign it to the point of their intersec-
tion:
Cuω // Cω
Lemma. Given four chambers as above, the order of taking cones of morphisms is irrelevant,
Cuω = Cωu.
Proof. see [GM].
Consider a point of selfintersection of the discriminant of codimension n. There are 2n
chambers adjacent to this point. Since the discriminant was resolved by Vassiliev [V], this
point can be considered as a point of transversal selfintersection of n hyperplanes in Rn, or an
origin of the coordinate system of Rn.
Now our local system looks as follows. On chambers of our space we have the local system of
Khovanov complexes, to any point t of the generic wall between chambers containing X• and
Y • (corresponding to a singular knot), we assign the cone of the morphism X• → Y • (with
the specialization maps from the cone to the small neighborhoods of t containing X• and Y •).
To the point of codimention n we assign the nth cone, 2n-graded complex, etc.
Definition. The Khovanov homology of the singular knot (with a single double point ) is a
bigraded complex
X• ⊕ Y •[1] with the matrix differential dCω =
0 dY [1]
where X• is Khovanov complex of the knot with overcrossing, Y • is the Khovanov complex of
the knot with undercrossing and ω is the wall-crossing morphism.
In [S4] we give the geometric interpretation of the above definition.
4.4. The definition of a theory of finite type.
Once we extended the local system to the singular locus, it is natural to ask if such an
extension will lead to the categorification of Vassiliev theory.
The first natural guess is that the theory, set up on some space of objects, which has quasi-
isomorphic complexes on all chambers is a theory of order zero. Such theory will consist of
trivial distinguished triangles as in (a) of the axiomatics of the triangulated category. When
complexes, corresponding to adjacent chambers are quasiisomorphic, the cone of the morphism
is an acyclic complex.
Baby example of a theory of order 0.
Let M be an n-dimensional compact oriented smooth manifold. Consider the space of func-
tions on M. This is an infinite-dimensional Euclidean space. The chambers of the space will
correspond to Morse functions on M, the walls of the discriminant - to simple degenerations
when two critical points collide, etc. Let’s consider the Morse complex, generated by the crit-
ical points of a Morse function on M. As it was shown by many authors, such complex is
isomorphic to the CW complex, associated with M. Since we are calculating the homology of
M via various Morse functions, complexes may vary, but will have the same homology and
Euler characteristics.
Then we can proceed according to our philosophy and assign cones of morphisms to the
walls and selfintersections of the discriminant. Since complexes on the chambers of the space
of functions are quasiisomorphic, all cones are acyclic.
Now we can introduce the main definition of a Floer-type theory being of finite type n:
Main Definition. The local system of (Floer-type) complexes, extended to the discriminant
of the space of manifolds via the cone of morphism, is a local system of order n if for any
selfintersection of the discriminant of codimension (n + 1), its (n+1)st cone is an acyclic
complex.
How one shows that an 2n-graded complex is acyclic? For example, if one introduces inverse
maps to the wall-crossing morphisms and construct the homotopy H, s.t.:
dH−Hd = I
It is easy to check that the existence of such homotopy H implies, that the complex doesn’t
have homology. Suppose dc = 0, i.e. c is a cycle, then:
dHc−Hdc = dHc = I
Example. Suppose some local system is conjectured to be of finite type 3. How one would
check this? By our definition, we should consider 23 chambers adjacent to the every point
of selfintersection of the discriminant of codimension 3, and 8 complexes, representing the
local system in the small neighbourhood of this point. This will correspond to the following
commutative cube:
}}}}}}}} g
||||||||
l // G•
}}}}}}}}
m // H•
||||||||
Let’s write the homotopy equation in the matrix form.
Consider dual maps f∗, g∗, ..., w∗. Then we get formulas for d and H as 8× 8 matrices:
dA f g 0 a 0 0 0
0 dB 0 h 0 b 0 0
0 1 dD w 0 0 e 0
0 0 1 dC 0 0 0 c
0 0 0 0 dE k m 0
0 0 0 0 1 dF 0 l
0 0 0 0 0 1 dH n
0 0 0 0 0 0 1 dG
dA 0 0 1 0 0 0 0
f∗ dB 0 0 0 0 0 0
g∗ 0 dD 0 0 0 0 0
0 h∗ w∗ dC 0 0 0 0
a∗ 0 0 0 dE 0 0 1
0 b∗ 0 0 k∗ dF 0 0
0 0 e∗ 0 m∗ 0 dH 0
0 0 0 c∗ 0l∗ h∗ dG
After substituting these matrices into the equation dH − Hd = I we obtain the diagonal
matrix which must be homotopic to the identity matrix:
ff∗ + gg∗ + aa∗ 0 0 0 0 0 0 0
0 −”− 0 0 0 0 0 0
0 0 −”− 0 0 0 0 0
0 0 0 −”− 0 0 0 0
0 0 0 0 −”− 0 0 0
0 0 0 0 0 −”− 0 0
0 0 0 0 0 0 −”− 0
0 0 0 0 0 0 0 cc∗ + nn∗ + ll∗
Thus the condition for the local to be of finite type n can be interpreted as follows. For
any selfintersection of the discriminant of codimension n + 1 consider 2n complexes, forming
a commutative cube (representatives of the local system in the chambers adjacent to the self-
intersection point). Then the naive geometrical interpretation of the local system being of
finite type n is the following: each complex can be ”split” into n+1 subcomplexes, which map
quasiisomorphically to n+ 1 neighbours, at least no homologies die or being generated.
5. Knots: theories of finite type. Further directions.
5.1. Examples of combinatorially defined theories.
In the following table we give the examples of theories, which are the categorifications of
classical invariants. All these theories fit into our framework and may satisfy the finitness
condition.
λ λ = χH∗(M)
Jones polynomial Khovanov homology [Kh]
Alexander polynomial Ozsvath-Szabo knot homology [OS2]
sl(n) invariants Khovanov - Rozhansky homology [KhR]
Casson invariant Instanton Floer homology [F]
Turaev’s torsion Ozsvath-Szabo 3 manifold theory [OS1]
Vafa invariant Gukov-Witten categorification [GW]
Note, that the only theory which is not combinatorially defined is the original Instanton
Floer homology [F]. The fact that it’s Euler characteristics is Casson’s invariant was proved by
C.Taubes [T].
5.2. Generalization to dimension 3 and 4.
In our paper [S1] we generalized Vassiliev’s construction to the case of 3-manifolds. In [S2]
we construct the space of parallelizable 4-manifolds and consider the paramentrized version of
the Refined Seiberg-Witten invariant [BF].
a). The space of 3-manifolds and invariants of finite type.
Note that all 3-manifolds are parallelizable and therefore carry spin-structures.
Following Vassiliev’s approach to classification of knots, we constructed spaces E1 and E2
of 3-manifolds by a version of the Pontryagin-Thom construction.
Our main results are as follows:
Theorem [S1]. In E1 − D each connected component corresponds to a homeomorphism
class of 3-dimensional framed manifold. For any connected framed manifold as above there is
one connected component of E1 −D giving its homeomorphism type.
Theorem [S1]. In E2 − D each connected component corresponds to a homeomorphism
class of 3-dimensional spin manifold. For any connected spin manifold there is one connected
component of E2 −D giving its homeomorphism type.
By a spin manifold we understand a pair (M,θ) where M is an oriented 3-manifold, and θ
is a spin structure on M . Two spin manifolds (M,θ) and (M ′, θ′) are called homeomorphic, if
there exists a homeomorphism M →M ′ taking θ to θ′.
The construction of the space naturally leads to the following definition:
Definition. A map I : (M,θ) → C is called a finite type invariant of (at most) order k if it
satisfies the condition:
(−1)#L
I(ML′) = 0
where L′ is a framed sublink of link L with even framings, L corresponds to the self-intersection
of the discriminant of codimension k+1, #L′ - the number of components of L′, ML′ - spin
3-manifold obtained by surgery on L′.
We introduced an example of Vassiliev invariant of finite order. Given a spin 3-manifold
M3 we consider the Euler characteristic of spin 0-cobordism W . Denote by I(M,spin) =
(sgn(W, spin) − 1)(mod2).
Theorem [S1]. Invariant I(M,spin) is finite type of order 1.
The construction of the space of 3-manifolds chambers of which correspond to spin 3-
manifolds is important for understanding, which additional structures one needs in order to
build the theory of finite-type invariants for homologically nontrivial manifolds. It suggests
that one should consider spin ramifications of known invariants.
In the following paper we will generalize our constructions and the main definition to the
case of 3-manifolds. We will construct a local system of Ozsvath-Szabo homologies, extend it
to the singular locus via the cone of morphism and find examples of theories of finite type.
b). Stably parallelizable 4-manifolds.
In this section we modify the previous construction [S1] to get the space of parallelizable
4-manifolds.
By the definition the manifold is parallelizable if it admits the global field of frames, i.e. has
a trivial tangent bundle. In the case of 4-manifolds this condition is equivalent to vanishing of
Euler and the second Stieffel-Whitney class. In particular signature and the Euler characterictic
of such manifolds will be 0.
We will use the theorem of Quinn:
Theorem Any punctured 4-manifold posesses a smooth structure.
Recall also the result of Vidussi, which states that manifolds diffeomorphic outside a point
have the same Seiberg-Witten invariants,so one cannot use them to detect eventual inequivalent
smooth structures. Thus for the purposes of constructing the family version of the Seiberg-
Witten invariants, it will be sufficient for us to consider “asymptotically flat” 4-manifolds, i.e.
such that outside the ball BR of some large radius R they will be given as the set of common
zeros of the system of linear equations (e.g. fi(x1, ...xn+4) = xi for i = 1, ...n.)
By Gromov’s h-principle any smooth 4-manifold (with all of its smooth structures and met-
rics) can be obtained as a common set of zeros of a system of equations in RN for sufficiently
large N.
Theorem [S2]. Any parallelizable smooth 4-manifold can be obtained as a set of zeros
of n functions on the trivial (n+4)-bundle over Sn. Each manifold will be represented by
|H1(M,Z2)⊕H
3(M,Z)| chambers.
There is a theory which also fits into our template - Ozsvath-Szabo homologies for 3-
manifolds, Euler characteristic of which is Turaev’s torsion. It would be interesting to show
that this theory is also of finite type or decomposes as the Khovanov theory.
c). Ozsvath-Szabo theory as triangulated category.
In [S3] we put the theory developed by P.Ozsvath and Z. Szabo into the context of homo-
logical algebra by considering a local system of their complexes on the space of 3-manifolds
and extending it to the singular locus. We show that for the restricted category the Heegaard
Floer complex CF∞ is of finite type one. For other versions of the theory we will be using the
new combinatorial formulas, obtained in [SW].
Recall that the categorification is the process of replacing sets with categories, functions
with functors, and equations between functions by natural isomorphisms between functors.
On would hope that after establishing this correspondence, homological algebra will provide
algebraic structures which one should assign to geometrical objects without going into the
specifics of a given theory. One can see that this approach is very useful in topological category,
in particular we will be getting knot and link invariants of Ozsvath and Szabo after setting up
their local system on the space of 3-manifolds.
Note Floer homology can be also considered as invariants for families, so it would be in-
teresting to connect our work to the one of M.Hutchings [H]. His work can be interpreted as
construction of local systems corresponding to various Floer-type theories on the chambers of
our spaces. Then we extend them to the discriminant and classify according to our definition.
6.3. Further directions.
1. There is a number of immediate questions from the finite-type invariants story:
a). What will substitute the notion of the chord diagram? What is the ”basis ” in the
theories of finite type?
b). What are the ”dimensions” of the spaces of theories of order n?
2. What is the representation-theoretical meaning of the theory of finite type?
a). Is it possible to construct a ”universal” knot homology theory in a sense of T.Lee [L] ?
b). Is it possible to rise such a ”universal” knot homology theory to the Floer-type theory
of 3-manifolds?
3. There are ”categorifications” of other knot invariants: Alexander polynomial [OS], HOM-
FLY polynomial [DGR ]. These theories also fit into our setting and it will interesting to show
that they decompose into the series of theories of finite type or that their truncations are of
finite type.
4. The next step in our program [S3] is the construction of the local system of Ozsvath-Szabo
homologies on the space of 3-manifolds introduced in [S1]. We also plan to raise Khovanov
theory to the homological Floer-type theory of 3-manifolds.
5. It should be also possible to generalize our program to the study of the diffeomorphism
group of a 4-manifold by considering Gukov-Witten [GW] categorification of Vafa invariant on
the moduli space constructed in [S2].
5. Bibliography.
[BF] Bauer S., Furuta M., A stable cohomotopy refinement of Seiberg-Witten invariants: I,
II, math.DG/0204340.
[BN1] Bar-Natan D.,On Khovanov’s categorification of the Jones polynomial, math.QA/0201043.
[BN2] Bar-Natan D., Vassiliev and Quantum Invariants of Braids, q-alg/9607001.
[Bu] R. Budney, Topology of spaces of knots in dimension 3, math.GT/0506524.
[CJS] Cohen R., Jones J.,Segal G., Morse theory and classifying spaces, preprint 1995.
[CKV] Champanerkar A., Kofman I., Viro O., Spanning trees and Khovanov homology,
preprint.
[D] Donaldson S., The Seiberg-Witten Equations and 4 manifold topology, Bull.AMS, v. 33,
1, 1996.
[DGR] Dunfield N., Gukov S., Rasmussen J., The Superpolynomial for Knot Homologies ,
math.GT/0505662.
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  In this paper we outline a program for the classification of Floer-type
theories, (or defining invariants of finite type for families). We consider
Khovanov complexes as a local system on the space of knots introduced by V.
Vassiliev and construct the wall-crossing morphism. We extend this system to
the singular locus by the cone of this morphism and introduce the definition of
the local system of finite type. This program can be further generalized to the
manifolds of dimension 3 and 4.

Introduction.
2. Vassiliev’s and Hatcher’s theories.
2.1. The space of knots, coorientation.
2.2. Vassiliev derivative.
2.3. The topology of the chambers of the space of knots.
3. Khovanov homology.
3.1. Jones polynomial as Euler characterictics. Skein relation.
3.2. Reidemeister and Jacobsson moves.
3.3. Wall-crossing morphisms.
3.4. The local system of Khovanov complexes on the space of knots.
4. Main definition, invariants of finite type for families.
4.1. Some homological algebra.
4.2. Space of knots and the classifying space of the category.
4.3. Vassiliev derivative as a cone of the wall-crossing morphism.
4.4. The definition of a theory of finite type.
5. Theories of finite type. Further directions.
5.1. Examples of combinatorially defined theories.
5.2. Generalizations to dimension 3 and 4.
5.3. Further directions.
6. Bibliography.
1. Introduction.
Lately there has been a lot of interest in various categorifications of classical scalar invariants,
i.e. homological theories, Euler characteristics of which are scalar invariants. Such examples
include the original instanton Floer homology, Euler characteristic of which, as it was proved by
C.Taubes [T], is Casson’s invariant. Ozsvath-Szabo [OS] 3-manifold theory categorifies Turaev’s
torsion, the Euler characteristic of their knot homologies [OS] is the Alexander polynomial. The
theory of M. Khovanov categorifies the Jones polynomial [Kh] and Khovanov-Rozhansky theory
categorifies the sl(n) invariants [KR] .
The theory that we are constructing will bring together theories of V. Vassiliev, A. Hatcher
and M. Khovanov, and while describing their results we will specify which parts of their con-
structions will be important to us.
The resulting theory can be considered as a ”categorification of Vassiliev theory” or a clas-
sification of categorifications of knot invariants. We introduce the definition of a theory of
finite type n and show that Khovanov homology theory in a categorical sense decomposes into
a ”Taylor series” of theories of finite type.
The Khovanov functor is just the first example of a theory satisfying our axioms and we
believe, that all theories mentioned above will fit into our template.
Our main strategy is to consider a knot homology theory as a local system, or a constructible
sheaf on the space of all objects (knots, including singular ones), extend this local system to
the singular locus and introduce the analogue of the ”Vassiliev derivative” for categorifications.
By studying spaces of embedded manifolds we implicitly study their diffeomorphism groups
and invariants of finite type. In his seminal paper [V] Vassiliev introduced finite type invariants
by considering the space of all immersions of S1 into R3 and relating the topology of the singular
locus to the topology of its complement via Alexander duality. He resolved and cooriented the
discriminant of the space and introduced a spectral sequence with a filtration, which suggested
the simple geometrical and combinatorial definition of an invariant of finite type, which was
later interpreted by Birman and Lin as a ”Vassiliev derivative” and led to the following skein
relation.
If λ be an arbitrary invariant of oriented knots in oriented space with values in some Abelian
group A. Extend λ to be an invariant of 1-singular knots (knots that may have a single
singularity that locally looks like a double point  ), using the formula
λ( ) = λ(!) − λ(")
Further extend λ to the set of n-singular knots (knots with n double points) by repeatedly
using the skein relation.
Definition We say that λ is of type n if its extension to (n + 1)-singular knots vanishes
identically. We say that λ is of finite type if it is of type n for some n.
Given the above formula, the definition of an invariant of finite type n becomes similar to
that of a polynomial: its (n+1)st Vassiliev derivative is zero.
It was shown that all known invariants are either of finite type, or are infinite linear combi-
nations of those, e.g. in [BN1] it was shown that the nth coeffitient of the Conway polynomial
is a Vassiliev invariant of order ≤ n.
In this paper we are working with Khovanov homology, which will be our main example,
however the latest progress in finding the combinatorial formula for the differential of the
Ozsvath-Szabo knot complex [MOS], makes us hopeful that more and more examples will be
coming.
For the construction of the local system it is important to understand the topological type
of the base. The topology of the connected components of the complement to the discriminant
in the space of knots, called chambers, was studied by A. Hatcher and R.Budney [H], [B].
They introduced simple homotopical models for such spaces. Recall that the local system is
well-defined on a homotopy model of the base, so Hatcher’s model is exactly what is needed to
construct the local system of Khovanov complexes.
Throughout the paper the following observation is the main guideline for our constructions:
local systems of the classifying space of the category are functors from this category to the
triangulated category of complexes.
It would be very interesting to understand the relation between the Vassiliev space of knots
is the classifying space of the category, whose objects are knots and whose morphisms are knot
cobordisms.
Our construction provides a Khovanov functor from the category of knots into the triangu-
lated category of complexes.
This allows us to translate all topological properties of the space of knots and Khovanov
local system on it into the language of homological algebra and then use the methods of
triangulated categories and homological algebra to assign algebraic objects to topological ones
(singular knots and links).
Recall that in his paper [Kh] M.Khovanov categorified the Jones polynomial, i.e. he found
a homology theory, the Euler characteristics of which equals the Jones polynomial. He starts
with a diagram of the knot and constructs a bigraded complex, associated to this diagram,
using two resolutions of the knot crossing:
0-resolution 1-resolution
The Khovanov complex then becomes the sum of the tensor products of the vector space
V, where the homological degree is given by the number of 1’s in the complete resolution of
the knot. The local system of Khovanov homologies on the Vassiliev’s space of knots can be
considered as invariants of families of knots.
The discriminant of Vassiliev’s space corresponds to knots with transversal self-intersection,
i.e. moving from one chamber to another we change overcrossing to undercrossing by passing
through a knot with a single double point. We study how the Khovanov complex changes under
such modification and find the corresponding morphism.
After defining a wall-crossing morphism we can extend the invariant to the singular locus
by the cone of a morphism which is our ”categorification of the Vassiliev derivative”. Then we
introduce the definition of a local system of finite type: the local system is of finite type n if for
any selfintersection of the discriminant of codimension n, its n’th cone is an acyclic complex.
The categorification of the Vassiliev derivative allows us to define the filtration on the Floer
- type theories for manifolds of any dimension.
In [S4] we prove the first finiteness result:
Theorem [S4]. Restricted to the subcategory of knots with at most n crossing, Khovanov
local system is of finite type n, n ≥ 3 and of type zero n = 0, 1, 2.
This definition can be generalized to the categorifications of the invariants of manifolds of
any dimension: we construct spaces of 3 and 4-manifolds by a version of a Pontryagin-Thom
construction, consider homological invariants of 3 and 4-manifolds as local systems on these
spaces and extend them to the discriminant.
In subsequent papers our main example will be the Heegaard Floer homology [OS], the Euler
characteristic of which is Turaev’s torsion. We show that local systems of such homological
theories on the space of 3 - manifolds [S1] will carry information about invariants of finite type
for families and information about the diffeomorphism group. We also have a construction [S2]
for the refined Seiberg-Witten invariants on the space of parallelizable 4-manifolds.
Acknowledgements. My deepest thanks go to Yasha Eliashberg for many valuable discus-
sions, for inspiration and for his constant encouragement and support. I want to thank Maxim
Kontsevich who suggested that I work on this project, for his attention to my work during my
visit to the IHES and for many important suggestions.
I want to thank graduate students Eric Schoenfeld and Isidora Milin for reading the paper
and making useful comments.
This paper was written during my visits to the IAS, IHES, MPIM and Stanford and I am
grateful to these institutions for their exceptional hospitality. This work was partially supported
by the NSF grant DMS9729992.
2. Vassiliev theory, invariants of finite type.
2.1. The space of knots, coorientation
Vassiliev considered the space of all maps E = f : S1 → R3. This space is a space of
functions, so it is an infinite-dimensional Euclidean space. It is linear, contractible, and consists
of singular (D) and nonsingular(E - D) knots. The discriminant D forms a singular hypersurface
in E and subdivides into chambers, corresponding to different isotopy types of knots. To move
from one chamber to another one has to change one overcrossing to undercrossing, passing
through a singular knot with one double point.
The discriminant of the space of knots is a real hypersurface, stratified by the number of the
double points, which subdivides the infinite-dimensional space into chambers, corresponding
to different isotopy types of knots.
Vassiliev resolved and cooriented the discriminant, so we can assume that all points of
selfintersection are transversal, with 2n chambers adjacent to a point of selfintersection of the
discriminant of codimension n.
To study the topology of the complement to the discriminant, Vassiliev wrote a spectral
sequence, calculating the homology of the discriminant and then related it to the homology of
its complement via Alexander duality. His spectral sequence had a filtration, which suggested
the simple geometrical and combinatorial definition of an invariant of finite type: an invariant
is of type n if for any selfintersection of the discriminant of codimension (n+1) its alternated
sum over the 2n+1 chambers adjacent to a point of selfintersection is zero.
For our constructions it will be very important to have a coorientation of the discriminant,
which was introduced by Vassiliev.
Definition. A hypersurface in a real manifold is said to be coorientable if it has a non-zero
section of its normal bundle, i.e. if there exists a continuous vector field which is not tangent
to the hypersurface at any point and doesn’t vanish anywhere.
So there are two sides of the hypersurface : one where this vector field is pointing to and
the other is where it is pointing from. And there are two choices of such vector field. The
coorientation of a coorientable hypersurface is the choice of one of two possibilities.
For example, Mobius band in R3 is not coorientable.
Vassiliev shows [V] that the discriminant of the space of knots has a coorientation, the
conistent choice of normal directions.
Recall that the nonsingular point ψ ∈ D of the discriminant is a map S1 → R3, gluing to-
gether 2 distinct points t1, t2 of S
1, s.t. derivatives of the map ψ at those points are transversal.
Coorientation of the discriminant. Fix the orientation of R3 and choose positively
oriented local coordinates near the point ψ(t1) = ψ(t2). For any point ψ1 ∈ D close to ψ define
the number r(ψ1) as the determinant:
|t2, ψ1(t1)− ψ1(t2))
with respect to these coordinates. This determinant depends only of the pair of points t1, t2,
not on their order. A vector in the space of functions at the point ψ ∈ D, which is transversal
to the discriminant, is said to be positive, if the derivative of the function r along this vector
is positive and negative, if this derivative is negative.
This rule gives the coorientation of the hypersurface D at all its nonsingular points and also
of any nonsingular locally irreducible component of D at the points of selfintersection of D.
The consistent choice of the normal directions of the walls of the discriminant will give the
”directions” of the cobordisms (which are embedded into E× I) between knots of the space E.
Note. It is interesting to compare this construction with the result of E.Ghys [Gh], who
introduced a metric on the space of knots and 3-manifolds.)
2.3. The topology of the chambers of the space of knots.
The study of the topology of the chambers of the space of knots was started by A. Hatcher
[H], who found a simple homotopy models for these spaces.
The main result is based on an earlier theorem regarding the topology of the classifying
space of diffeomorphisms of an irreducible 3-manifold with nonempty boundary.
In the following theorem A. Hatcher and D. McCullough answered the question posed by M.
Kontsevich [K], regarding the finiteness of the homotopy type of the classifying space of the
group of diffeomorphisms [HaM]:
Theorem [HaM]. Let M be an irreducible compact connected orientable 3-manifold with
nonempty boundary. Then BDiff(M, rel∂) has the homotopy type of a finite aspherical CW-
complex.
The proof of this theorem uses the JSJ-decomposition of a 3-manifold.
When applied to knot complements, the JSJ-decomposition defines a fundamental class
of links in S3, the ”knot generating links” (KGL). A KGL is any (n + 1)-component link
L = (L0, L1, · · · , Ln) whose complement is either Seifert fibred or atoroidal, such that the
n-component sub-link (L1, L2, · · · , Ln) is the unlink. If the complement of a knot f contains
an incompressible torus, then f can be represented as a ‘spliced knot’ f = J�L in unique way,
where L is an (n + 1)-component KGL, and J = (J1, · · · , Jn) is an n-tuple of non-trivial long
knots.
The spliced knot J�L is obtained from L0 by a generalized satellite construction. For any
knot there is a representation of a knot as an iterated splice knot of atoroidal and hyperbolic
KGLs. The order of splicing determines the ”companionship tree” of f , Gf , and is a complete
isotopy invariant of long knots.
Given a knot f ∈ K, denote the path-component of K containing f by Kf . The topology
of the chambers Kf was further studied by R. Budney The main result of his paper [Bu] is
the computation of the homotopy type of Kf if f is a hyperbolically-spliced knot ie: f = J�L
where L is a hyperbolic KGL.
The combined results can be summarized in the following theorem:
Theorem [Bu, H].
If f = J�L where L is an (n+ 1)-component hyperbolic KGL, then
Kf ⋍ S
SO2 ×Af
Af is the maximal subgroup of BL such that induced action of Af on K
n preserves
i=1KLi .
The restriction map Af → Diff(S
3, L0) → Diff(L0) is faithful, giving an embedding Af →
SO2, and this is the action of Af on SO2.
This result completes the computation of the homotopy-type of K since we have the prior
results:
H1 If f is the unknot, then Kf is contractible.
H2 If f is a torus knot, then Kf ≃ S
H3 If f is a hyperbolic knot, then Kf ⋍ S
1 × S1
H4 If a knot f is a cabling of a knot g then Kf ⋍ S
1 ×Kg.
B5 If the knot f is a connected sum of n ≥ 2 prime knots f1, f2, · · · , fn then Kf ⋍
((C2(n)×
i=1Kfi) /Σf . Here Σf ⊂ Sn is a Young subgroup of Sn, acting on C2(n)
by permutation of the labellings of the cubes, and similarly by permuting the factors
of the product
i=1Kfi . The definition of Σf ⊂ Sn is that it is the subgroup of Sn
that preserves a partition of {1, 2, · · · , n}, the partition being given by the equivalence
relation i ∼ j ⇐⇒ Kfi = Kfj .
B6 If a knot has a non-trivial companionship tree, then it is either a cable, in which case
H4 applies, a connect-sum, in which case B5 applies or is hyperbolically spliced. If
a knot has a trivial companionship tree, it is either the unknot, in which case H1
applies, or a torus knot in which case H2 applies, or a hyperbolic knot, in which case
H3 applies. Moreover, every time one applies one of the above theorems, one reduces
the problem of computing the homotopy-type of Kf to computing the homotopy-type
of knot spaces for knots with shorter companionship trees, thus the process terminates
after finitely-many iterations.
For constructing a local system we need only the homotopy type of the chamber. The
theorem of Hatcher and Budney provides us with a complete classification of homotopy types
of chambers, corresponding to all possible knot types.
3. Khovanov’s categorification of Jones polynomial.
3.1. Jones polynomial as Euler characterictics. Skein relation.
In his paper [Kh] M. Khovanov constructs a homology theory, with Euler characteristics
equal to the Jones polynomial.
He associated to any diagram D of an oriented link with n crossing points a chain complex
CKh(D) of abelian groups of homological length (n+1), and proved that for any two diagrams
of the same link the corresponding complexes are chain homotopy equivalent. Hence, the
homology groups Kh(D) are link invariants up to isomorphism.
His construction is as follows: given any double point of the link projection D, he allows two
smoothings:
0-resolution 1-resolution
If the the diagram has n double points, there are 2n possible resolutions. The result of each
complete smoothing is the set of circles in the plane, labled by n-tuples of 1’s and 0’s:
CKh(©, ...,©
︸ ︷︷ ︸
ntimes
) = V ⊗n
The cobordisms between links, i.e., surfaces embedded in R3 × [0, 1], should provide maps
between the associated groups. A surface embedded in the 4-space can be visualized as a
sequence of plane projections of its 3-dimensional sections (see [CS]). Given such a presentation
J of a compact oriented surface S properly embedded in R3 × [0, 1] with the boundary of S
being the union of two links L0 ⊂ R
3 × {0} and L1 ⊂ R
3 × {1}, , Khovanov associates to J a
map of cohomology groups
θJ : Kh
i,j(D0) → Kh
i,j+χ(S)(D1), i, j ∈ Z
The differential of the Khovanov complex is defined using two linear maps m : V ⊗ V → V
and ∆ : V → V ⊗ V given by formulas :
V ⊗ V
v+ ⊗ v− 7→ v− v+ ⊗ v+ 7→ v+
v− ⊗ v+ 7→ v− v− ⊗ v− 7→ 0
→ V ⊗ V
v+ 7→ v+ ⊗ v− + v− ⊗ v+
v− 7→ v− ⊗ v−
The differential in Khovanov complex can be informally described as ”all the ways of changing
0-crossing to 1-crossing”.
Homological degree of the Khovanov complex in the number of 1’s in the plane diagram
resolution. The sum of ”quantum” components of the same homological degree i gives the ith
component of the Khovanov complex.
One can see that the i-th differential di is the sum over ”quantum” components, it will
map one of the quantum components in homological degree i to perhaps several quantum
components of homological degree i+1.
Khovanov theory can be considered as a (1+1) dimensional TQFT. The cubes, that are
used in it’s definition come from the TQFT corresponding to the Frobenius algebra defined
by V,m,∆. As we will see later, our constructions will give the interpretation of Khovanov
local system as a topological D-brane and will suggest to study the structure of the category
of topological D-branes as a triangulated category.
We prove the following important property of the Khovanov’s complex:
Theorem 1. Let k denote the kth crossing point of the knot projection D, then for any k
the Khovanov’s complex C decomposes into a sum of two subcomplexes C = Ck0 ⊕ C
1 with
matrix differential of the form
d0 d0,1
Proof. Let Ck0 denote the subcomplex of C, consisting of vector spaces, which correspond
to the complete resolutions of D, having 0 on the kth place. The differential d0 obtained by
restricting d only to the arrows between components of Ck0 . We define C
1 the same way, by
restricting to the complete resolutions of D, having 1 on the kth place.
The only components of the differential, which are not yet used in our decomposition, are
the ones which change 0-resolution on the kth place of Ck0 to 1 on the kth place in C
1 , we
denote them d0,1.
One can easily see from the definition of the Khovanov’s differential (which can be intuitivly
described as ”all the ways to change 0-resolution in the ith component of the complex to the
1-resolution in the (i+1)st component”), that there is no differential mapping ith component
of Ck1 to the (i+1)st component of C
Mirror images and adjoints. Taking the mirror image of the knot will dualize Khovanov
complex. So if we want to invert the cobordism between two knots, we should consider the
”dual” cobordism between mirror images of these knots.
3.2. Reidemeister and Jacobsson moves.
A cobordism (a surface S embedded into R3 × [0, 1]) between knots K0 and K1 provide a
morphism between the corresponding cohomology:
FS : Kh
i,j(D0) → Kh
i,j+χ(S)(D1)
where D0 and D1 are diagrams of the knots K0 and K1 and χ(S) is the Euler characteristic
of the surface.
We will distinguish between two types of cobordisms - first, corresponding to the wall crossing
(and changing the type of the knot). And second, corresponding to nontrivial loops in chambers
which will reflect the dependence of Khovanov homologies on the selfdiffeomorphisms of the
knot, similar to the Reidemeister moves. In this paragraph we will discuss the second type of
cobordisms.
By a surface S in R4 we mean an oriented, compact surface S, possibly with boundary,
properly embedded in R3 × [0, 1]. The boundary of S is then a disjoint union
∂S = ∂0S ⊔ −∂1S
of the intersections of S with two boundary components of R3 × [0, 1]:
∂0S = (S ∩R
3 × {0})
−∂1S = (S ∩R
3 × {1})
Note that ∂0S and ∂1S are oriented links in R
The surface S can be represented by a sequence J of plane diagrams of oriented links where
every two consecutive diagrams in J are related either by one of the four Reidemeister moves
or by one of the four moves birth, death, fusion described by Carter-Saito [CS].
To each Reidemeister move between diagrams D0 and D1 Khovanov [Kh] associates a quasi-
isomorphism map of complexes C(D0) → C(D1).
Given a representation J of a surface S by a sequence of diagrams, we can associate to J a
map of complexes
ϕJ : C(J0) → C(J1)
Any link cobordism can be described as a one-parameter family Dt, t ∈ [0, 1] of planar dia-
grams, called a movie. The Dt are link diagrams, except at finitely many singular points which
correspond to either a Reidemeister move or a Morse modification. Away from these points
the diagrams for various t are locally isotopic . Khovanov explained how local moves induce
chain maps between complexes, hence homomorphisms between homology groups. The same is
true for planar isotopies. Hence, the composition of these chain maps defines a homomorphism
between the homology groups of the diagrams of links.
In his paper [Ja] Jacobsson shows that there are knots, s.t. a movie as above will give a
nontrivial morphism of Khovanov homology:
Theorem [Ja] For oriented links L0 and L1, presented by diagrams D0 and D1, an oriented
link cobordism Σ from L0 to L1, defines a homomorphism H(D0) → H(D1), invariant up
to multiplication by -1 under ambient isotopy of Σ leaving ∂Σ setwise fixed. Moreover, this
invariant is non-trivial.
Jacobsson constructs a family of derived invariants of link cobordisms with the same source
and target, which are analogous to the classical Lefschetz numbers of endomorphisms of man-
ifolds.
The Jones polynomial appears as the Lefschetz polynomial of the identity cobordism.
From our perspective the Jacobsson’s theorem shows that the Khovanov local system will
have nontrivial monodromies on the chambers of the space of knots.
3.3. Wall-crossing morphisms.
In 3.2 we described what kind of modifications can occur in the cobordism, when we consider
the ”movie” consisting only of manifolds of the same topological type. These modifications
implied corresponding monodromies of the Khovanov complex.
However, morphisms that are the most important for Vassiliev-type theories are the ”wall-
crossing” morphisms. We will define them now (locally).
Consider two complexes A• and B• adjacent to the generic wall of the discriminant. Recall,
that the discriminant is cooriented ( 2.2). If B• is ”right” via coorientation (or ”further in the
Ghys metric form the unknot) of A•, then we shift B•’s grading up by one and consider B•[1]:
A•|B•[1]
Note. In general, and this will be very important for us in subsequent chapters, if the complex
K• is n steps (via the coorientation) away from the unknot, we shift its grading up by n.
Thus adjacent complexes will have difference in grading by one (as above), defined by the
coorientation.
Now we want to understand what happens to the Khovanov complex when we change the
kth over-crossing (in the knot diagram D) to an under-crossing.
We will illustrate these changes on one of the Bar-Natan’s trademark diagrams (with his
permission)[BN1].
By ”I” we mark the arrow , connecting components of the complex which will exchange
places under wall-crossing morphisms when we change over-crossing to under-crossing for the
self-intersection point 1. By ”II” when we do it for point 2 and ”III” when we do it for 3:
2 V {1}
V ⊗2{2}
==zzzzzzzzzzzzzzzzzz
V {1}
<<yyyyyyyyyyyyyyyyyy
V ⊗2{2}
V ⊗3{3}
V {1}
wwwwwwww
;;wwwwwwww
V ⊗2{2}
;;vvvvvvvvvvvvvvvvv
d0 // J&K1
d1 // J&K2
d2 // J&K3
(0.1)
Now recall the theorem proved in (3.1): for any k, where k is the number of crossings of the
diagram D, the Khovanov complex can be split into the sum of two subcomplexes with the
uppertriangular differential.
Notice from the diagram above that when we change kth overcrossing to an undercrossing,
0 and 1-resolutions are exchanged , so A• = A•0 ⊕ A
•[1] = B•0 [1] ⊕ B
1 [1], thus for every k
we can define the wall-crossing morphism ω as follows:
Theorem 2. The map defined as the identity on A•0 and as a trivial map on A
ω : A•0
Id // B•0 [1]
ω : A•1
∅ // B•1 [1]
is the morphism of complexes.
Proof. From the Theorem 1 we know that for any crossing k the Khovanov complex can be
decomposed as a direct sum with uppertriangular differential:
d0 d0,1
It is an easy check that the wall-crossing morphism defined as above is indeed a morphism
of complexes (i.e. it commutes with the differential):
ω // B•
ω // B•
Since we defined the morphism as 0 on A•1, the diagram above becomes the following com-
mutative diagram:
Id // B•0 [1]
Id // B•0 [1]
3.4. The local system of Khovanov complexes on the space of knots.
In this paragraph we introduce the Khovanov local system on the space of knots.
Definition. A local system on the locally connected topological space M is a fiber bundle
over M, the sections of which are abelian groups. The fiber of the bundle depend continuously
on the point of the base (such that the group structure on the set of fibers can be extended
over small domains in the base).
Any local system on M with fiber A defines a representation π1(M) → Aut(A). To any loop
there corresponds a morphism of the fibers of the bundle over the starting point of the loop.
The set of isomorphism classes of local systems with fiber A are in one-to-one corresondence
with the set of such representations up to conjugation. For example any representation of an
arbitrary group π in Aut(A) uniquely (up to isomorphism) defines a local system on the space
K(π, 1) [GM].
Morphisms of local systems are morphisms of fiber bundles, preserving group structure in
the fibers. Thus introducing the continuation functions (maps between fibers) over paths in
the base will define a local system over the manifold M.
Next we set up the Khovanov complexes as a local system on the space of knots. If we
were doing it ”in coordinates”, we would introduce charts on the chambers of the space of
knots and define our local system via transition maps, starting with some ”initial” point . This
would be a very interesting and realistic approach, since the homotopy models for chambers
are understood [H], [B], e.g. we would have just one chart for the chamber, containing the
unknot (since that chamber is contractible), two for a torus knot, four for a hyperbolic one,
etc. Then monodromies of the Khovanov local system along nontrivial loops in the chamber
will be given via Jacobsson movies.
It would be also very interesting to find a unique special point in every chamber of the space
E and study monodromies of the local system with respect to this point.
The candidate for such point is introduced in the works of J. O’Hara, who studied the minima
of the electrostatical energy function of the knot [O’H]:
E(K) =
|(x− y)|−2dxdy
It was shown that under some assumptions and for perturbation of the above functional, its
critical points on the space of knots will provide a ’distinguished” point in the chamber.
The first natural question for this setup is: which nontrivial loops in the chamber EK ,
corresponding to the knot K are distinguised by Khovanov homologies and which are not?
However, assuming Khovanov’s theorem [Kh] (that his homology groups are invariants of
the knot, independent on the choices made) and assuming also the results of Jacobsson [J] , it
is enough for us to introduce the continuation maps, along any path γ in the chamber of the
space of knots.
These methods were developed by several authors (see [Hu]):
Let K1 and K2 be two knots in the same chamber of the space E, let Ki be generic, and let
γ = {Kt | t ∈ [0, 1]} be any path of equivalent objects in E from K1 to K2. Then a generic
path γ
induces a chain map
F (γ) : CKh∗(K1)−→CKh∗(K2)
called the “continuation” map, which has the following properties:
• 1)Homotopy A generic homotopy rel endpoints between two paths γ1 and γ2 with
associated chain maps F1 and F2 induces a chain homotopy
H : HKh∗(K1)−→HKh∗+1(K2)
∂H +H∂ = F1 − F2
• 2)Concatenation If the final endpoint of γ1 is the initial endpoint of γ2, then F (γ2γ1)
is chain homotopic to F (γ2)F (γ1).
• 3)Constant If γ is a constant path then F (γ) is the identity on chains.
These three properties imply that ifK1 andK2 are equivalent, thenHKh∗(K1) ≃ HKh∗(K2).
(Khovanov’s theorem).
This isomorphism is generally not canonical, because different homotopy classes of paths
may induce different continuation isomorphisms on Khovanov homology (Jacobsson moves).
However, since the loop is contractible, we do know that HKh∗(K) depends only on K, so we
denote this from now on by HKh∗(K).
We now define the restriction of the Khovanov local system to finite-dimensional subspaces
of the space of knots.
Note that in the original setting our complexes may have had different length. For example,
the complex corresponding to the standard projection of the unknot will have length 1, however,
we can consider very complicated ”twisted” projections of the unknot with an arbitrary large
number of crossing points. The corresponding complexes will be quasiisomorphic to the original
This construction resembles the definition of Khovanov homology introduced in [CK], [W].
They define Khovanov homology as a relative theory, where homology groups are calculated
relative to the twisted unknots.
When considering the restrictions of the Khovanov local system to the subcategories of knots
with at most n crossings, we would like all complexes to be of length n+ 1.
This can be achieved by ”undoing” the local system, starting with the knots of maximal
crossing number n and then using the wall-crossing morphisms, define complexes of length
(n + 1), quasiisomorphic to the original ones, in all adjacent chambers. We continue this
process till it ends, when we reach the chamber containing unknot.
Recall that Khovanov homology is defined for the knot projection (though is independent
of it by Khovanov’s theorem). So we will consider a ramification of Vassiliev space, a pair, the
embedding of the circle into R3 and its projection on (x,y)-plane. Then each chamber will be
subdivided into ”subchambers” corresponding to nonsingular knot projections and the ”sub-
discriminant” will consist of singular projections of the given knot. The local system, defined
on such ramification will live on the universal cover of the base, the original Hatcher chamber
corresponding to knot K and morphisms of the local system between ”subchambers” are given
by Reidemaister moves. The composition of such moves may constitute the Jacobsson’s movie
and will give nontrivial monodromies of the local system within the original chamber.
Note. As we will see later, if one assines cones of Reidemeister morphisms to the walls of
the ”subdiscriminant”, all such cones will be acyclic complexes. This statement in a different
form was proved in the original Khovanov [Kh] paper.
4. The main definition, invariants of finite type for families.
4.1. Some homological algebra.
We describe results and main definitions from the category theory and homological algebra
which will be used in subsequent chapters. The standard references on this subject are [GM],
[Th].
By constructing the local system of (3.4) we introduced the derived category of Khovanov
complexes. The properties of the derived category are summarized in the axiomatics of the
triangulated category, which we will discuss in this chapter.
Definition. An additive category is a category A such that
• Each set of morphisms Hom(A,B) forms an abelian group.
• Composition of morphisms distributes over the addition of morphisms given by the
abelian group structure, i.e. f ◦ (g + h) = f ◦ g + f ◦ h and (f + g) ◦ h = f ◦ h+ g ◦ h.
• There exist products (direct sums) A×B of any two objects A,B satisfying the usual
universal properties.
• There exists a zero object 0 such that Hom(0, 0) is the zero group (i.e. just the identity
morphism). ThusHom(0, A) = 0 = Hom(A, 0) for all A, and the unique zero morphism
between any two objects is the one that factors through the zero object.
So in an abelian category we can talk about exact sequences and chain complexes, and
cohomology of complexes. Additive functors between abelian categories are exact (respectively
left or right exact) if they preserve exact sequences (respectively short exact sequences 0 →
A→ B → C or A→ B → C → 0).
Definition. The bounded derived category Db(A) of an abelian category A has as objects
bounded (i.e. finite length) A-chain complexes, and morphisms given by chain maps with quasi-
isomorphisms inverted as follows. We introduce morphisms f for every chain map between
complexes f : Xf → Yf , and g
−1 : Yg → Xg for every quasi-isomorphism g : Xg
→ Yg. Then
form all products of these morphisms such that the range of one is the domain of the next.
Finally identify any combination f1f2 with the composition f1 ◦ f2, and gg
−1 and g−1g with
the relevant identity maps idYg and idXg .
Recall that a triangulated category C is an additive category equipped with the additional
data:
Definition. Triangulated category is an additive category with a functor T : X → X[1]
(where Xi[1] = Xi+1) and a set of distinguished triangles satisfying a list of axioms.
The triangles include, for all objects X of the category:
1) Identity morphism
X → X → 0 → X[1],
2) Any morphism f : X → Y can be completed to a distinguished triangle
X → Y → C → X[1],
3) There is also a derived analogue of the 5-lemma, and a compatibility of triangles known
as the octahedral lemma, which can be understood as follows:
If we naively interprete property 1) as the difference X −X = 0, property 2) as C = X −Y ,
then the octahedron lemma says:
(X − Y )− Z = C − Z = X − (Y − Z)
When topological spaces considered up to homotopy there is no notion of kernel or cokernel.
The cylinder construction shows that any map f : X → Y is homotopic to an inclusion
X → cyl (f) = Y ⊔ (X × [0, 1])/f(x) ∼ (x, 1), while the path space construction shows it is also
homotopic to a fibration.
The cone Cf on a map f : X → Y is the space formed from Y ⊔ (X × [0, 1]) by identifying
X × {1} with its image f(X) ⊂ Y , and collapsing X × {0} to a point.
It can be considered as a cokernel, i.e. if f : X → Y
is an inclusion, then Cf is homotopy equivalent to Y/X.
Taking the ith cohomologyHi of each term, and using the suspension isomorphismHi(ΣX) ∼=
Hi−1(X) gives a sequence
Hi(X) → Hi(Y ) → Hi(Y,X) → Hi−1(X) → Hi−1(Y ) → . . .
which is just the long exact sequence associated to the pair X ⊂ Y .
Up to homotopy we can make this into a sequence of simplicial maps, so that taking the
associated chain complexes we get a lifting of the long exact sequence of homology to the level
of complexes. It exists for all maps f , not just inclusions, with Y/X replaced by Cf .
If f is a fibration, Cf can act as the “kernel” or fibre of the map. If f : X →point, then
Cf = ΣX, the suspension of the fibre X.
Thus Cf acts as a combination of both cokernel and kernel, and if f : X → Y is a map
inducing an isomorphism of homology groups of simply connected spaces then the sequence
Hi(X) → Hi(Y ) → Hi(Cf ) → Hi−1(X) → Hi−1(Y ) → . . .
implies H∗(Cf )=0. Then Cf homotopy equivalent to a point. Thus we can give the following
definition.
Definition. If X and Y are simplicial complexes, then a simplicial map f : X → Y ,
defines (up to isomorphism) an object in triangulated category, called the cone of morphism
f, denoted Cf .
C•X ⊕ C
Y [1] with differential dCf =
0 dY [1]
where [n ] means shift a complex n places up.
Thus we can define the cone Cf on any map of chain complexes f : A
• → B• in an abelian
category A by the above formula, replacing C•X by A
• and C•Y by B
•. If A• = A and B• = B
are chain complexes concentrated in degree zero then Cf is the complex {A
→ B}. This has
zeroth cohomology h0(Cf ) = ker f , and h
1(Cf ) = coker f , so combines the two (in different
degrees). In general it is just the total complex of A• → B•.
So what we get in a derived category is not kernels or cokernels, but “exact triangles”
A• → B• → C• → A• [ 1 ].
Thus we have long exact sequences instead of short exact ones; taking ith cohomology hi of
the above gives the standard long exact sequence
hi(A•) → hi(B•) → hi(C•) → hi+1(A•) → . . .
The cone will fit into a triangle:
u // B
The “[1]” denotes that the map w increases the grade of any object by one.
4.2. Space of knots as a classifying space of the category.
In this paragraph we will construct the Khovanov functor from the category of knots into
the triangulated category of Khovanov complexes.
Definition. The category of knots K is the category, the objects of which are knots,
S1 → S3, morphisms are cobordisms, i.e. surfaces Σ properly embedded in R3 × [0, 1] with the
boundary of Σ being the union of two knots K1 ⊂ R
3 × {0} and K2 ⊂ R
3 × {1}.
We denote Kn the subcategory of knots with at most n crossings.
(Recall that a knot’s crossing number is the lowest number of crossings of any diagram of
the knot. )
Note that our cobordisms (morphisms in the category of knots) are directed via the coori-
entation of the discriminant of the space of knots.
Note that to reverse cobordism, we can consider the same cobordism between mirror images
of the knots.
Definition. The nerveN (C) of a category C is a simplicial set constructed from the objects
and morphisms of C, i.e. points of N (C) are objects of C, 1-simplices are morphisms of C,
2-simplices are commutative triangles, 3-simplices are commutative tetrahedrons of C, etc.
N (C) = (limN i(C))
The geometric realization of a simplicial set N (C) is a topological space, called the classi-
fying space of the category C, denoted B(C).
The following observation is the main guideline for our constructions: sheaves on the classi-
fying space of the category are functors on that category [Wi].
Once we prove that the Vassiliev space of knots is a classifying space of the category K, our
local system will provide a representation of the Khovanov functor.
Let C be a category and let Set be the category of sets. For each object A of C let Hom(A,)
be the hom functor which maps objects X to the set Hom(A,X).
Recall that a functor F : C → Set is said to be representable if it is naturally isomorphic
to Hom(A, ) for some object A of C. A representation of F is a pair (A,Ψ) where
Ψ : Hom(A, ) → F
is a natural isomorphism.
If E - the space of knots, denote KE the category of knots, whose objects are points in E
and morphisms Mor(x, y) = {γ : [0, 1] → X; s.t.γ(0) = x, γ(1) = y} and KK - subcategory
corresponding to knots of the same isotopy type K.
Proposition. The chamber EK of the space of long knots forK - unknot, torus of hyperbolic
knot is the classifying space of the category KK .
Proof. By Hatcher’s theorem [H] the chambers of the space of knots EK , corresponding to
unknot, torus or hyperbolic knot are K(π, 1).
By definition the space of long knots is E = {f : R1 → R3}, nonsingular maps which are
standard outside the ball of large radius. If f1, f2 are vector equations giving knots K1,K2,
then tf1 + (1 − t)f2 is a path in the mapping space, defining a knot for each value of t. The
cobordism between two embeddings is given by equations in R3× I. All higher cobordisms can
be contracted, since there are no higher homotopy groups in EK . So both the classifying space
of the category and the chamber of the space of knots are K(π, 1) with the same π. They are
the same as simplicial complexes.
Note, that in the case of hyperbolic knots one can choose the distinguished point in the
chamber - corresponding to the hyperbolic metric on the complement to the knot.
4.3. Vassiliev derivative as a cone of the wall-crossing morphism.
To be able to construct a categorification of Vassiliev theory, we have to extend the local
system, which we defined on chambers, to the discriminant of the space of knots.
Recall that according to the axiomatics of the triangulated category, described in (4.1), we
assign an new object to every morphism in the category:
for a complex X = (Xi, dix) define a complex
X[1] by
(X[1])i = Xi+1, dX[1] = −dX
For a morphism of complexes f : X → Y let f [1] : X[1] → Y [1] coincide with f component-
wise.
Let f : X → Y be a wall-crossing morphism. The cone of f is the following complex C(f):
X → Y → Z = C(f) → X[1]
C(f)i = X[1]i ⊕ Y i, dC(f)(x
i+1, yi) = (−dXx
i+1, f(xi+1)− dY y
Recall, that we set up the local system on the space of knots (3.4) s.t. if the complex X• is n
steps (via the coorientation) away from the unknot, we shift its grading up by n. So complexes
in adjacent chambers will have difference in grading by one, defined by the coorientation.
Thus, given a bigraded complex, associated to the generic wall of the discriminant, we get
two natural specialization maps into the neighbourhoods, containing X• and Y •:
So with any morphism f we associate the triangle:
// Y •
aaBBBBBBBB
With any commutative cube
u // •
u // •
(in the space of knots the above picture corresponds to the cobordism around the self-
intersection of the discriminant of codimension two), we associate the map between cones,
corresponding to the vertical and horisontal walls, and assign it to the point of their intersec-
tion:
Cuω // Cω
Lemma. Given four chambers as above, the order of taking cones of morphisms is irrelevant,
Cuω = Cωu.
Proof. see [GM].
Consider a point of selfintersection of the discriminant of codimension n. There are 2n
chambers adjacent to this point. Since the discriminant was resolved by Vassiliev [V], this
point can be considered as a point of transversal selfintersection of n hyperplanes in Rn, or an
origin of the coordinate system of Rn.
Now our local system looks as follows. On chambers of our space we have the local system of
Khovanov complexes, to any point t of the generic wall between chambers containing X• and
Y • (corresponding to a singular knot), we assign the cone of the morphism X• → Y • (with
the specialization maps from the cone to the small neighborhoods of t containing X• and Y •).
To the point of codimention n we assign the nth cone, 2n-graded complex, etc.
Definition. The Khovanov homology of the singular knot (with a single double point ) is a
bigraded complex
X• ⊕ Y •[1] with the matrix differential dCω =
0 dY [1]
where X• is Khovanov complex of the knot with overcrossing, Y • is the Khovanov complex of
the knot with undercrossing and ω is the wall-crossing morphism.
In [S4] we give the geometric interpretation of the above definition.
4.4. The definition of a theory of finite type.
Once we extended the local system to the singular locus, it is natural to ask if such an
extension will lead to the categorification of Vassiliev theory.
The first natural guess is that the theory, set up on some space of objects, which has quasi-
isomorphic complexes on all chambers is a theory of order zero. Such theory will consist of
trivial distinguished triangles as in (a) of the axiomatics of the triangulated category. When
complexes, corresponding to adjacent chambers are quasiisomorphic, the cone of the morphism
is an acyclic complex.
Baby example of a theory of order 0.
Let M be an n-dimensional compact oriented smooth manifold. Consider the space of func-
tions on M. This is an infinite-dimensional Euclidean space. The chambers of the space will
correspond to Morse functions on M, the walls of the discriminant - to simple degenerations
when two critical points collide, etc. Let’s consider the Morse complex, generated by the crit-
ical points of a Morse function on M. As it was shown by many authors, such complex is
isomorphic to the CW complex, associated with M. Since we are calculating the homology of
M via various Morse functions, complexes may vary, but will have the same homology and
Euler characteristics.
Then we can proceed according to our philosophy and assign cones of morphisms to the
walls and selfintersections of the discriminant. Since complexes on the chambers of the space
of functions are quasiisomorphic, all cones are acyclic.
Now we can introduce the main definition of a Floer-type theory being of finite type n:
Main Definition. The local system of (Floer-type) complexes, extended to the discriminant
of the space of manifolds via the cone of morphism, is a local system of order n if for any
selfintersection of the discriminant of codimension (n + 1), its (n+1)st cone is an acyclic
complex.
How one shows that an 2n-graded complex is acyclic? For example, if one introduces inverse
maps to the wall-crossing morphisms and construct the homotopy H, s.t.:
dH−Hd = I
It is easy to check that the existence of such homotopy H implies, that the complex doesn’t
have homology. Suppose dc = 0, i.e. c is a cycle, then:
dHc−Hdc = dHc = I
Example. Suppose some local system is conjectured to be of finite type 3. How one would
check this? By our definition, we should consider 23 chambers adjacent to the every point
of selfintersection of the discriminant of codimension 3, and 8 complexes, representing the
local system in the small neighbourhood of this point. This will correspond to the following
commutative cube:
}}}}}}}} g
||||||||
l // G•
}}}}}}}}
m // H•
||||||||
Let’s write the homotopy equation in the matrix form.
Consider dual maps f∗, g∗, ..., w∗. Then we get formulas for d and H as 8× 8 matrices:
dA f g 0 a 0 0 0
0 dB 0 h 0 b 0 0
0 1 dD w 0 0 e 0
0 0 1 dC 0 0 0 c
0 0 0 0 dE k m 0
0 0 0 0 1 dF 0 l
0 0 0 0 0 1 dH n
0 0 0 0 0 0 1 dG
dA 0 0 1 0 0 0 0
f∗ dB 0 0 0 0 0 0
g∗ 0 dD 0 0 0 0 0
0 h∗ w∗ dC 0 0 0 0
a∗ 0 0 0 dE 0 0 1
0 b∗ 0 0 k∗ dF 0 0
0 0 e∗ 0 m∗ 0 dH 0
0 0 0 c∗ 0l∗ h∗ dG
After substituting these matrices into the equation dH − Hd = I we obtain the diagonal
matrix which must be homotopic to the identity matrix:
ff∗ + gg∗ + aa∗ 0 0 0 0 0 0 0
0 −”− 0 0 0 0 0 0
0 0 −”− 0 0 0 0 0
0 0 0 −”− 0 0 0 0
0 0 0 0 −”− 0 0 0
0 0 0 0 0 −”− 0 0
0 0 0 0 0 0 −”− 0
0 0 0 0 0 0 0 cc∗ + nn∗ + ll∗
Thus the condition for the local to be of finite type n can be interpreted as follows. For
any selfintersection of the discriminant of codimension n + 1 consider 2n complexes, forming
a commutative cube (representatives of the local system in the chambers adjacent to the self-
intersection point). Then the naive geometrical interpretation of the local system being of
finite type n is the following: each complex can be ”split” into n+1 subcomplexes, which map
quasiisomorphically to n+ 1 neighbours, at least no homologies die or being generated.
5. Knots: theories of finite type. Further directions.
5.1. Examples of combinatorially defined theories.
In the following table we give the examples of theories, which are the categorifications of
classical invariants. All these theories fit into our framework and may satisfy the finitness
condition.
λ λ = χH∗(M)
Jones polynomial Khovanov homology [Kh]
Alexander polynomial Ozsvath-Szabo knot homology [OS2]
sl(n) invariants Khovanov - Rozhansky homology [KhR]
Casson invariant Instanton Floer homology [F]
Turaev’s torsion Ozsvath-Szabo 3 manifold theory [OS1]
Vafa invariant Gukov-Witten categorification [GW]
Note, that the only theory which is not combinatorially defined is the original Instanton
Floer homology [F]. The fact that it’s Euler characteristics is Casson’s invariant was proved by
C.Taubes [T].
5.2. Generalization to dimension 3 and 4.
In our paper [S1] we generalized Vassiliev’s construction to the case of 3-manifolds. In [S2]
we construct the space of parallelizable 4-manifolds and consider the paramentrized version of
the Refined Seiberg-Witten invariant [BF].
a). The space of 3-manifolds and invariants of finite type.
Note that all 3-manifolds are parallelizable and therefore carry spin-structures.
Following Vassiliev’s approach to classification of knots, we constructed spaces E1 and E2
of 3-manifolds by a version of the Pontryagin-Thom construction.
Our main results are as follows:
Theorem [S1]. In E1 − D each connected component corresponds to a homeomorphism
class of 3-dimensional framed manifold. For any connected framed manifold as above there is
one connected component of E1 −D giving its homeomorphism type.
Theorem [S1]. In E2 − D each connected component corresponds to a homeomorphism
class of 3-dimensional spin manifold. For any connected spin manifold there is one connected
component of E2 −D giving its homeomorphism type.
By a spin manifold we understand a pair (M,θ) where M is an oriented 3-manifold, and θ
is a spin structure on M . Two spin manifolds (M,θ) and (M ′, θ′) are called homeomorphic, if
there exists a homeomorphism M →M ′ taking θ to θ′.
The construction of the space naturally leads to the following definition:
Definition. A map I : (M,θ) → C is called a finite type invariant of (at most) order k if it
satisfies the condition:
(−1)#L
I(ML′) = 0
where L′ is a framed sublink of link L with even framings, L corresponds to the self-intersection
of the discriminant of codimension k+1, #L′ - the number of components of L′, ML′ - spin
3-manifold obtained by surgery on L′.
We introduced an example of Vassiliev invariant of finite order. Given a spin 3-manifold
M3 we consider the Euler characteristic of spin 0-cobordism W . Denote by I(M,spin) =
(sgn(W, spin) − 1)(mod2).
Theorem [S1]. Invariant I(M,spin) is finite type of order 1.
The construction of the space of 3-manifolds chambers of which correspond to spin 3-
manifolds is important for understanding, which additional structures one needs in order to
build the theory of finite-type invariants for homologically nontrivial manifolds. It suggests
that one should consider spin ramifications of known invariants.
In the following paper we will generalize our constructions and the main definition to the
case of 3-manifolds. We will construct a local system of Ozsvath-Szabo homologies, extend it
to the singular locus via the cone of morphism and find examples of theories of finite type.
b). Stably parallelizable 4-manifolds.
In this section we modify the previous construction [S1] to get the space of parallelizable
4-manifolds.
By the definition the manifold is parallelizable if it admits the global field of frames, i.e. has
a trivial tangent bundle. In the case of 4-manifolds this condition is equivalent to vanishing of
Euler and the second Stieffel-Whitney class. In particular signature and the Euler characterictic
of such manifolds will be 0.
We will use the theorem of Quinn:
Theorem Any punctured 4-manifold posesses a smooth structure.
Recall also the result of Vidussi, which states that manifolds diffeomorphic outside a point
have the same Seiberg-Witten invariants,so one cannot use them to detect eventual inequivalent
smooth structures. Thus for the purposes of constructing the family version of the Seiberg-
Witten invariants, it will be sufficient for us to consider “asymptotically flat” 4-manifolds, i.e.
such that outside the ball BR of some large radius R they will be given as the set of common
zeros of the system of linear equations (e.g. fi(x1, ...xn+4) = xi for i = 1, ...n.)
By Gromov’s h-principle any smooth 4-manifold (with all of its smooth structures and met-
rics) can be obtained as a common set of zeros of a system of equations in RN for sufficiently
large N.
Theorem [S2]. Any parallelizable smooth 4-manifold can be obtained as a set of zeros
of n functions on the trivial (n+4)-bundle over Sn. Each manifold will be represented by
|H1(M,Z2)⊕H
3(M,Z)| chambers.
There is a theory which also fits into our template - Ozsvath-Szabo homologies for 3-
manifolds, Euler characteristic of which is Turaev’s torsion. It would be interesting to show
that this theory is also of finite type or decomposes as the Khovanov theory.
c). Ozsvath-Szabo theory as triangulated category.
In [S3] we put the theory developed by P.Ozsvath and Z. Szabo into the context of homo-
logical algebra by considering a local system of their complexes on the space of 3-manifolds
and extending it to the singular locus. We show that for the restricted category the Heegaard
Floer complex CF∞ is of finite type one. For other versions of the theory we will be using the
new combinatorial formulas, obtained in [SW].
Recall that the categorification is the process of replacing sets with categories, functions
with functors, and equations between functions by natural isomorphisms between functors.
On would hope that after establishing this correspondence, homological algebra will provide
algebraic structures which one should assign to geometrical objects without going into the
specifics of a given theory. One can see that this approach is very useful in topological category,
in particular we will be getting knot and link invariants of Ozsvath and Szabo after setting up
their local system on the space of 3-manifolds.
Note Floer homology can be also considered as invariants for families, so it would be in-
teresting to connect our work to the one of M.Hutchings [H]. His work can be interpreted as
construction of local systems corresponding to various Floer-type theories on the chambers of
our spaces. Then we extend them to the discriminant and classify according to our definition.
6.3. Further directions.
1. There is a number of immediate questions from the finite-type invariants story:
a). What will substitute the notion of the chord diagram? What is the ”basis ” in the
theories of finite type?
b). What are the ”dimensions” of the spaces of theories of order n?
2. What is the representation-theoretical meaning of the theory of finite type?
a). Is it possible to construct a ”universal” knot homology theory in a sense of T.Lee [L] ?
b). Is it possible to rise such a ”universal” knot homology theory to the Floer-type theory
of 3-manifolds?
3. There are ”categorifications” of other knot invariants: Alexander polynomial [OS], HOM-
FLY polynomial [DGR ]. These theories also fit into our setting and it will interesting to show
that they decompose into the series of theories of finite type or that their truncations are of
finite type.
4. The next step in our program [S3] is the construction of the local system of Ozsvath-Szabo
homologies on the space of 3-manifolds introduced in [S1]. We also plan to raise Khovanov
theory to the homological Floer-type theory of 3-manifolds.
5. It should be also possible to generalize our program to the study of the diffeomorphism
group of a 4-manifold by considering Gukov-Witten [GW] categorification of Vafa invariant on
the moduli space constructed in [S2].
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