数理解析研究所講究録
第 1707 巻 2010 年 37-55

37

Vassiliev-type invariants revisited
Toru Ohmoto (Hokkaido Univ.)
Dec. 8, 2009 at Nihon Univ. Sakura-Jousui, Tokyo.

In this talk, we revisit an’old and new’ theme on the topology ofbifircation locus in the infinite dimensional space of smooth mappingsfrom a m-fold$M$ to Euclidean space, imitiated by R. Thom [27],$\mathcal{M}$$\Gamma$$\mathcal{M}:=C^{\infty}(M,\mathbb{R}^{n})\supset\Gamma:=$(A) Study$C^{\infty}$

unstable maps}.

as the space of all isotopy invariants ofVass licv-type invariants...

$H^{0}(\mathcal{M}-\Gamma)$

stable maps

(B) Study{$\Rightarrow$$C^{\infty}$

structural$i$

as a representation space of the diffeomorphism group$\mathcal{M}$$Diff(M)\Rightarrow$

Thom polynomials... .

As for (A), I will describe a general framework based on the ThomMather theory, and state an elementary observation (Theorem 3.3 below):
where $m=\dim M\geq 2$, a naive analog to finite typefor generic maps
knot invariants is not so fruiful. The order-one invariants were studied inseveral cases (recent works are, e.g., [30], [31], [22]), however we have still
missed a proper definition of Vassiliev-type invariants of higher order for suchmaps. As for (B), I will only comment about a few examples and propose
a further direction.
thoughout.
All spaces and mappings are of class$Marrow \mathbb{R}^{n}$$C^{\infty}$

1 Mapping space
1.1 A-equivalence and invariant stratification
Let $M$ be a compact manifold of dimension without boundary, and amanifold of dimension without boundary. Denote the space of smooth$:=C^{\infty}(M,N)$ .
topology, bymaps, equipped withif
( being finite dimensional) is called of type
A map :map (Frechet manifold structure
the evaluation map $M\cross Uarrow N$ is a
$N$

$m$

$n$

$C^{\infty}$$Uarrow \mathcal{M}$$\varphi$$\mathcal{M}$$C^{\infty}$$U$$C^{\infty}$on$\mathcal{M})$.

The A-equivalence group or right-left group is the direct product of dif-feomorphism groups$ffl_{M,N}$ $:=$

Also put

Di $ff(M)\cross Diff(N)$ acting on$\ovalbox{\tt\small REJECT}_{M,N}^{0}$$:=$$\mathcal{M}$by$(\varphi,\tau).f$

the connected component containing$:=\tau\circ f\circ\varphi^{-1}$$(id_{M}, id_{N})$

.

.

38
Definition 1.1 $f,g$ : $Marrow N(\in \mathcal{M})$ areョ

$(\varphi,\tau)\in\ovalbox{\tt\small REJECT}_{M,N}$$\ovalbox{\tt\small REJECT}$-equivalent if$\ovalbox{\tt\small REJECT}_{M,N}.f=\ovalbox{\tt\small REJECT}_{M,N}.g$i.e.,s.t. $g=(\varphi,\tau).f$$Marrow^{f}N$$\varphi\{$$\simeq\downarrow$$\tau$$\simeq g$$M$ $arrow N$) ’ sutble map, if the orbit ofis an open set in
We say $f$ is a
-stru
. If $(m,n)$ is in so-called Mather’s nice mnge, structurally stable maps form
-Structural Stability Theorem, proved by Johnan open dense subset in
Mather). Often say generic maps for short.
$\backslash tumll$$C^{\alpha)}$$ffl_{M_{I}N}.f$$\vee$$\mathcal{M}$$\mathcal{M}(C^{\infty}$-invariant Frechet submanifolds of
becomeAny orbits
We are interested in finite codimensional orbits (or families of orbits).
$\ovalbox{\tt\small REJECT}_{M,N}$$\ovalbox{\tt\small REJECT}_{M_{I}N}.f$

Deflnition 1.2 In this note, a nulti-singula rity tvpe means a$\dagger$$f$: $M,Sarrow N,f(S)$($S$$\ovalbox{\tt\small REJECT}$$\mathcal{M}$.

-type of germs

and $f(S)$ are finite),i.e., an unordered l-tuple $a=$of -classes of multi-germs :may be a family (moduli) of
where $l=|f(S)|$ and $|S|= \sum|S_{j}|$ . Here
codimension of is defined by $s=|a|$-classes of multi-germs. The-codimension).
(codim means
codim$(\alpha_{1}, \cdots , \alpha_{l})$$\ovalbox{\tt\small REJECT}$$\alpha_{j}$$\mathbb{R}^{n},0$$\mathbb{R}^{m},S_{j}arrow$$\alpha_{j}$$\ovalbox{\tt\small REJECT}$$\sum$$:=$$a$$(\ovalbox{\tt\small REJECT}_{e^{-}})$$\ovalbox{\tt\small REJECT}_{e}$$\alpha_{j}$Assume that $|a|=s<\infty$ . We put.
$\Gamma(a):=$ Closurc {. has multi-singularities of typc
$f\in\Gamma_{\iota}\backslash ^{\backslash }$

$f$

$a$

}

$\subset\Gamma_{\iota}\backslash \cdot$.

in , and if is a multi-singularity. The set
is contained in
becomes ain the sense of [17]: In particular, its smooth
pseudo-algebraic subset inpart is a Frechet submanifold of codimension .
.
and
codim
Put

By definition,
type adjacent to , then$\Gamma(a)=\Gamma(\alpha l)\cap\cdots\cap\Gamma(\alpha_{l})$$b$$\mathcal{M}$$a$$\Gamma(a)$$\Gamma(b)$$\Gamma(a)$$\mathcal{M}$$s$$\Gamma_{\infty}$$:=\{f\in \mathcal{M}|$$\ovalbox{\tt\small REJECT}_{M_{\dagger}N}.f=\infty\}$$u_{0}:=\mathcal{M}-\Gamma_{\infty}$$(i.e., f\in\prime u_{0})$Lemma 1.3 codim
if and only if there exists a finite$M$
subset in such that
1 the germ $f$ : $M,Sarrow N,f(S)$ is of -finite codimension;
2 each point $x\in f^{-1}(f(S))-S$ is not critical;
3 For anyfinite ‘ away from , the germ of $f$ at ‘ is stable.
$\ovalbox{\tt\small REJECT}_{M_{I}N}.f<\infty$

$S$

$)$

$\ovalbox{\tt\small REJECT}_{e}$

$)$

$)$

$S$

$S$

$S$

39
Theorem 1.4 (Mather [17])Assume that $(m,n)$ belongs to the nice range. Then$\Gamma_{\infty}$

and there exists a filtration$\mathcal{M}\supset(\Gamma=)\Gamma_{1}\supset\Gamma_{2}\supset\cdots\supset\Gamma_{s}\supset\cdots\supset\Gamma_{\infty}$

has infinite codimension,
codim

,

$\Gamma_{s}=s$-invariant closed pseudo-algebraic subsets such that there is a topologby-orbitwhere each fibre is an
ically locally trival fibmtion :and is afinite dimensional manifold.
$\Gamma_{s}$$\ovalbox{\tt\small REJECT}_{M,N}$$\Gamma_{s}-\Gamma_{s+1}arrow Y_{s}$$\pi_{s}$$\ovalbox{\tt\small REJECT}_{M,N}$$Y_{s}$.
-structurally stable maps up to
Remark 1.5 1)classifies allis at most countable.
also
2 The rank of
3 Each component of corresponds to the moduli space of certain multi-singularities.
becomes an
4 For a multi-singularity type of codimension
is a union ofinvariant Frechet submanifold of codimension in(countably many) such Frechet submanifolds.
may have finite codimension. In this case,
5 lf $(m, n)$ is out of the nice range,.
byTheorem 1.4 holds when replacing$H^{0}(\mathcal{M}-\Gamma)$$H^{0}(\mathcal{M}-\Gamma)$

$)$

$)$

$C^{c}$$\vee 7t_{M,N}^{()}$$\mathscr{K}(\Gamma_{s}-\Gamma_{s+1}))$$($$Y_{s}$$s,$ $\Gamma(a)-\Gamma_{s+1}$

$a$

$)$

$\mathcal{M};\Gamma_{s}-\Gamma_{s+1}$$s$$\Gamma_{\infty}$$)$$\mathcal{M}$

1.2 Contact equivalence for$\prime u:=\mathcal{M}-\Gamma_{\infty}$$\mathcal{M}$be a p-dimensional closed submanifold andprojection. Put
Let$\ovalbox{\tt\small REJECT}_{M_{I}N^{-}}$$B\subset N$$’\kappa_{M,N,B}:=${ $H\in Diff(M\cross N)|H$ preserves

Deflnition 1.6 $f,g:Marrow N$ areョ$H\in’\kappa_{B}$Theorem 1.7$\hat{\Gamma}_{\infty}$$:=\{f\in \mathcal{M}$$q\zeta_{B}$$M\cross B$$p$: $M\cross Narrow M$ theand fibers of

$p$

}.

-equivalent if

s.t. $H(graph(f))=$ graph $(g)$ .

, codim $JC_{B}.f=\infty\}$ has

Moreover, there exists a filtmtion,$\mathcal{M}\supset(\hat{\Gamma}=)\hat{\Gamma}_{1}\supset\overline{\Gamma}_{2}\supset\cdots\supset\overline{\Gamma}_{s}\supset\cdots\supset\overline{\Gamma}_{\infty}$

infinite codimension in .
$\mathcal{M}$codim$\hat{\Gamma}_{s}=s$40-invariant closed pseudo-algebraic subsets such that there is a topologicallywhere each fibre is an -orbit andlocally trtval fibmtion :is a finite dimensional manifold.
b$y^{t}K_{B}$$\hat{\Gamma}_{s}-\hat{\Gamma}_{s+1}arrow Z_{s}$$’\kappa_{B}$$\pi_{s}$iffNotice thatdiffeo-typcs of submanifolds$f\in \mathcal{M}-\hat{\Gamma}$$f$

is transverse to$\gamma-\downarrow(B)$in $M$ .

$B$

. So,

$Z_{s}$$H^{0}(\mathcal{M}-\hat{\Gamma})$classifies

2 Vassiliev complex
2.1 Spectral sequence
Take Mather’s filtration of$\prime u$(Theorem [17]) and its ’dual filtration’: Put$21_{s}:=\mathcal{M}-\Gamma_{s+1}$

then we have invariant open subsets of$\prime v$,:$\mathcal{M}-\Gamma=^{t}\mathcal{U}_{0}\subset u_{1}\subset T1_{2}\subset\cdots\subset u_{s}\subset\cdots\subset\bigcup_{s=0}^{\infty}u_{s}\subset \mathcal{M}$.

We think of a spectral sequence associated to this filtration: the firstterms are$E_{1^{-}}$$E_{1}^{s,t}:=H^{s+\iota}(u_{s},u_{s-1})arrow H^{t}\sigma_{s}-\Gamma_{s+1})$

with a coefficient ring . The arrow indicates the Ale.xander duality. ) frrn
tionalin the sense of Eells [7], that is, the Thom isomorphism for(for acoorientable components of the s-codimensional mamifold
non-coorientable connected component the Thom class within integer coefficients vanishes, but it works within -coefficients). Thusis the, especially,
R-module generalted by coorientable components in$R$$f\dot{r}$

$r$

$-$

$\iota vpc|ces$$\Gamma_{s}-\Gamma_{s+1}$$E_{1}^{s,0}$$\mathbb{Z}_{Q}$$\Gamma_{s}-\Gamma_{s+1}$$E_{l}^{0,0}=H^{0}(u_{0})=H^{0}(\mathcal{M}-\Gamma)$.

The first cochain complex is$0arrow E_{1}^{0,t}arrow E_{1}^{1,t}arrow E_{l}^{2,t}arrow...$:is the connection homomorphism
where the operatorof cohomology exact sequence for the triple $(u_{s+1},u_{s},u_{s-1})$ . As usual, we
put for $r\geq 1$ ,$d_{1}$$E_{1}^{s,t}arrow E_{1}^{+1,\iota}$$E_{r+1}^{s,t}:= \frac{ker[d_{r}:E_{r}^{t}arrow E_{r}^{+r,t-r+1}]}{{\rm Im}[d_{r}:E_{r’}^{\sigma-r,t+r-1}’arrow E_{r}^{s,t}]}$$\partial$

41
and we have a natural homomorphism
Instead, we may takewith$d_{r+1/}$$E_{\infty}^{s,t}arrow H^{*}(\mathcal{M})$.

$0arrow E_{1}^{1,t}arrow E_{1}^{2,t}arrow E_{1}^{3,t}arrow\cdots$

and$E_{r}^{s,t}(s\geq 1)$, that approximates $H^{*}(u,u_{0})=H^{*}(\mathcal{M},\mathcal{M}-\Gamma)$, the coho-

mology with support on F.
2.2 Vassiliev complex
Defineand for$C^{0}(\ovalbox{\tt\small REJECT})=0$$s\geq 1$,$C^{s}(\ovalbox{\tt\small REJECT})=\oplus R\cdot a$

the vector space generated by coorientable ffl-classes of multi-singularities
of codimension (precisely saying, it is defined as an inductive limit ofvector spaces generated by coorientable strata of codimension in someinvariant Wluitney stratifications of multi-jet spaces, cf. [28], [19] . The
is regarded as
coboundary is defined by using versal unfoldings.
a submodule ofby a simple identification
$a$

$s$

$s$

$)$

$\partial$$C^{s}(\ovalbox{\tt\small REJECT})$$E_{1}^{s,0}$,$C^{S}(\ovalbox{\tt\small REJECT})\subset E_{1}^{s,0}$The coboundary$\partial:C^{s}(\ovalbox{\tt\small REJECT})arrow C^{s+1}(\ovalbox{\tt\small REJECT})$

Definition 2.1 The cochain complex
plex. for$\prime n_{-(}\cdot lo^{i\backslash }!J\grave{t}cc\iota tlollO\backslash /ll\iota ltl$

$a\mapsto\Gamma(a)$.

is induced from$(C^{*}(\ovalbox{\tt\small REJECT}),\partial)$$d_{1}$.

is called the 10$al$Vassilie$|’ c\cdot 0\iota-$-singularities.

(ffl) can explicitly be written down as
The operator :. Take a versal deformation of .
andfollows. Letof type is defined:
On the parameter space, the bifurcation diagram
It is either empty or l-dimensional semi-algebraic curves approaching theorigin. Count the incidence oefJicienr $[a;b]$ , defined by the algebraic interwith a small oriented sphere centered at the origin:section number ofThen $\partial a=\Sigma[a;b]b$ , the sum taken over all generators . An example isshown in subsection 2.4 described below.
$\partial$$C^{s}(\ovalbox{\tt\small REJECT})arrow C^{s+1}$$b$$b\in C^{s+1}(\ovalbox{\tt\small REJECT})$$a\in C^{s}(\ovalbox{\tt\small REJECT})$$\Psi(a)$

$a$

$c$

$\Psi(a)$$b$Remark 2.2 Notice that the Vassiliev complex is determined only by the local
classification of singularities. In fact, although there are possibly many connected, they are regarded as just ’one stratum’ in thecomponents in each
Vassiliev complex,
complex. A more finer subcomplex, say an$\Gamma(a)-\Gamma_{s+1}$$e\tau iri^{1}hed$$C(\ovalbox{\tt\small REJECT})$$C_{en}^{\sigma}(\ovalbox{\tt\small REJECT})\subset E_{1}^{\tau,0}$

42
may be obtained by some additional ‘non-local’ data to. Some choices are
to input: the data of configurations of,on at which a multi-signularity
occurs (when $m=1$ , these data are called cord-diagrams or weight systems), the$C^{*}(\ovalbox{\tt\small REJECT})$$S_{1},$$M$$S_{k}$$\cdots$

placement of the singular point locus in $M$ $(if m\geq 2)$ , the topological types of
singular fibers $(if m\geq n)$ , and so on.

2.3 Local invariants for generic maps
There is a natural homomorphism, for$s\geq 1$,$H^{s}(C(ffl))(=ker\partial)arrow E_{2}^{s,0}arrow E_{\infty}^{s.0}arrow H^{s}(\mathcal{M},\mathcal{M}-Darrow H^{s}(\mathcal{M})$In particular, if$H^{1}(\mathcal{M})=0$, we have.

.

$H^{1}(C(\ovalbox{\tt\small REJECT}))arrow H^{0}(\mathcal{M}-\Gamma)$

Deflnition 2.3 A function
is called a 10 invariant for genericmaps if it comes from(cf. Goryunov [6]). lf we take some ennichedcomplex instead, then we say is semi-local or enriched-local.
$v\in H^{0}(\mathcal{M}-\Gamma)$$al$$H^{1}(C(\ovalbox{\tt\small REJECT}))$$v$

Local in.variants should be

Euler characterislic associated to stable
which are determined only by local modifications and initial data, e.g.
’’‘

$ob.ie.\cdot l_{I}\backslash ^{\backslash }$number of individual singular points,, or more generally,
Euler characteristics of singular point sets of several types,
Euler characteristics of images, etc,
index (rotation number), normal Eulcr numbers, Smalc invariants for generic immersions;-total linking number for oriented]inks; Bennequin invariants for criticalvalue sets (a sort of linking numbers)... ...
-$\#L(f\cdot)$$\gamma(\Sigma(f\cdot))$$\lambda^{r}(.f\cdot(M)),\chi(f(\Sigma(f)))$$-Wl\urcorner itncy$

Some related works are listed below:

43
2.4 Examples of local invariants$\bullet$$\underline{(m,n)}.=(1,2)$(Arnold [2])$J^{-},St$ for generic immersed plane curves (Precisely
Basic mvariants
saying, $St$ is not local in the above sense, but semi-local; are local).
$J^{+},$$\Rightarrow$$J^{\pm}$$Cod\frac{(m,n)=(2,4)}{im.0:imme}rsions\bullet$

with transverse double pts ( $=generic$ immersion).
Codim. 1: -singularity, and tangency of two sheets.
Codim. 2: -singularity, and triple points
are of double
Local invariants of generic immersion $f$ :points $d(J)$ and normal Euler number $e(f)$ .
$A_{1}$$A_{2}$$M^{2}arrow \mathbb{R}^{4}$$\Rightarrow$$\bullet$$\#$$(m,n)=(2,3)$ (Goryunov [6])maps with Cross-cap and transverse double/triple pts $(=generic$$Cod\overline{im.0:}$maps).

Codim. 1: There are 12 types
are of triple pts,
Local invariants of generic maps
Cross-caps, and a new invariant (relating inverse self-tangency)$M^{2}arrow \mathbb{R}^{3}$$\Rightarrow$$\bullet$

$\#$

$\#$

of

$(m,n)=(2,2)$ (Ohmoto-Aicardi [20], [18])maps with fold, cusps and double folds $(=generic$ maps .

$Cod\overline{im.0:}$$)$

The apparent contour ( critical value set (discriminant)) of a generic map
looks like$=$$M^{2}arrow \mathbb{R}^{2}$,of (multi-)germs: Here three
Codim. 1: there are 10 types $\alpha=L,B,$
are depicted below (These are coorientable: an
examples named byorientation of parameter is defined as the number of double pts (or cusps)
increases- from the picture on the left to the picture on the right).
$\cdots$$S,$ $B,$ $K_{0}$$C_{1}$

44

of (multi-)germs: The type
Codim. 2: there are 20 types $\beta=I,II,III,A_{1}^{4},$III has the following bifurcation diagram (2-parameter):$\cdots$

Then incidence coefficients $[a;b](b=I\Pi)$ are counted as follows:$[S;III]=2,$ $[B;III]=-2$ .

$[K_{1};III]=-1$.

The same computations for any $a,b$ can be done, and then we have$0arrow C^{1}(\ovalbox{\tt\small REJECT}_{2,2})\simeq \mathbb{Z}^{10}arrow^{\partial}C^{2}(\ovalbox{\tt\small REJECT}_{2,2})\simeq \mathbb{Z}^{20}arrow^{\partial}$$\Rightarrow$rank

and local invariants over$H^{1}(C(\ovalbox{\tt\small REJECT});\mathbb{Z})=3$$($$R\ni\eta 1)$

are generated

modulo constants by$\Delta c=2\Delta I_{1}$:$\Delta d=\Delta I_{2}+\Delta I_{3}$

$\#$

:

$\Delta v=\Delta I_{1}-\Delta_{2}+\Delta I_{3}$$\#$of cusps;of double fOlds;: projective Bennequin invariant.

45. Further, noteThose generates local invariants of generic mapsthat type (beak-to-beak) can be separated into two types according to
how components of contour curves are mutually cormected, that yields ansemi-local invariant, see Hacon-Mendes-Romero Fuster [10]:$M^{2}arrow \mathbb{R}^{2}$$B$$\Delta I_{4}=\Delta l-\Delta b_{1}+\Delta b_{2}$

:

$\#$

of components of critical set $C(f)$ .

Remark 2.4 The projective Bennequin number itself is an interesting invaniant forapparent contours. Although it is not easily computed, there is a nice algorithm ofBellettini et al [4].

3 Finite type invariants of mappings-classification

3.1 Finite type invariants of mappings: GlobalLeta normal slice to$\mathcal{M}=C^{\infty}(M,N)$. For$\Gamma(a)$$a=(\alpha_{1}, \cdots , \alpha_{k})$where all$\ovalbox{\tt\small REJECT}$$\alpha_{j}$are of codimension 1,

is denoted by$\Xi^{a}:[-1,1]^{k}arrow \mathcal{M}$,We define.
commutative ring.

$(t_{1}, \cdots, t_{k})\mapsto\Xi_{t_{1}\cdots t_{k}}^{a}$$[^{\backslash }n\downarrow(t\mathfrak{l}\cdot iaflls$,$\Xi_{0}^{a}\in\Gamma_{k}$in a $mj_{1)}e$ sense as follows. Let

$R$

be a

is an invariant of order . if the following
Deflnition 3.1 A function
having typeequality holds for any $k\geq r+1$ and for any k-parameter family,$a=$with codim$v:\mathcal{M}-\Gammaarrow R$$f$$\Xi^{a}$$\alpha J=1$$(\alpha 1, \cdots , \alpha k)$$(*)$$\sum_{\epsilon}\epsilon_{1}\cdots\epsilon_{k}v(\Xi_{\epsilon_{I}\cdots\epsilon_{k}}^{a})=0$

where the sum is taken over$2^{k}$

combinations of$\epsilon_{i}=\pm 1$.

denote the R-module generated by invariants of order . By
definition, we have a filtration
Let

$r$

$V_{r}$.

$V_{0} \subset V_{1}\subset\cdots\subset V_{\infty};=\bigcup_{r=0}^{\infty}V_{r}\subset H^{0}(\mathcal{M}-\Gamma)$, constant functions over connected components
Obviouslyof . In a natural way,
becomes a graded ring: the multiplication
is defined by $v1\mathcal{V}2(f):=v1(f)v2(f)(f\in \mathcal{M}-\Gamma)$.
$V_{0}=H^{0}(\mathcal{M})$$\mathcal{M}$$V,$ $\cross V_{l}arrow V_{r+l}$$V_{\infty}$

46, Vassiliev constructed a simplicial
Remark 3.2 In case of $M=S^{1}$ and, and introduced a spectral sequence for the resolution
resolution of inspace, which producesof knot invariants (For simplicity, let the coefficient). In our terminology there is some enriched Vassiliev complex soringthat $H^{s}(C_{en}(n))\simeq V_{s}/V_{s-1}$ and ’initial data’ (called an acmality table) gives an
inductive construction of an injective homomorphism
for each$N=\mathbb{R}^{3}$$\Gamma$$\mathcal{M}\cross \mathbb{R}^{\infty}$$\{V_{s}\}_{s=0}^{\infty}$$R=\mathbb{Q}$$H^{s}(C_{en}(\ovalbox{\tt\small REJECT}))arrow V_{s}$$s=0,1,2,$.

Let $(m, n)$ be in the nice range. Let the source }ithout boundary and of dimension $ffl\geq 2$ . Assume that$r1^{\eta}h\iota^{1}0$$\tau 0$$\cdots$

1 $\iota^{1}m3.3$$f$(for example,$H^{1}(\mathcal{M})=0$$N=\mathbb{R}^{;l})$be closed,$\Omega l\iota ifoldM$.

1. If $m+]$ . $<n$ , any finite ) invariants
local invaria $nts$ modulo consta , i.e.
$f\backslash 1/\dagger$$C$$v:\mathcal{M}-\Gammaarrow R$

are polynomials in$nt_{b^{\neg}}$$V_{r}/V_{0}=\oplus_{k-\cdot 1}’$Sy$m^{}$$(V_{1}/V_{()})$,$V_{1}/\mathcal{V}_{0}=H^{1}(C(_{\vee}^{t}fl))$

2. If $m+]\geq n,$
typechange along any $’\gamma()f1-t.)$ stra ta” inlnvaria $ntsrn$ odulo consta $nts$ .
$in_{t)}’arim\iota ts\downarrow:$$.\beta nite$$\mathcal{M}-\Gammaarrow R$ノ$\Gamma$.

, whose valne do not, are }

$y\prime^{\neg}$in local

Theorem says that the above naive definition is somehow irrelevant for
generic maps with source dimension greater than 1. A similar statement as
the claim 1 in Theorem has already been known in particular dimension:$n=2m$ (Kamada [12], Januszkiewicz-Swiatkowski [11]) and $n=2m-1(m\geq$
4 (Ekholm [9]).
, and recall
Sketch of the pmof. Let $a=$
where codim$)$$(\alpha 1, \cdots , \alpha_{s})$$\Gamma(a)=\cap\Gamma(\alpha_{j})$ $:=$$\bullet$$\prime J^{\backslash }/|e${$f\in\Gamma_{s}|f$$bi$self-transverse focns$1\cdots(a)$$\alpha_{j}=1$

has multi-singularities of type

$a$

}.

is iriedncible, that is, any $f,g\in\Gamma(a)-\Gamma_{+l}$can be $|oined$ by some generic pathth im
in $\Gamma(a);\gamma(0)=f,$ $\gamma(1)=g$ .
This is because $m\geq 2$ and $M$ is connected. (Remark that the case of $m=1$
is completely different:
has many irred. components labeled by ’nonlocal’ data called cord-diagrams.)
If $m+1<n,$ meets only self-transverse locifor of codimension 1.
Roughly saying, codimension one invariant cyclesinare determined by the same type coherent system for local invariants, and an
estimate about the dimension of invariants of order leads us to conclude
the assertion stated in the theorem.
$\gamma$$\Gamma(a)$$\bullet$$\gamma$$\Gamma(a)\cap\Gamma(\beta)$$\Sigma a\beta_{i}$$s$$\beta$$\Gamma(a)$47for ofIf $m+1\geq n,$ also meets $nonarrow transverst^{J}$ loci’
codimension 2. For example, in case of $m=n=2$ , there is a self-tangency
locus $\Gamma(B)\cap\Gamma(IV)(a=(B), \beta=(IV))$ :$\Gamma(a)\cap\Gamma(\beta)$$\bullet$$\beta$$\gamma$(red sheet) passes through the beaks point, our invariWhen our curve
ant may change. In order to define ’non trivial higher order’ Vassilievtype invariants, we can not ignore these kinds of jumps of invariants.
$\gamma(t)$$v$

3.2 Finite type invariants for closed n-folds : Global$q\zeta$-classification

Recall the classical Thom-Pontjagin construction. Let $M$ be a compact ori: The classifying map ofented n-manifold, and embed it in
the normal bundle of rank is : $Marrow B$ where $B:=BSO(\ell)$ is the, it is naturally extended to a map
Grassmanian of oriented -planes inpreserving base points to the Thom space $N_{t}:=MSO(f))$ :$\mathbb{R}^{n+\mathcal{E}}(l\gg n)$$\ell$$\rho$$\mathbb{R}^{\infty}$$\ell$$f:S^{n+\mathcal{E}}arrow N_{\ell}$Take$\rho$,

to be generic: $f$ is tmnsverse to$f(p_{0})=\infty\in N_{l}$$B$.

and $M=f^{-1}(B)$ .

48
The cobordism group of oriented n-manifolds is.

$\Omega^{ori}(n)\simeq\pi 0(\lim_{tarrow\infty}C^{\infty}(S^{n+\ell},N_{l})_{base})=\pi 0(\Omega^{\infty}N_{\infty})$

We put$\hat{\mathcal{M}}:=C^{\infty}(.S^{r\iota+t},N_{I})_{l)_{l}1se}$,$B:=BSO(r,)\subset N_{I}$$(\ell\gg 0)$

is the space of closed orientcd n.mfds (see R. Thom, secNote thattion 3 in [27] . It follows from Theorem 1.7 that there is a-invarimt
stratification$\hat{\mathcal{M}}-\hat{\Gamma}$$K_{B}$$)$$\overline{\mathcal{M}}\supset\hat{\Gamma}=\hat{\Gamma}_{l}\supset\hat{\Gamma}_{2}\supset\cdots$

A global version of Maninet’s versality theorem is stated as follows (Kazarbe the evaluation map $e(p,f)$ $:=f(p)$ , andian [15] : Let :denote by the restriction to $e^{-1}(B)$ of the projection to the second factor. Then, we may regard as tlie ”universal
stabl.e map“, and as:
the ”discriminant set of
$)$

$e$

$S^{n+l}\cross\overline{\mathcal{M}}arrow N\ell$$\pi_{B}$$\hat{\mathcal{M}}$$\overline{\Gamma}$$C^{\infty}$$\pi_{I;}$$\pi_{li^{\dagger}’}$$s^{n+\ell_{\cross\overline{\mathcal{M}}arrow N\ell}^{e}}$$arrow$$e^{-1}(.fJ)\downarrow\pi_{IJ}\subset$$Q^{n+p}\downarrow$$P^{p}$$arrow$$\hat{\mathcal{M}}$In fact, any-structural stable maps $Qarrow P$ with $\dim Q-\dim P=n$ canbe obtained as a fiber product of and a smooth map
transverseto (uniquely up to isotopies).
$C^{\infty}$$Parrow\overline{\mathcal{M}}$$\pi_{B}$$\hat{\Gamma}$

be a connected component of . Note that$al/(oml?$ act oriemed n-m anifo
to a fixedLet$\hat{\mathcal{M}}$$\mathcal{M}$$ld\backslash bCon_{:}fl\uparrow g$$H^{()}(\mathcal{M}-\hat{\Gamma}\tilde{)}(..\cdot la_{k}\backslash ^{\backslash }sifi^{}e_{\iota}t^{\backslash }$$(,\cdot obo\prime dis/,\iota(/\subset lS,\uparrow’’$

Remark 3.4
The set of diffeomorphism types of compact n-dimensional$C^{\infty}$.

-manifolds is at

49
most countable (classical, Milnor etc).
(2) The ‘null-cobordant’ component in$\mathcal{M}_{0}=C^{\infty}(S^{n+t},S^{l})_{base}$$\overline{\mathcal{M}}$is,$B=\{0\}\subset S^{t}$,$\ell\gg 0$.

Note $\Omega^{ori}(1)=\Omega^{ori}(2)=\Omega^{ori}(3)=0$. So, in these cases,$(n=1)$ Saeki [23] introduced a cochain complex for topological types of singular-inv. filtration, which is equivalent to a (enriched) Vassiliev complex forof .
$(n=3)$ Sirokova [25] dealt with “the space of closed ori. 3-mfds”.
$\overline{\mathcal{M}}=\mathcal{M}0\cdot$$’\kappa_{B}$$fib\underline{rc^{1}}$$\Gamma$

Remark 3.5
consists of maps $f$ having one Morse singularity
1,
handle surgery: for$)$$\hat{\Gamma}_{1}-\hat{\Gamma}_{2}$$A_{1}$

on $f^{-1}(B)$ , i.e., a$0 \leq k\leq[\frac{n+1}{2}]$.

$A_{1,k}:(x_{1}, \cdots, x_{n+1}, z)\mapsto(-F_{1}-\cdots-x_{k}^{2}+x_{k+1}^{2}+\cdots+x_{n+1}^{2}, z)$.
This is cooricntablc, except for odd,
consists of maps $f$ having either of two Morse singularities, or one
2-singularity ( $=cancelation$ of surgeries) :$k=[ \frac{n+1}{2}]$

$n$

$)$

$\hat{\Gamma}_{2}-\hat{\Gamma}_{3}$-$A_{2}$$A_{2,k}:(x_{1,;}\cdot,x_{n+1},y,z)\mapsto(x_{1}^{3}+yX1\pm x_{2}^{2}\pm\cdots\pm x_{n+1}^{2},y, z)$

3 The ’self-tangential locus’ belongs to$\hat{\Gamma}_{3}$

$)$

.

.

Naive finite type invariants are defined in the same way as before:
Deflnition 3.6 A locally constant functionring) is of order if$v:\mathcal{M}-\hat{\Gamma}arrow R(R$being a commutative$r$$\sum_{\epsilon}\epsilon\cdots\epsilon v_{-\epsilon_{1}\cdots\epsilon_{k}}(^{-A_{1}^{k}})=0$

for any k-tuple self-intersection $(k\geq r+1)$ , i.e., any connected components of$\hat{\Gamma}(A_{1}^{k})-\hat{\Gamma}_{k+1}$.

Theorem 3.7 (Folklore)
1 In case of even, finite type invariants
$)$

$n$

$teristics,\gamma:\mathcal{M}-\hat{\Gamma}arrow \mathbb{Z},$$f\mapsto\chi(f^{-1}(B))$

2 In case of $n(>1)$ odd, finite:
Euler chamcteristics$)$$t_{W}e$generated only by the Euler charac-, modulo constants.

invariants$\mathcal{M}-\hat{\Gamma}arrow \mathbb{Z}_{2},$$\lambda’2$$a\gamma e$$a\gamma e$

generated only by the semi-$f\mapsto\lambda’2(f^{-1}(B))$, modulo constants.

50
Thuis is almost trivial and well-known perhaps. The proof is the same,
, that means
is irreducible for )’as before: lre locus
that the above naive definition allows us to forget any information about$nn$$\Gamma(a)$$\iota$$a=(./t_{1,k_{I}},$ $\cdots$$A_{1.k,)}$glueing maps of a specified surgery of type .
$a$Remark 3.8 In particular, Betti number functions $f\mapsto b_{k}(f^{-1}(B))$ are not finite
type invariants in the $nve$ sense.

Any n-tuple self-intersection of this type has non-zero values.

In order to keep the information of glueing maps of surgeries, we needmore restrictions, i.e., not to be allowed to make other surgeries freely. A
way to make such a restriction is to consider smaller mapping space: For
instance, take an open subset ofBeni numbers, e.g., theof. That is the case as the theory of finite type invariants for
hontology
homology 3-spheres (Ohtsuki [21]) and its generalization (Cochran-Melvin$\mathcal{M}with.r\iota_{\vee}ved$$1\backslash ^{\backslash }pa(:e$$\cdot$$\backslash pl,ere_{\iota}\backslash ^{\backslash }$

[5] .
$)$

Let

$\mathcal{M}_{ZHS}$

be the space of -homology 3-sphere$\mathbb{Z}$$\mathcal{M}_{ZHS}\subset \mathcal{M}=C^{\infty}(S^{3+\ell},S^{\ell})_{base}$.

in MHS corresponds to the Dehn
A codimension 1 stratum ofsurgery along a framed knot with framing $coefficient\pm 1$ .
in
has quite $malty$ connected
The self-transverse locus
components, each of which is labeled by an ’algebraically split’ framed
lin . Further, many components become coorientable.
The picture below dipicts strata adjacent to the stratum labeled by theof theBorromean lin (a component of the triple-point locus). Whuite walls (labeled by the trivial knot) are
discriminant in
non-coorientable; On the other hand, colored walls (labeled by the trefoil, which distinguishes the Poincar\’eknot) form a coorientable cycle in
3-fold from the standard 3-sphere .
$\hat{\Gamma}(A_{1,2})$$I^{rightarrow}(a)-\Gamma_{\backslash ^{\backslash }+1}$$1\vee\uparrow/IJ_{\iota}S$$k$$\hat{\Gamma}(A_{1}A_{1}A_{1})$$k$$\hat{\Gamma}$$\mathcal{M}_{ZHS}$$\mathcal{M}_{ZHS}$$S^{3}$

51$C))\circ_{c_{c}^{\prime’}}$triple pt locus

4 Characteristic classes for fiber bundles
Let $M$ be a compact, connected oriented manifold. We regard the affine
as a representation of the diffeomorphism gmup $G=$
space
Diff $M$.
First, recall the classifying space of the topological group $G=$ DiffM oforientation preserving diffeomorphisms. If is quite high,, we, the space of all embeddings of $M$ in . SendingEmb
may identify the classifying space of with the topological quotient$\mathcal{M}=C^{\infty}(M,\mathbb{R}^{\ell})$$C^{\infty}(M,\mathbb{R}^{\mathcal{E}})-\Gamma=$$n$$parrow\infty$$\mathbb{R}^{t}$$(M,\mathbb{R}^{\mathcal{E}})$

$G$

$B$

Diff $M=$ Emb$(M, \mathbb{R}^{\infty})/$Diff $M$.

. Since $EG$ is highly
Denote it by $BG$ for short and put $EG=$ Emb
connected, the canonical map $EGarrow BG$ gives the universal principal bundle for the group . Let $BM$ $:=(EGxM)/G$, the associated bundle with fibre$M$ , then any smooth fiber bundle $Earrow B$ (paracompact), with fiber $M$ andstructure group $G=DiffM$, can be obtained, up to isomorphisms, from theuniversal bundle $BMarrow BG$ via the classifying map : $Barrow BG$ . Any element of $H^{*}(BG)$ is called a universal G-characteristic class: G-characteristicclasses of $Earrow B$ are defined by their -image in $H^{*}(B)$ .
Now thin of the composition of an embedding of $M$ and a fixed ’profor some small ,jection’ onto$(M,\mathbb{R}^{\infty})$

$G$

$B$

$\rho$$\rho^{*}$$k$$\mathbb{R}^{n}$

$n$

incl

$Marrow$$\mathbb{R}^{\infty}$$\mathbb{R}^{n}$52
Then the map must admit unavoidable (structurally stable) singularities
and by using these data let us try to characterize the topology of $M$, that
was an idea of R. Thom.
to be the associated bundle
So we put $M=C^{\infty}(M,R^{n})$ andwith fibre . Now is a contractible space, hence the Borel cohomology
is isomorphic to $H_{G}^{*}(pt)=H^{*}(BG)$ .
The Vassiliev complex has much meanings in this equivarant setting:),
We then have (under assumption$B\mathcal{M}arrow BG$$\mathcal{M}$$\mathcal{M}$$H_{G}^{*}(\mathcal{M});=H^{*}(B\mathcal{M})$$N=\mathbb{R}^{n}$.

$H^{s}(C(\ovalbox{\tt\small REJECT}))arrow E_{\infty}^{s,0}arrow H_{G}^{s}(\mathcal{M})\simeq H^{*}(BG)$

We denote by $Tp_{c}\in H^{*}(BG)$ the G-characteristic class associated to a cocy. In fact, it holds (Kazarian [14]) thatis written
cle
as a universal polynomial in the relative Novikov-Landweber classes$c=\Sigma\lambda_{i}a_{i}\in C^{s}(ffl)$$Tp_{c}$$\pi_{*}cl^{I}(T_{\pi})=\pi_{*}(cl_{1}^{i_{1}}(T_{\pi})\cdots cl_{k}^{i_{k}}(T_{\pi}))$is the relative tangent bundle of : $Earrow B$ (see below) andmeans Pontrjagin class, Euler class (with rational coefficients) or StiefelWhuitney class (coefficients in ). This means the following: Suppose that
we are given a fiber bundle $\pi:Earrow B$ with fiber $M$ over a manifold andover the total space of the bundle.
a smooth map $f$ :
where

$cl$

$\pi$

$T_{\pi}$$\mathbb{Z}_{0}$$B$$Earrow \mathbb{R}^{n}$$Earrow^{f}\mathbb{R}^{n}$$\approx$$B\mathcal{M}$

$\{$

$B$

$Barrow BG\rho$,
To any multi-singularity type and appropriately generic $f$ :we associate the bifurcation locus $Ba(f)(\subset B)$ , which is a locally closed:
submamifold consisting of points $b\in B$ over which the map
admits the multi-singularity of type at some finite points of .
and a generic $f$, we define
Given a Vassiliev cocylce
in : It is a
to be the geometric cycle
the bifurcation cycle
geometric presentation of the G-characteristic class$Earrow \mathbb{R}^{n}$$a$$f_{b}$$Marrow \mathbb{R}^{n}$

$a$

$c$

$E_{b}\simeq$$E_{b}$$:=\Sigma\lambda_{i}a_{i}\in C^{s}(\ovalbox{\tt\small REJECT})$$B_{c}(f)$$\Sigma\lambda_{i}Ba_{i}(f)$Dual$[\overline{B_{c}(f)}]=\rho^{*}Tp_{c}$$B$.

Here should be reasonably small: For if we take to , cocycles of $BG$.
live inThus interesting problems from this singularity approach would be:
for given classes-Find the precise forms of universal polynomials$\infty$$n$$n$$\mathcal{M}-\Gamma$$Tp_{c}$53,-Find nontrivial relations among those G-characteristic classes,-Find elements inrepresenting torsion parts of G-characteristicclasses (as geometric realization), etc.
$[c]\in H^{*}(C(\mathscr{S}))$$Tp_{c}’s$$H^{*}(C(\ovalbox{\tt\small REJECT}))$

Example 4.1 For example, in case that $M$ is oriented circle$S^{1}$,$H^{*}(BS^{1})=H^{*}(BU(1))=\mathbb{Z}[c_{1}]$where is the first Chem class of complex line bundles. In [13] Kazarian observedthat the class can be realized by some bifurcation locus of functionsormaps
over total space of -bundles (Also, for a classification of singularities of bifurcation loci, he computed the corresponding universal polynomials
in . But if one takes
as the target space, can not be realized by any bifurcation points, i.e., lives in the space of knots (embeddings).
$c_{1}$$Earrow \mathbb{R}$$cl$$Earrow \mathbb{R}^{2}$$E$$S^{1}$$\mathbb{R}^{3}$$c_{1})$$c_{1}$$c_{1}$

Example 4.2 Recall that for an oriented-surface bundle $\pi:Earrow B$ with fibre aclosed oriented surface $M$, the r-th Morita-Miller-Munford classis defined to be the pushforward
whereis the relative tangent bundle
over the total space andis the Euler class. It is obvious that
the MMM classis realized by the -bifurcation locus of generic maps $Earrow$, wheremeans singularities :of dim ker $d\varphi=2$ :$C^{\infty}$$e_{r}(E)\in H^{2r}(B;\mathbb{Z})$$\pi_{*}e(T_{\pi})^{r+1}$$E$$\Sigma^{2}$$e_{r}(E)$$\mathbb{R}^{r+1}(r\geq 1)$$T_{\pi}$$e(T_{\pi})\in H^{2}(E;\mathbb{Z})$$\Sigma^{2}$$\mathbb{R}^{2},0arrow \mathbb{R}^{r+1},0$$\varphi$$[B_{\overline{\Sigma 2(f)}]=\pi_{*}[\overline{\Sigma 2_{(f)]=\pi_{*}e(T_{\pi}\otimes f^{*}\epsilon^{r+1})=\pi_{*}e(T_{\pi})^{\gamma+1}=e_{r}(E)}}}$.

$(i.e., r=1)$ with $\dim B=2,$
In case of generic maps $f$ :consistsof discrete points in , over which there is a point $p\in E_{b}$ such that the germ$E,$of $f$ at is -equivalent to$Earrow \mathbb{R}^{2}$$b$$parrow \mathbb{R}^{2}$$p$$B_{\Sigma^{2}}(f)$$B$$\ovalbox{\tt\small REJECT}$$I_{22}+II_{22}$: $(x^{2}\pm y^{2}+x^{3}+ay, xy+bx)$

where $x,y$ are local coordinates of fibre andare local coordinates of $B(=de-$formation parameters). As another example, there is a work by Saeki-Yamamoto[24] which shows that $e_{1}(E)$ is realized by the codimension 2 bifurcation locus
conespondin to a special topological type of singular fiber of generic functions$f$ :: The singular fiber consists of 3 circle components each two of whichmeet at 2 nodal points.
$a,$ $b$$g$$Earrow \mathbb{R}$

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