Cofibrations in the Category of Frölicher Spaces:
Part I
Brett Dugmore
Cadiz Financial Strategists (Pty) Ltd, Cape Town, South Africa
Email: Brett.Dugmore@cadiz.co.za
Patrice Pungu Ntumba
Department of Mathematics and Applied Mathematics
University of Pretoria
Hatfield 0002, Republic of South Africa
Email: patrice.ntumba@up.ac.za
Abstract
Cofibrations are defined in the category of Frölicher spaces by weak-
ening the analog of the classical definition to enable smooth homotopy
extensions to be more easily constructed, using flattened unit intervals.
We later relate smooth cofibrations to smooth neighborhood deforma-
tion retracts. The notion of smooth neighborhood deformation retract
gives rise to an analogous result that a closed Frölicher subspace A of the
Frölicher space X is a smooth neighborhood deformation retract of X if
and only if the inclusion i : A →֒ X comes from a certain subclass of
cofibrations. As an application we construct the right Puppe sequence.
Subject Classification (2000): 55P05.
Key Words:Frölicher spaces, Flattened unit intervals, Smooth neighborhood de-
formation retracts, Smooth cofibrations, Cofibrations with FCIP, Puppe se-
quence.
1 Preliminaries
The purpose of this section is to survey brielfy the notion of Frölicher spaces.
Frölicher spaces arise naturally in physics, and do generalize the concept of
smooth manifolds. A Frölicher space, or smooth space as initially called by
Frölicher and Kriegl [7], is a triple (X, CX ,FX) consisting of a setX , and subsets
CX ⊆ XR, FX ⊆ RX such that
• FX ◦ CX = {f ◦ c| f ∈ FX , c ∈ CX} ⊆ C∞(R)
• ΦCX := {f : X → R| f ◦ c ∈ C∞(R) for all c ∈ CX} = FX
http://arxiv.org/abs/0704.0342v1
• ΓFX := {c : R → X | f ◦ c ∈ C∞(R)for all f ∈ FX} = CX
Frölicher and Kriegl [7], and Kriegl and Michor [10] are our main reference for
Frölicher spaces. The following terminology will be used in the paper: Given a
Frölicher space (X, CX ,FX), the pair (CX ,FX) is called a smooth structure; the
elements of CX and FX are called smooth curves and smooth functions respec-
tively. The topology assumed for a Frölicher space (X, CX ,FX) throughout the
paper is the initial topology TF induced by the set FX of functions. When there
is no fear of confusion, a Frölicher space (X, CX ,FX) will simply be denoted X .
The most natural Frölicher spaces are the finite dimensional smooth manifolds,
where if X is such a smooth manifold, then CX and FX consist of all smooth
curves R → X and smooth functions X → R. Euclidean finite dimensional
smooth manifolds Rn, when viewed as Frölicher spaces, are called Euclidean
Frölicher spaces. In the sequel, by Rn, n ∈ N, we mean the Frölicher space Rn,
equipped with its usual smooth manifold structure.
A Frölicher space X is called Hausdorff if and only if the smooth real-valued
functions on X are point-separating, i.e. if and only if TF is Hausdorff.
A Frölicher structure (CX ,FX) on a set X is said to be generated by a set
F0 ⊆ RX (resp. C0 ⊆ XR) if CX = ΓF0 and FX = ΦΓF0 (resp. FX = ΦC0
and CX = ΓΦC0 ). Note that different sets F0 ⊆ RX on the same set X may
give rise to a same smooth structure on X . A set mapping ϕ : X → Y between
Frölicher spaces is called a map of Frölicher spaces or just a smooth map if for
each f ∈ FY , the pull back f ◦ϕ ∈ FX . This is equivalent to saying that for each
c ∈ CX , ϕ ◦ c ∈ CY . For Frölicher spaces X and Y , C∞(X,Y ) will denote the
collection of all the smooth maps X → Y . The resulting category of Frölicher
spaces and smooth maps is denoted by FRL.
Some useful facts regarding Frölicher spaces can be gathered in the following
Theorem 1.1 The category FRL is complete (i.e. arbitrary limits exist ), co-
complete (i.e. arbitrary colimits exist), and Cartesian closed.
Given a collection of Frölicher spaces {Xi}i∈I , let X =
i∈I Xi be the set
product of the sets {Xi}i∈I and πi : X → Xi, i ∈ I, denote the projection map
(xi)i∈I 7→ xi. The initial structure on X is generated by the set
{f ◦ πi : f ∈ FXi}.
The ensuing Frölicher space (X,ΓF0, ϕΓF0) is called the product space of the
family {Xi}i∈I . Clearly,
ΓF0 = {c : R → X | if c(t) = (ci(t))i∈I , then ci ∈ CXi for every i ∈ I}.
Now, let
i∈I Xi be the disjoint union of sets {Xi}i∈I , and ιXi : Xi →
i∈I Xi
the inclusion map. Place the smooth final structure on
i∈I Xi corresponding
to the family {ιXi}i∈I . The resulting Frölicher space is called the coproduct of
{Xi}i∈I , and denoted
i∈I Xi, and
Xi = {f :
Xi → R| for each i ∈ I, f |Xi ∈ FXi}
is the collection of smooth functions for the coproduct.
Corollary 1.1 Let X, Y , and Z be Frölicher spaces. Then the following canon-
ical mappings are smooth.
• ev: C∞(X,Y )×X → Y , (f, x) 7→ f(x)
• ins:X → C∞(Y,X × Y ), x 7→ (y 7→ ins(x)(y) = (x, y))
• comp:C∞(Y, Z)× C∞(X,Y ) → C∞(X,Z), (g, f) 7→ g ◦ f
• f∗ : C∞(X,Y ) → C∞(X,Z), f∗(g) = f ◦ g, where f ∈ C∞(Y, Z)
• g∗ : C∞(Z, Y ) → C∞(X,Y ), g∗(f) = f ◦ g, where g ∈ C∞(X,Z).
Given Frölicher spaces X , Y , and Z; in view of the cartesian closedness of
the category FRL, the exponential law
C∞(X × Y, Z) ∼= C∞(X,C∞(Y, Z))
holds. Because FX = C∞(X,R), it follows by cartesian closedness of FRL that
the collection FX can be made into a Frölicher space on its own right.
Finally we would like to show how to construct smooth braking functions,
following Hirsch [8]. Smooth braking functions are tools that are behind most
results in this paper. In [11], it is shown that the function ϕ : R → R given by
ϕ(u) =
0 if u ≤ 0
u if u > 0
is smooth. Substituting x2 for u in the above function, one sees that the function
ψ : R → R, given by
ψ(x) =
0 if x ≤ 0
x2 if u > 0
is smooth. Now, let us construct a smooth function α : R → R with the following
properties. Let 0 ≤ a < b. α(t) should satisfy:
• α(t) = 0 for t ≤ a,
• 0 < α(t) < 1 for a < t < b,
• α is strictly increasing for a < t < b,
• α(t) = 1 for t ≥ b.
Define α : R → [0, 1] by
α(t) =
γ(x)dx
γ(x)dx
where γ(x) = ψ(x− a)ψ(b − x).
In the sequel, the notation αǫ, 0 < ǫ <
, will refer to a smooth braking
function with the following properties
• αǫ(t) = 0 for t ≤ ǫ,
• 0 < αǫ(t) < 1 for ǫ < t < 1− ǫ,
• α strictly increasing for ǫ < t < 1− ǫ,
• αǫ(t) = 1 for 1− ǫ ≤ t.
2 Basic Constructions of Homotopy Theory in
In this section, we define the fundamental notions of homotopy theory in the
category FRL, such as the homotopy relation and the mapping cylinder. We
begin with an overview of our approach to homotopy in FRL, and then discuss
alternate Frölicher structures on the unit interval which are used in this and
subsequent sections.
2.1 Our Approach to Homotopy Theory in FRL
One might begin investigating homotopy theory in FRL by simply following
the homotopy theory of topological spaces, replacing continuous functions with
smooth ones. One can certainly define the notion of a homotopy H : I×X → Y
between smooth maps H(0,−) and H(1,−) in this way (which we do). One can
even get as far as the left Puppe sequence (see [4]), but eventually difficulties
begin to arise.
Extending functions defined on a subspace of a Frölicher space tends to be
a little tricky, and so the definition of a cofibration in FRL is one that needs
careful consideration. We envisage to construct the right Puppe sequence in a
future paper. To do this we define a slightly weaker notion of cofibration than
the notion obtained from topological spaces. In addition, we define the mapping
cylinder of a smooth map f : X → Y using not the unit interval, but a modified
version called the weakly flattened unit interval, denoted I, which, as one
can show, is topologically homeomorphic to the unit interval. This modified
structure on the unit interval allows us to show that the inclusion of a space X
into the mapping cylinder of f : X → Y is a cofibration (in our weaker sense ).
The weakly flattened unit interval is useful, but it also has its drawbacks.
It would be ideal to have a single structure on the unit interval that can be
used throughout out homotopy theory, but the weakly flattened unit interval
is not suitable, because it has the rather restrictive property that a smooth
map f : I → I on the usual unit interval often does not define a smooth map
f : I → I unless the endpoints of the interval are mapped to the endpoints. This
restrictive property means that we only use the flattened unit intervals where
they are absolutely necessary.
In our future work, we will investigate whether with our modified notions of
cofibration and mapping cylinder, Baues’ cofibration axioms are satisfied.
2.2 Flattened Structures on the Unit Interval
We define two main Frölicher structures which we call the flattened unit in-
terval and the weakly flattened unit interval . Let (CI ,FI) be the subspace
structure induced on I by the inclusion I →֒ R.
Definition 2.1 The Frölicher space (I, CI,FI), where the structure (CI,FI) is
the structure generated by the set
F = {f ∈ FI| there exists 0 < ǫ < 14 with f(t) = f(0) for t ∈ [0, ǫ) and
f(t) = f(1) for t ∈ (1− ǫ, 1]},
is called the flattened unit interval.
It is easy to see that any continuous map c : R → [0, 1] defines a structure
curve on I if and only if it is smooth at every point t ∈ R, where c(t) ∈ (0, 1), .
We define the left (resp. right) flattened unit interval, denoted by I−
(resp. I+), to be the Frölicher space whose underlying set is the unit interval
[0, 1], and structure is the structure generated by the structure functions in FI
that are constant near 0 (resp. 1).
Definition 2.2 The Frölicher space (I, CI,FI), with the structure defined below
is called the weakly flattened unit interval. The underlying set is the unit
interval; the structure (CI,FI) is generated by the family
F = {f ∈ FI | lim
f(t) = 0, lim
f(t) = 0, n ≥ 1}.
We call the property, for all f ∈ F ,
f(t) = 0, lim
f(t) = 0, n ≥ 1,
the zero derivative property of f .
We shall prove that all structure functions on I have the zero derivative
property, in other words, FI = F . To that effect, we need the following lemma.
Lemma 2.1 Let c : R → R be a smooth real-valued function at t = t0, and let
f : R → R be a smooth real-valued function at t = c(t0). Then,
(f ◦ c)(t0) = f (n)(c(t0))(c′(t0))n + terms of the form
af (k)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(n−1)(t0))
mn−1 ,
where k < n and a ∈ R. In addition, if a 6= 0 then at least one ofm2,m3, . . . ,mn−1
is also non-zero.
Proof. The proof is done by induction. For the sake of brevity, we call the
term f (n)(c(t0))(c
′(t0))
n the primary term for n, and the terms of the form
af (k)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(n−1)(t0))
mn−1 the lower order terms for n.
The statement is true for n = 1 and for n = 2. Suppose the result is true for
n = k. To show that the result holds for n = k + 1, since
dtk+1
(f ◦ c)(t0) =
(f (k)(c(t0))(c
′(t0))
+terms of the form d
(af (j)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(k−1)(t0))
mk−1),
where j < k + 1 and a ∈ R, we need only show that
(af (j)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(k−1)(t0))
mk−1)
gives rise to lower terms for n = k + 1, which is by the way straightforward. �
Theorem 2.1 FI = {f ∈ FI | limt→0+ d
f(t) = 0 = limt→1−
f(t)} =: F
Proof. That F ⊆ FI is evident. We must show the reverse inequality. Let
0 < ǫ < 1
, and 0 < M < 1. Consider the function cM : R → R, given by
cM (t) = (1− αǫ(|t|))βM (t) + αǫ(|t|),
where αǫ : R → R is a smooth braking function as defined in the Preliminaries,
and βM : R → R is given by
βM (t) =
−Mt if t ≤ 0
t if t > 0
It is easily seen that cM is continuous over all R, and smooth over all R except
at t = 0. Also note that 0 < cM (t) < 1 for all t ∈ R, and cM (t) = βM (t) = 0 for
all 0 ≤ t < ǫ. Now,
cM (t) =
βM (t) = −M, for −ǫ < t < 0
cM (t) =
βM (t) = 1, for 0 < t < ǫ
For n > 1, we have
cM (t) =
βM (t) = 0, for t ∈ (−ǫ, 0) ∪ (0, ǫ).
We now show that for cM ∈ ΓF . To this end, let f ∈ F . To show that
f ◦ cM : R → R is smooth, it is obvious that we need only concentrate on the
point t = 0, because f ◦ c is smooth at every t 6= 0. It follows for t 6= 0, and
n ∈ N that Lemma 2.1 applies. But as t → 0, cM (t) → 0+, and so, letting
s = cM (t), we have
f (j)(cM (t)) = lim
f (j)(s) = 0,
for all j ∈ N, by the zero derivative property of f . Thus, as t approaches
the value 0, the primary term and all the lower order terms of d
(f ◦ cM )(t)
vanish, and we have shown that f ◦ cM is smooth at t = 0. This implies that
f ◦ cM ∈ C∞(R,R) for all f ∈ F . It follows that cM ∈ ΓF .
We are now ready to show that FI ⊆ F . To this end, suppose that we
are given a structure function f ∈ FI. We shall show that this f has the zero
derivative property, and is thus an element of F .
Since f ∈ FI, we know that f ◦ c is a smooth real-valued function for every
c ∈ ΓF . In particular, f ◦ cM is smooth for all 0 < M < 1. Thus, for any n ∈ N,
(f ◦ cM )(t) = lim
(f ◦ cM )(t).
As t→ 0−, cM (t) → 0+; let us consider the lower order terms for n. Each term
of the form
af (k)(cM (t))(c
M (t))
m1(c′′M (t))
m2 . . . (c
(n−1)
M (t))
has some term (c
(t))mi , for some i > 1, with mi 6= 0. But limt→0− c
(t) = 0,
if i > 1, and so
af (k)(cM (t))(c
M (t))
m1(c′′M (t))
m2 . . . (c
(n−1)
(t))mn−1 = 0.
So all the lower order terms fall away, therefore
limt→0−
(f ◦ cM )(t) = limt→0− f (n)(cM (t))(c′M (t))n
= limt→0− f
(n)(cM (t))(−M)n
= lims→0+ f
(n)(s)(−M)n,
where s = cM (t). In a similar way one shows that
(f ◦ cM )(t) = lim
f (n)(s).
But f◦cM is smooth, therefore lims→0+ f (n)(s)(−M)n = lims→0+ f (n)(s), which
implies that lims→0+ f
(n)(s) = 0.
We have shown that the zero derivative property of f holds for the left
endpoint of the unit interval. To show that the zero derivative property of f
holds for the right endpoint of f , note that dM : R → R, dM (t) = 1− cM (t), is
a smooth real-valued function with d(0) = 1, and 0 ≤ dM (t) ≤ 1 for all t ∈ R.
One can follow a similar procedure to the above, using dM instead of cM to
show that lims→1− f
(n) = 0. �
2.3 Some Properties of Smooth Functions between the
Flattened Unit Intervals
One has to be careful when dealing with the various flattened unit intervals. A
smooth function f : I → I from the R- Frölicher subspace unit interval I to
itself need not define a smooth function f : I → I, for example. Conversely,
not every smooth function f : I → I defines a smooth function f : I → I. In
particular, we need to be aware of the fact that addition and multiplication of
functions when defined between the various flattened unit intervals does not
preserve smoothness, as is the case with the usual unit interval.
Example 2.1
The function f : I → I, f(t) = 1
t is clearly smooth, but the corresponding
function f : I → I, given by the same formula, is not smooth. To see this, let
α : R → R be a smooth braking function with the properties that
• α(t) = −1, for t < − 3
• α(t) = t, for − 1
< t < 1
• α(t) = 1, for t > 3
Define c : R → I by c(t) = 1− |α(t)|. The curve c is smooth everywhere except
at t = 0, where c(0) = 1. However, every generating function f on I is constant
near 1, and so the composite f ◦ c is smooth. Thus c is a structure curve on I.
Now, f ◦ c : R → I is given by (f ◦ c)(t) = 1
(1 − |α(t)|). Let h : I → R be a
structure function with the properties that
• h(s) = 0, for s < 1
• h(s) = s, for 1
< s < 3
• h(s) = 1, for 7
Then (h ◦ f ◦ c)(t) = 1
(1− |α(t)|) for t near 0, and is not smooth at t = 0. Thus
f does not define a smooth function from I to I.
Example 2.2
The function f : I → I, f(t) =
t, is smooth, but the corresponding f : I → I,
given by the same formula, is not smooth. This follows from the fact that f is
smooth on the open interval (0, 1), and a generating function g on I is constant
near 0 and 1. On the side, f : I → I is not smooth, because if c : R → I is a
structure curve with c(t) = t2 near t = 0, then (f ◦ c)(t) = |t| near t = 0, which
is not smooth on I at t = 0.
Example 2.3
The functions f, g : I− → I−, given by f(t) = 1
t and g(t) = 1
are both
smooth, but the sum f(t) + g(t) = 1
is not smooth.
The following lemma follows from the definition of the Frölicher structures
on the various flattened unit intervals.
Lemma 2.2 Let f : I → I be a smooth function with the properties that f(0) =
0 and f(1) = 1. Then the following maps are smooth:
• f : I → I±,
• f : I → I,
• f : I± → I,
• f : I → I,
• f : I → I.
The function defined in the following example is for later reference.
Example 2.4
Let H : I × I− → I− be given by H(t, s) = (1 − α(t))s, where α : R → R is a
smooth braking function with the properties that
• α(t) = 0 for t < 1
• 0 ≤ α(t) ≤ 1 for all t ∈ R,
• α(t) = 1 for t > 3
We show that H is smooth. To see this, let f : I− → R be a generating function
on I−. So f is constant near 0. Now, let c : R → I × I− be a structure curve,
given by c(v) = (t(v), s(v)). The curve t is a structure curve on I, and so is
a smooth real-valued function for all v ∈ R, except possibly when t(v) = 0 or
t(v) = 1. Similarly, the curve s is a structure curve on I−, and so is smooth
for all v ∈ R except possibly when s(v) = 0. Now consider the composite
H ◦ c : R → I−. Clearly, α(t(v)) is smooth for all v, since the only possible
points for non-smoothness occur when t(v) = 0 or t(v) = 1, and α(t(v)) is
locally constant near these points. Consequently, H ◦ c is smooth everywhere
except possibly when s(v) = 0. Now, let’s consider f ◦H ◦ c : R → R; the only
possible points for non-smoothness are those in which s is 0, i.e. H◦ = 0. But f
is a structure generating function on I−, and so is locally constant near 0. This
shows that f ◦H ◦ c is smooth for all v ∈ R, and thus H is smooth.
2.4 Homotopy in FRL and Related Objects
Definition 2.3 (1) Let X be a Frölicher space, and x0, x1 ∈ X. We say that
x0 is smoothly path-connected to x1 if there is a smooth path c : I → X such
that c(0) = x0 and c(1) = x1. We write x0 ≃ x1. The relation ≃ is called
smooth homotopy when it is applied to hom-sets.
(2) Let f : X → Y be a map of Frölicher spaces. f is called a smooth
homotopy equivalence provided there exists a smooth map g : Y → X such that
f ◦ g ≃ 1Y and g ◦ f ≃ 1X .
One can show that smooth homotopy is a congruence in RFL. In practice, we
say that smooth maps f, g : X → Y are smoothly homotopic if there exists a
smooth map H : I ×X → Y with H(0,−) = f and H(1,−) = g. If A ⊆ X is
subspace of X , then we say that H is a smooth homotopy (rel A) if the map
H has the additional property that H(t, a) = a for each t ∈ I and a ∈ A. See
Cherenack [5] and Dugmore [6] for more detail regarding smooth homotopy.
The notion of deformation retract is fundamental to topological homotopy
theory. The following definitions are adapted for smooth homotopy, and will be
needed at a later stage.
Definition 2.4 Let A ⊆ X be a subspace of a Frölicher space X, and let i :
A →֒ X denote the inclusion map. Then
• We say that A is a retract of X if there exists a smooth map r : X → A
such that ri = 1A. We call r a retraction.
• We call A a weak deformation retract of X if the inclusion i is a smooth
homotopy equivalence.
• The subspace A is called a deformation retract of X if there exists a re-
traction r : X → A such that ir ≃ 1X .
• The subspace A is called a strong deformation retract of X if there exists
a retraction r : X → A such that ir ≃ 1X(relA).
Definition 2.5 The mapping cylinder If of f : X → Y is defined by the fol-
lowing pushout
I ×X // If
where i1 : X → I ×X is given by i1(x) = (1, x), for any x ∈ X. We denote the
elements of If by [t, x] or [y], where (t, x) ∈ I ×X and y ∈ Y .
Replacing I ×X in the above pushout diagram by I×X or I×X, we obtain
the flattened mapping cylinder If and weakly flattened mapping cylinder If of f
respectively. We use the same notation for elements of these flattened mapping
cylinders as described above for the mapping cylinder.
There is also a map i0 : X → I ×X , defined by i0(x) = (0, x) for x ∈ X . This
induces an inclusion map i′0 : X → If , which identifies X with the Frölicher
subspace i′0(X) of If . An inclusion is induced in a similar way for the flattened
mapping cylinders. If one identifies {0}×X to a point in the mapping cylinder
If of a map f : X → Y , then one obtains the mapping cone Tf of the
map f . In a similar fashion, we define the flattened mapping cone Tf and
weakly flattened mapping cone Tf of a smooth map f : X → Y .
2.5 Cofibrations in FRL
A cofibration is a map i : A→ X for which the problem of extending functions
from i(A) to X is a homotopy problem. In other words, if a map f : i(A) → Z
can be extended to a map f∗ : X → Z, then so can any map homotopic to f . For
topological spaces, the usual definition is phrased in a slightly more restrictive
way. The extension of a map g ≃H f , for some homotopy H : I × i(A) → Z, is
required to exist at every level of the homotopy simultaneously. In other words,
one requires each H(t,−) to be extendable in such a way that the resulting
homotopy H∗ : I ×X → Z is continuous.
We weaken this definition somewhat, to enable smooth homotopy extensions
to be more easily constructed using a flattening at the endpoints of the homo-
topy. This enables us to characterize smooth cofibrations in terms of a flattened
unit interval, and then later to relate smooth cofibrations to smooth neigh-
borhood deformation retracts. Our definition of smooth cofibration, though
different from from Cap’s definition, see [1], leads to several classical results as
does Cap’s. As pointed out by Cap, the analogue of the classical definition of
cofibration would not allow even {0} →֒ I to be a smooth cofibration. So, we
have the following
Definition 2.6 A smooth map i : A → X is called a smooth cofibration if,
corresponding to to every commutative diagram of the form
(0,1A)
f // Z
66mmmmmmmmmmmmmm
there exists a commutative diagram in FRL of the form
(0,1X )
::tttttttttt
where G′ : I × A → Z is given by G′(t, a) = G(αǫ(t), a) for some 0 < ǫ < 12 ,
and each t ∈ I, a ∈ A.
The problem of extending a map smoothly from a subspace of a Frölicher
space to the whole space is a more difficult problem than simply extending con-
tinuously. It is mainly for this reason that the definition of smooth cofibration
differs somewhat from the corresponding definition of a topological cofibration.
Lemma 2.3 Let i : A → X be a smooth cofibration, then i is an initial mor-
phism in FRL. In addition, if A is Hausdorff, then i is injective. So in this
case A can be regarded as a subspace of X.
Proof. Let us show that every smooth map f : A→ R factors through i, that
is for every f ∈ FA, there exists f̃ ∈ FX such that f = f̃ ◦ i. To this end,
consider the smooth map G : I × A → R, given by H(t, a) = tf(a). Clearly,
0|A = G(0,−), where 0 : X → R is the constant map 0. It follows that there is
map F : I ×X → R such that F ◦ (1× i) = G′. Then, clearly f̃ := F (1,−) has
the desired property.
The remaining part of the proof of Proposition 3.3, in [1], holds verbatim
here as well. �
In this paper, we are interested only in cofibrations that are injective. Hence-
forth, all cofibrations are assumed to be injective.
All topological cofibrations are inclusions, and this result is true for smooth
cofibrations too. The proof of the following lemma is essentially the same as
the proof given by James [9] for the topological result, although James’s proof
is in some sense dual to ours, using path-spaces in place of cartesian products
and the adjoint versions of our homotopies.
Lemma 2.4 A cofibration
i // X
is a smooth inclusion.
Proof. Let Ii be a mapping cylinder of i, and let j : X → Ii be the standard
inclusion map. Consider the smooth map γ : I → I, γ(t) = 1 − t, for all t ∈ I,
and the quotient map q : (I ×A) ⊔X → Ii; we have the following commutative
diagram
(0,1A)
j // Ii
66mmmmmmmmmmmmmm
where G(t, a) = [(1 − t, a)]. Notice that the map G is smooth. Since i is a
cofibration, we have the commutative diagram
(0,1X )
::uuuuuuuuuu
where G′(t, a) = G(αǫ(t), a) for some 0 < ǫ <
. Define U : X → Ii by
U(x) = F (1, x). We have U ◦ i = G′(1,−), where G′(1, a) = [(0, a)], for every
a ∈ A. Thus the assignment a 7→ G′(1, a) defines the usual inclusion of A into
the mapping cylinder. From this we deduce that U ◦ i is an inclusion, and hence
i is an inclusion. �
There is an equivalent formulation of definition 2.6, given in the following
lemma.
Lemma 2.5 A smooth map
i // X
is a cofibration if and only if, for every smooth map h : (0×X)∪(I−×i(A)) → Z,
the following diagram
(0×X) ∪ (I− × i(A)) h //
I− ×X
77oooooooooooooo
where j is the evident inclusion, exists in FRL.
Proof. Suppose that the inclusion A //
i // X is a smooth cofibration, and
suppose that h : (0 × X) ∪ (I− × i(A)) → Z is a smooth map. We have the
diagram
(0×B) ∪ (I− × i(A)) h //
I− ×X
We need to fill in a smooth map G : I− × X → Z which makes the resulting
diagram commute. To do this, notice that h|I−×i(A) is smooth, and thus the
corresponding map h|I × i(A), using the usual unit interval, is also smooth. We
have the following diagram
(0,1A)
h|0×X // Z
66mmmmmmmmmmmmmm
where h|0×X(0,−) : X → Z is denoted as h|0×X . The fact that i is a smooth
cofibration yields the following FRL-commutative diagram:
h0×X //
(0,1A)
::tttttttttt
where (h|I−×A)′(t, a) = h|I−×A(αǫ(t), a), for some 0 < ǫ < 12 . Now, chose a
smooth braking function β : R → R with the following properties.
• α(t) = 0 for t < ǫ
• α(t) = t for ǫ < t.
F may not be smooth on I− × A due to the flattening requirements of the left
flattened unit interval. To correct this, set G(t, a) = F (β(t), a). Notice that the
insertion of this braking function does not affect the commutativity conditions
of G, since the only adjustments to F occur in the first coordinate where the
map (h|I−×X)′ is constant.
Now, assume the converse, i.e. to every smooth map h : (0 × X) ∪ (I− ×
i(A)) → Z, corresponds a commutative diagram
(0×X) ∪ (I− × i(A)) h //
I− ×X
77oooooooooooooo
We wish to show that the inclusion i : A → X is a cofibration; so assume we
have the following diagram
(0,1A)
f // Z
66mmmmmmmmmmmmmm
There exists the diagram
(0,1A)
f // Z
66mmmmmmmmmmmmmm
where G′(t, a) = G(αǫ(t), a). Our hypothesis allows us to construct the diagram
(0×X) ∪ (I− × i(A))
f∪G′ //
I− ×X
77oooooooooooooo
Note that f ∪ G′ is smooth since αǫ(t) is constant near 0. Since H is smooth
on I− ×X it defines a smooth map on I ×X . One can verify that the diagram
(0,1X)
::tttttttttt
commutes as required. �
3 Smooth Neighborhood Deformation Retracts
This section is concerned with the formulation of a suitable notion of smooth
neighborhood deformation retract. For topological spaces, the statement that a
closed subspace A of X is a neighborhood deformation retract of X is equivalent
to the statement that the inclusion i : A →֒ X is a closed cofibration. We show
that in the category of Frölicher spaces there is a notion of smooth neighborhood
deformation retract that gives rise to an analogous result that a closed Frölicher
subspace A of the Frölicher space X is a smooth neighborhood deformation
retract of X if and only if the inclusion i : A →֒ X comes from a certain subclass
of cofibrations. As an application, we construct the right Puppe sequence.
3.1 SNDR pairs and SDR pairs
The definition of ‘smooth neighborhood deformation retract’ that we adopt in
this paper is similar to the definition of ‘R-SNDR pair’suggested in [6], but we
have modified the definition in order to retain only the essential aspects of ‘first
coordinate independence’ defined in [6].
We begin by defining the ‘first coordinate independence property’ of a func-
tion on a product of a Frölicher space with I (or I−, I+).
Definition 3.1 Let i : A → X be a smooth map, and c : R → X a structure
curve on X. Define
Λ(c, i) = {t∗ ∈ c−1(i(A))| there exists a sequence {tn} of real numbers
with limn→∞ tn = t∗ and each tn ∈ c−1(X − i(A))}.
The points in Λ(c, i) are those values in R where the curve ‘enters’ i(A) from
X − i(A), or ‘touches’ a point in i(A) whilst remaining in X − i(A) nearby.
Now, we are ready to define the ‘first coordinate independence property’ for a
structure function on a product.
Definition 3.2 Let i : A→ X be a smooth map and suppose f : I×X → R is
a structure function on I ×X. Let c : R → I ×X, given by c(s) = (t(s), x(s))
have the following properties
• The map x(s) is a structure curve on X.
• For all ǫ > 0, t(s) is a smooth real-valued function on R−∪s∗∈Λ(x,i)[s∗ −
ǫ, s∗ + ǫ].
If, for every such map c, the composite f ◦ c is a smooth real-valued function,
then we say that f : I×X → R has the first independence property (FCIP) with
respect to i.
Extending the definition, we say that a map g : I × X → Y has the FCIP
with respect to i if the composite h ◦ g : I ×X → R has the FCIP with respect
to i for every h ∈ FY .
Notice that we can formulate a similar definition of the FCIP if we replace
I throughout by I− or I+, leaving the rest of the definition unchanged. We will
have occasion to use this type of first coordinate independence property in the
later part of this work.
Note. Let i : A→ X , and suppose that we are given a map g : I×X → Y . Let
f : Y → R be a structure function on Y , and suppose that f ◦ g : I ×X → R
has the FCIP with respect to i for any such f . Then, given a smooth map
h : Y → Z, the composite f ′ ◦ h ◦ g : I×X → R has the FCIP with respect to
i for any structure function f ′ on Z.
The above note applies equally well if g : I− ×X → Y or g : I+ ×X → Y
has the FCIP with respect to i when composed with a smooth function h on Y .
Example 3.1
1. For any i : A→ X , the projection onto the second coordinate πX : I×X → X
has the FCIP.
2. Let α : R → R be a smooth braking function with the properties that
• α(t) = 0 if t < 1
• 0 < α(t) < 1 if 1
≤ t ≤ 3
• α(t) = 1 if 3
Consider 0 →֒ I−. Let H : I× I− → I− be given by H(t, s) = (1−α(t))s. Then,
f ◦H : I× I− → R has the FCIP with respect to the inclusion 0 →֒ I−, for any
f ∈ FI− .
Definition 3.3 Consider a smooth inclusion i : A →֒ X. Suppose that there
exists a smooth map u : X → I, with u−1(0) = i(A). If there exists a smooth
map H : I×X → X that satisfies the following properties:
• H has the FCIP with respect to i.
• H(0, x) = x for all x ∈ X.
• H(t, x) = x for all (t, x) ∈ I× i(A).
• H(1, x) ∈ i(A) for all x ∈ X with u(x) < 1,
then the pair (X,A) is called a smooth neighborhood deformation retract pair,
or SNDR pair for short.
If, in addition, H is such that H(1 × X) ⊂ i(A), then the pair (X,A) is
called a smooth deformation retract pair, or an SDR pair for short.
The subspace A is called a smooth neighborhood deformation retract or smooth
deformation retract of X if (X,A) is an SNDR pair or SDR pair, respectively.
The pair (u,H) is called a representation for the SNDR (or SDR) pair.
Example 3.2
1. The pair (X, ∅) is an SNDR pair. A representation is u(x) = 1, H(t, x) = x,
for each t ∈ I and x ∈ X .
2. The pair (X,X) is an SNDR pair. A representation is u(X) = 0, H(t, x) = x,
for each t ∈ I and x ∈ X .
Lemma 3.1 The pair (I−, 0) is an SDR pair.
Proof. Let α : R → R be the smooth braking function of Examples 3.1. A
representation for (I−, 0) as an SDR pair is (u,H), where u : I− → I and
H : I× I− → I− are given by u(s) = s, and H(t, s) = (1 − α(t))s. Clearly, the
identity u : I− → I is smooth. And the map H , as shown in Example 2.4, is
smooth and clearly has the FCIP with respect to the inclusion, since whenever
v approaches a value for which s(v) = 0, one has
g((1− α(t(v)))s(v)) = g(0)
for v in a neighborhood of this value and g ∈ FI− . �
Lemma 3.2 The pair (I, {0, 1}) is an SNDR pair.
Proof. A representation (u,H) for the SNDR pair can be given as follows.
Define u : I → I to be a bump function such that
• u(t) = 0 for t = 0 or t = 1,
• u(t) = 1 for t ∈ [ 1
• 0 < u(t) < 1 otherwise,
and let β : I → I be a braking function with the properties that β(s) = 0 for
0 ≤ s ≤ 1
, and β(s) = 1 for 3
≤ s ≤ 1. Let 0 < ǫ 1
, and define H : I× I → I by
H(t, s) = (1− αǫ(t))s+ αǫ(t)β(s). It is clear that H(0, s) = s, H(t, 0) = 0, and
H(t, 1) = 1. Suppose that u(s) < 1. Then, s ∈ [0, 1
) ∪ (3
, 1]. This implies that
β(s) = 0 or β(s) = 1. We then have H(1, s) = 0 or H(1, s) = 1, which means
that H(1, s) ∈ {0, 1} if u(s) < 1.
To see that H is smooth, let f : I → R be a generating function for the
flattened unit interval. The only possible points of non-smoothness are points
where t = 0, 1 and s = 0, 1. The braking function αǫ ensures that H is locally
constant in the tb variable whenever t is near 0 or 1, so no problem arises from
the t component. When s is near s = 0, we have H(t, s) near 0, and so the
generating function f is locally constant. Similarly, when s is near s = 1, we
have H(t, s) near 1, and the generating function f is again locally constant. �
We now show that the product of SNDR pairs is again an SNDR pair.
Theorem 3.1 Let i : A →֒ X and j : B →֒ Y be inclusion mappings. If (X,A)
and (Y,B) are SNDR pairs, then so is
(X × Y, (X ×B) ∪ (A× Y )).
If one of (X,A) or (Y,B) is an SDR pair, then so is the pair
(X × Y, (X ×B) ∪ (A× Y )).
Proof. Let α : R → I be a smooth braking function with the properties that
α(t) = 0 for t ≤ 1
, and α(t) = 1 for t ≥ 3
, and let β : R → R be a smooth
increasing braking function with the properties that β(t) = t for t ≤ 1
, and
β(t) = 1 for t ≥ 3
. Suppose that (u,H) and (v, J) are representations for the
SNDR pairs (X,A) and (Y,B), respectively. Let u : X → I, and v : Y → I be
given by u(x) = β(u(x)) and v(y) = β(v(y)) respectively. Define w : X×Y → I
by w(x, y) = u(x)v(y). The braking function β ensures smoothness of u and
v, and consequently of w. We have w−1(0) = (X × B) ∪ (A × Y ), as required.
Define Q : I×X × Y → X × Y as follows .
Q(t, x, y) =
(H(α(t), x), J(α(t), y)) if u(x) = v(y) = 0
(H(α(t), x), J(α(
)α(t), y)) if v(y) ≥ u(x), v(y) > 0,
(H(α(
)α(t), x), J(α(t), y)) if u(x) ≥ v(y), u(x) > 0.
We must show that Q is a smooth map, with the first coordinate independence
property with respect to the inclusion (X × B) ∪ (A × Y ) →֒ X × Y . We first
consider each part of the definition of Q separately. The first part is clearly
smooth. Let us verify that Q is smooth on the second part of its definition; the
third part is similar.
We need only focus on the component J(α(
)α(t), y). Each function
making up J(α(
)α(t), y) is smooth individually, so we need only pay extra
attention to those parts that involve flattened unit intervals, remembering that
addition and multiplication on the flattened unit interval need not preserve
smoothness, as is the case for the usual unit interval.
So let us consider α(
); it is smooth except possibly when
approaches
0 or 1, since it is here that structure curves on the flattened unit interval need
not be smooth in the usual sense. Clearly, if u(x) approaches 0 and v(y) does
not approach 0, then the braking function α ensures that
= 0 near such
points. If v(y) approaches 0, then u(x) must approach 0 too. This situation is
dealt with later.
Thus, Q, in part two of the definition, is smooth, and one can show similarly
that Q in the third part of the definition is smooth as well.
Let us now consider the overlaps of the three parts of the definition of Q.
Observe that if u(x) is in a sufficiently small neighborhood of v(y), with u(x) 6= 0
and v(y) 6= 0, then we have α(u(x)
) = 1, and so the second and third
parts of the definition of Q coincide here. Thus, it remains only to show that Q
is smooth as u(x) and v(y) both approach 0.
If Q is smooth in each of its coordinates then it is smooth, so consider the
coordinate involving the map J . Let c : R → I×X × Y be a structure that is
given by c(s) = (t(s), x(s), y(s)). Then, the map c1 : R → I× Y , given by
c1(s) =
(α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0
u(x(s))
v(y(s))
)α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0
(α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0
is a map satisfying the conditions of Definition 3.2, since its second coordinate is
smooth, but its first coordinate may be singular as v(y(s)) ( and hence u(x(s)))
approaches 0. Since J has the first coordinate independence property, the map
(Joc1)(s) =
J(α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0
u(x(s))
v(y(s))
)α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0
J(α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0
is smooth. Thus, Q ◦ c is smooth, and since c is arbitrary, Q is smooth. In a
similar way, the coordinate of Q involving H can be shown to be smooth.
We now verify that Q satisfies the required boundary conditions. When t =
0, all three lines defining Q reduce to (H(0, x), J(0, y)) = (x, y). Let x ∈ A and
y ∈ B; then u(x) = v(y) = 0. Therefore, Q reduces to (H(α(t), x), J(α(t), y)) =
(x, y). If x ∈ A and y /∈ B, then Q is given by the second part of its definition,
which reduces to (H(α(t), x), J(0, y)). The case when x /∈ A and y ∈ B is
similar. If t = 1 and 0 < w(x, y) < 1 then either 0 < u(x) < 1 or 0 < v(y) < 1.
Suppose that 0 < u(x) < 1. Then either u(x) ≤ v(y) or v(y) < u(x). If
u(x) ≤ v(y), then Q is given by the second part of its definition, which reduces
to (H(1, x), J(α(
, y)) ∈ i(A)× Y . If v(y) < u(x), then the third part of the
definition of Q applies and Q reduces to (H(α(
), x), J(1, y)) ∈ X × j(B).
Finally, we must show that for any f ∈ FX×Y , f ◦Q has the first coordinate
independence property with respect to the inclusion (X×B)∪(A×Y ) →֒ X×Y .
To this end, consider a map c : R → I×X×Y , given by c(s) = (t(s), x(s), y(s)).
Let {sn} be a sequence of real numbers converging to s∗ with c(sn) ∈ (X×Y )−
((A× Y ) ∪ (X ×B)), and c(s∗) ∈ (A× Y ) ∪ (X ×B). There are three cases to
consider.
• Suppose that c(s∗) ∈ A×B. Then x(s∗) ∈ A and y(s∗) ∈ B. The fact that
H and J have the first coordinate independence property with respect to
i and j respectively means that each coordinate of Q is smooth, and so Q
is smooth.
• Suppose that c(s∗) ∈ A × Y , and that y(s∗) /∈ B. Then at each of the
points c(sn), (Q ◦ c)(sn) is given by the second part of the definition of
Q, for n large enough. Since x(s∗) ∈ A, the component of Q involving H
is smooth, since H has the first coordinate independence property. For
any s in a neighborhood of s∗, α(
u(x(s))
v(y(s))
) = 0. Thus, the component of
Q involving J is constant for s in a neighborhood of s∗, and so is smooth
there.
• The case with c(s∗) ∈ X ×B, and x(s∗) /∈ A is similar to the second case
above.
For the last part of the theorem, suppose that (u,H) represent (X,A) as an
SDR pair. If we replace u by u′ = 1
u, then (u′, H) also represent (X,A) as an
SDR pair. Making the above constructions now with u′ in place of u, it follows
that w(x, y) < 1 for all (x, y) and so Q(1, x, y) ∈ (X × B) ∪ (A × Y ). This
completes the proof. �
4 Cofibrations
In this section, we show that for a subspace A ⊆ X that is closed in the under-
lying topology, the inclusion i : A → X is a cofibration if and only if (X,A) is
an SNDR pair.
Definition 4.1 Let i : A→ X be a cofibration. We call i a cofibration with
FCIP if any homotopy extension can be chosen to have the FCIP with respect
to i.
Using the equivalent formulation of the notion of cofibration, given by Lemma
2.5, we may restate Definition 4.1 as follows: A cofibration i : A → X is a
cofibration with the FCIP if and only if the map G that we may fill in to
complete the commutative diagram
(0×X) ∪ (I− ×A) h //
I− ×X
may be chosen to have the FCIP with respect to the inclusion i.
We have the following result, which corresponds to a similar topological
result.
Lemma 4.1 A smooth map i : A → X is a cofibration (with the FCIP) if
and only if (0 × X) ∪ (I− × A) is a retract of I− × X, (where the retraction
r : I− ×X → (0×X) ∪ (I− ×A) has the FCIP ).
Proof. In the one direction, suppose that (0 × X) ∪ (I− × A) is a retract of
I− ×X . We wish to complete the following diagram:
(0×X) ∪ (I− ×A) h //
I− ×X
By hypothesis, there exists r : I−×X → (0×X)∪ (I− ×A) such that r ◦ j = 1.
Define G = h ◦ r. If r has the FCIP, then so does h ◦ r.
Conversely, suppose that i : A → X is a cofibration (with the FCIP). We
may find a map r such that the diagram
(0×X) ∪ (I− ×A) 1//
(0 ×X) ∪ (I− ×A)
I− ×X
commutes. Thus, r ◦ j = 1. If i is cofibration with the FCIP with respect to i,
then r can be chosen to have the FCIP. �
The next theorem shows the relationship between cofibrations, retracts and
SNDR pairs.
Theorem 4.1 Let i : A → X be an inclusion, with A closed in the underlying
topology of X. Then the following are equivalent.
(1) The pair (X,A) is an SNDR pair.
(2) There is a smooth retraction r : I− × X → (0 ×X) ∪ (I− × A) with the
FCIP.
(3) The map i : A→ X is a cofibration with the FCIP.
Proof. To show that (1) and (2) are equivalent, note that the pair (I−×X, (0×
X) ∪ (I− × A)) is an SDR pair, as a consequence of Lemma 3.1 and Theorem
3.1. Let (w,Q) be a representation for the pair (I− ×X, (0×X)∪ (I− ×A)) as
an SDR pair, and let Q be constructed as in Theorem 3.1. Define
r : I− ×X → (0×X) ∪ (I− ×A)
by r(t, x) = Q(1, t, x), where (t, x) ∈ I− ×X . We observe that r has the FCIP,
since Q has this property, and Q has this property since each of its components
has this property.
The equivalence of (2) and (3) is Lemma 4.1.
We need only show that (2) implies (1). Let r : I−×X → (0×X)∪ (I−×A)
be a retraction with the FCIP with respect to i. Define H : I × X → X
by H(t, x) = (πX ◦ r)(α(t), x), where πX is the projection onto the second
coordinate, and α : R → R is a braking function with the following properties:
α(t) = 0 for t ≤ 0, α(t) = 1 for t ≥ 3
, and 0 < α(t) < 1 for 0 < t < 3
. This
braking function is necessary to ensure smoothness at the right endpoint of the
flattened unit interval I. Smoothness at the left endpoint is already taken care
of by the fact that r is defined in terms of the left flattened unit interval. The
map H satisfies the following properties:
• H has the FCIP since r has this property.
• H(0, x) = (πX ◦ r)(0, x) = x, for x ∈ X .
• H(t, x) = (πX ◦ r)(α(t), x) = x, for x ∈ A.
We now construct u : X → I. Let πI : I×X → I denote the projection onto I.
Define a smooth function β : R → R by
β(t) =
0 if t ≤ 0
t2 if t > 0.
Now, define u : X → I by
u(x) =
β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt
β(α(t))dt
It is clear that u is a smooth mapping.
We now verify that (u,H) represents (X,A) as an SNDR pair.
(1) Let x ∈ A. Clearly, (πI ◦ r)(1, x) = 1 and πI ◦ r)(α(t), x) = α(t), and so
β(α(t)− (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt = 0. Thus, u(x) = 0, for all x ∈ A.
(2) Suppose that x ∈ X−A. Since 0×(X−A) is open in the underlying topology
on (0×X)∪ (I− ×A), we may choose an open neighborhood W ⊆ 0× (X −A)
of (0, x). Since r is continuous, there is a neighborhood V ⊆ I− ×X such that
r(V ) ⊆W ⊆ 0× (X −A). Now, consider the mapping qx : I → I×X , given by
qx(t) = (α(t), x), for each x ∈ X . This is clearly smooth. Thus, there exists a
neighborhood U ⊆ I− such that qx(U) ⊆ V . In other words, U × {x} ⊆ V . So,
we have (πI ◦ r)(α(t), x) = 0, for all t ∈ U . Thus, we have
u(x) =
β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt +
β(α(t))dt
β(α(t))dt
Combining this with part (1), we deduce that u−1(0) = A.
(3) Suppose that x is such that u(x) < 1. There must be a neighborhood U of I
such that (πI◦r)(1, x)(πI ◦r)(α(t), x) > 0, for t ∈ U . Thus (πI◦r)(1, x) > 0, but
this implies that r(1, x) ∈ I×A, and hence H(1, x) ∈ A. The proof is complete.
5 The Mapping Cylinder
In this section we show that the inclusion of X into the flattened mapping
cylinder If of a map f : X → Y is a cofibration with the FCIP.
Theorem 5.1 Let f : X → Y be a smooth map. Then, the pair (If , X) is an
SNDR pair.
Proof. Let α : I → R be a smooth braking function with the following proper-
ties: α(t) = 0 if 0 ≤ t ≤ 1
, α(t) = 1 if 3
≤ t ≤ 1, 0 < α(t) < 1, otherwise. Define
two more braking functions α1, α2 : I → R as follows: α1(0) = 0, 0 < α1(t) < 1
if 0 < t < 3
, α1(t) = 1 if
≤ t ≤ 1, and α2(t) = 0 if 0 ≤ t ≤ 34 , α2(t) = 1
≤ t ≤ 1. Now, define u : If → I by u([t, x]) = α1(t) and u([y]) = 1, for
(t, x) ∈ I×X and y ∈ Y . Define H : I× If → If by
H(s, [t, x]) = [(1 − α(s))t+ α(s)α2(t), x] if (t, x) ∈ I×X
H(s, [y]) = [y] if y ∈ Y .
That u is smooth comes from the fact that it is smooth when restricted to
each component of the coproduct (I×X)⊔Y ; it is thus smooth on the quotient
To see that the map H : I × If → If is smooth, note that since we are
working in a cartesian closed category, products commute with quotients, i.e. if
q is quotient, then so is 1× q, where 1 is an identity map. Thus, we may think
of H as being defined on the space
(I× I×X) ⊔ (I× Y )
where ∼ is the identification (t, 1, x) = (t, f(x)) for t ∈ I, and x ∈ X . Since H is
smooth when restricted to each component of the coproduct (I×I×X)⊔(I×Y ),
H is smooth on the quotient I× If .
We now verify that (u,H) is a representation for (If , X) as an SNDR pair.
• u−1(0) = [0, x] = i0(X).
• H(0, [t, x]) = [t, x] and H(0, [y]) = [y].
• H(s, [0, x]) = [0, x].
• If u[t, x] < 1, then t < 3
and so α2(t) = 0. Thus, H(1, [t, x]) = [0, x].
This completes the proof. �
Finally, we have the following important corollary.
Corollary 5.1 Given any smooth map f : X → Y , the inclusion X →֒ If is a
cofibration with the FCIP.
6 The Exact Sequence of a Cofibration
Our aim in this section is to show how one can use SNDR pairs to prove the
existence of the right exact Puppe sequence. We state the result in Theorem
6.1 and break the proof of the result up into a number of lemmas. We follow
the method used by Whitehead [12] for the topological case.
Throughout this section we work in the category FRL∗ of pointed Fr—’olicher
spaces, and basepoint preserving smooth maps.
Theorem 6.1 Let W be an object in FRL∗, and suppose that i : A →֒ X is a
cofibration in FRL∗. For any basepoint x0 ∈ A ⊆ X there is a sequence
. . . // [
A,W ]
Ti,W ]
X,W ]
A,W ] // . . .
. . . // [
A,W ]
// [Ti,W ]
// [X,W ]
// [A,W ]
which is an exact sequence in SETS∗, where j : X → Ti is the inclusion dis-
cussed in Paragraphe 2.4 and k : Ti →
A is the quotient map defined below.
It is, in fact, possible to prove that the sequence above is an exact sequence
of groups as far as
A,W ] and that the morphisms to this point are group
homomorphisms, but we shall not do so here.
The reduced(flattened)suspension of a pointed Frölicher space X is de-
fined as
X = (I/{0, 1}) ∧X,
where the reduced join is defined as for topological spaces with the identified
set taken as basepoint, and with 0 the basepoint of I.
In this section, whenever we refer to the suspension of a space , we mean
the reduced flattened suspension defined above.
Lemma 6.1 If (x,A) is an SNDR pair and p : X → X/A the quotient map,
then the sequence
i // X
p // X/A
is right exact.
Proof. To show that the given sequence is right exact we must show that for
any Frölicher space W the following sequence is exact in SETS:
[X/A,W ]
// [X,W ]
// [A,W ] .
It is easy to see that im p∗ ⊆ ker i∗. To see the reverse inclusion, let g : X →
W be an element of [X,W ], with g|A ≃ w0 (rel w0), where w0 ∈ W . Since
i // X is an SNDR pair, the map i is a cofibration, and so we may extend
w0 to a smooth map g
′ : X → W such that g′ ≃ g. But g′ is constant on A,
and so there exists a smooth map g1 : X/A → W such that p∗(g1) = g′. This
shows that ker i∗ ⊂ im p∗. �
Lemma 6.2 For any smooth map f : X → Y , the sequence
f // Y
l // Tf
is right exact, where l is the usual inclusion of Y into the mapping cone; i.e.
y 7→ [y] ∈ Tf .
Proof. One can show that there is a homotopy commutative diagram
i ��@
// Tf
where i, j, and l are the usual inclusions, and p is the quotient map that collapses
away {0} ×X to a point. Since, by Theorem 5.1, (If , X) is an SNDR pair, it
follows from Lemma 6.1 that the sequence
i // If
p // Tf
is right exact. It is fairly easy to show that j : Y → If is a homotopy equivalence.
Therefore, the sequence
f // Y
l // Tf
is right exact. �
Lemma 6.3 For any smooth map i : A → X, there is an infinite right exact
sequence
i // X
// Ti
// . . . i
// Tin−2
// Tin−1
// . . .
where in, n ≥ 1, are inclusion maps.
Proof. The pair (Ti, X) is an SNDR pair. The representation for the pair
(If , X) in Theorem 5.1 can be adapted to show this. One iterates the procedure
of Lemmas 6.1 and 6.2. �
One can easily see that there is an isomorphism between Ti/X and
Define q : Ti →
A to be the map which identifies X ⊂ Ti to a point, followed
by the isomorphism Ti/X →
Lemma 6.4 The sequence
// Ti
is right exact.
Proof. As noted above the pair (Ti, X) is an SNDR pair. We have the com-
mutative diagram
// Ti
where p : Ti → Ti/X is the identification map, and q0 : Ti/X →
A is an
isomorphism. The top line of the diagram is right exact, by Lemma 6.1, and so
the sequence
// Ti
is right exact. �
There is a commutative diagram
// Ti
where q1 is a homotopy equivalence. ( See Whitehead [12] for more details of
this map. ) Using commutative diagrams of this form, one can now proceed
almost exactly as one does in the topological situation, as in Whitehead [12] for
example, to get the following infinite right exact sequence:
i // X
// Ti
// . . .
. . . //
// . . .
The definition of right exactness now gives us the exact sequence of Theorem
References
[1] A. Cap, K-Theory for Convenient Algebra, Dissertationen, Faculty of
Mathematics, University of Vienna, 1993.
[2] Cherenack P., Applications of Frölicher Spaces to Cosmology, Ann. Univ.
Sci. Budapest 41(1998), 63-91.
[3] P. Cherenack , Frölicher versus Differential Spaces: A prelude to Cosmol-
ogy, Kluwer Academic Publishers 2000, 391-413.
[4] P. Cherenack, The Left Exactness of the Smooth Left Puppe Sequence.
In L. Tamassy and J. Szenthe, editors, New Developments in Differential
Geometry, (Proceedings of the Colloquium on Differential Geometry, De-
brecen, Hungary, July 26-30, 1994), Mathematics and Its Applications.
Kluwer Academic Publishers, 1996.
[5] P. Cherenack, Smooth Homotopy, Topology with Applications, (18):27-41,
1984.
[6] B. Dugmore, The Right Exactness of the Smooth Right Puppe Sequence.
Master’s Thesis, University of Cape Town, 1996.
[7] A. Frölicher, A. Kriegl, Linear Spaces and Differentiation Theory, J. Wiley
and Sons, New York, 1988.
[8] M.W. Hirsch, Differential Topology, GTM 33, Springer-Verlag, New York,
1976.
[9] I.M. James, General Topology and Homotopy Theory, Springer-Verlag,
Berlin, 1984.
[10] A. Kriegl, P. Michor, Convenient Settings of Global Analysis, Am. Math.
Soc., 1997.
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York, 1978.
	Preliminaries
	Basic Constructions of Homotopy Theory in FRL
	Our Approach to Homotopy Theory in FRL
	Flattened Structures on the Unit Interval
	Some Properties of Smooth Functions between the Flattened Unit Intervals
	Homotopy in FRL and Related Objects
	Cofibrations in FRL
	Smooth Neighborhood Deformation Retracts
	SNDR pairs and SDR pairs
	Cofibrations
	The Mapping Cylinder
	The Exact Sequence of a Cofibration

  Cofibrations are defined in the category of Fr\"olicher spaces by weakening
the analog of the classical definition to enable smooth homotopy extensions to
be more easily constructed, using flattened unit intervals. We later relate
smooth cofibrations to smooth neighborhood deformation retracts. The notion of
smooth neighborhood deformation retract gives rise to an analogous result that
a closed Fr\"olicher subspace $A$ of the Fr\"olicher space $X$ is a smooth
neighborhood deformation retract of $X$ if and only if the inclusion $i:
A\hookrightarrow X$ comes from a certain subclass of cofibrations. As an
application we construct the right Puppe sequence.

Cofibrations in the Category of Frölicher Spaces:
Part I
Brett Dugmore
Cadiz Financial Strategists (Pty) Ltd, Cape Town, South Africa
Email: Brett.Dugmore@cadiz.co.za
Patrice Pungu Ntumba
Department of Mathematics and Applied Mathematics
University of Pretoria
Hatfield 0002, Republic of South Africa
Email: patrice.ntumba@up.ac.za
Abstract
Cofibrations are defined in the category of Frölicher spaces by weak-
ening the analog of the classical definition to enable smooth homotopy
extensions to be more easily constructed, using flattened unit intervals.
We later relate smooth cofibrations to smooth neighborhood deforma-
tion retracts. The notion of smooth neighborhood deformation retract
gives rise to an analogous result that a closed Frölicher subspace A of the
Frölicher space X is a smooth neighborhood deformation retract of X if
and only if the inclusion i : A →֒ X comes from a certain subclass of
cofibrations. As an application we construct the right Puppe sequence.
Subject Classification (2000): 55P05.
Key Words:Frölicher spaces, Flattened unit intervals, Smooth neighborhood de-
formation retracts, Smooth cofibrations, Cofibrations with FCIP, Puppe se-
quence.
1 Preliminaries
The purpose of this section is to survey brielfy the notion of Frölicher spaces.
Frölicher spaces arise naturally in physics, and do generalize the concept of
smooth manifolds. A Frölicher space, or smooth space as initially called by
Frölicher and Kriegl [7], is a triple (X, CX ,FX) consisting of a setX , and subsets
CX ⊆ XR, FX ⊆ RX such that
• FX ◦ CX = {f ◦ c| f ∈ FX , c ∈ CX} ⊆ C∞(R)
• ΦCX := {f : X → R| f ◦ c ∈ C∞(R) for all c ∈ CX} = FX
http://arxiv.org/abs/0704.0342v1
• ΓFX := {c : R → X | f ◦ c ∈ C∞(R)for all f ∈ FX} = CX
Frölicher and Kriegl [7], and Kriegl and Michor [10] are our main reference for
Frölicher spaces. The following terminology will be used in the paper: Given a
Frölicher space (X, CX ,FX), the pair (CX ,FX) is called a smooth structure; the
elements of CX and FX are called smooth curves and smooth functions respec-
tively. The topology assumed for a Frölicher space (X, CX ,FX) throughout the
paper is the initial topology TF induced by the set FX of functions. When there
is no fear of confusion, a Frölicher space (X, CX ,FX) will simply be denoted X .
The most natural Frölicher spaces are the finite dimensional smooth manifolds,
where if X is such a smooth manifold, then CX and FX consist of all smooth
curves R → X and smooth functions X → R. Euclidean finite dimensional
smooth manifolds Rn, when viewed as Frölicher spaces, are called Euclidean
Frölicher spaces. In the sequel, by Rn, n ∈ N, we mean the Frölicher space Rn,
equipped with its usual smooth manifold structure.
A Frölicher space X is called Hausdorff if and only if the smooth real-valued
functions on X are point-separating, i.e. if and only if TF is Hausdorff.
A Frölicher structure (CX ,FX) on a set X is said to be generated by a set
F0 ⊆ RX (resp. C0 ⊆ XR) if CX = ΓF0 and FX = ΦΓF0 (resp. FX = ΦC0
and CX = ΓΦC0 ). Note that different sets F0 ⊆ RX on the same set X may
give rise to a same smooth structure on X . A set mapping ϕ : X → Y between
Frölicher spaces is called a map of Frölicher spaces or just a smooth map if for
each f ∈ FY , the pull back f ◦ϕ ∈ FX . This is equivalent to saying that for each
c ∈ CX , ϕ ◦ c ∈ CY . For Frölicher spaces X and Y , C∞(X,Y ) will denote the
collection of all the smooth maps X → Y . The resulting category of Frölicher
spaces and smooth maps is denoted by FRL.
Some useful facts regarding Frölicher spaces can be gathered in the following
Theorem 1.1 The category FRL is complete (i.e. arbitrary limits exist ), co-
complete (i.e. arbitrary colimits exist), and Cartesian closed.
Given a collection of Frölicher spaces {Xi}i∈I , let X =
i∈I Xi be the set
product of the sets {Xi}i∈I and πi : X → Xi, i ∈ I, denote the projection map
(xi)i∈I 7→ xi. The initial structure on X is generated by the set
{f ◦ πi : f ∈ FXi}.
The ensuing Frölicher space (X,ΓF0, ϕΓF0) is called the product space of the
family {Xi}i∈I . Clearly,
ΓF0 = {c : R → X | if c(t) = (ci(t))i∈I , then ci ∈ CXi for every i ∈ I}.
Now, let
i∈I Xi be the disjoint union of sets {Xi}i∈I , and ιXi : Xi →
i∈I Xi
the inclusion map. Place the smooth final structure on
i∈I Xi corresponding
to the family {ιXi}i∈I . The resulting Frölicher space is called the coproduct of
{Xi}i∈I , and denoted
i∈I Xi, and
Xi = {f :
Xi → R| for each i ∈ I, f |Xi ∈ FXi}
is the collection of smooth functions for the coproduct.
Corollary 1.1 Let X, Y , and Z be Frölicher spaces. Then the following canon-
ical mappings are smooth.
• ev: C∞(X,Y )×X → Y , (f, x) 7→ f(x)
• ins:X → C∞(Y,X × Y ), x 7→ (y 7→ ins(x)(y) = (x, y))
• comp:C∞(Y, Z)× C∞(X,Y ) → C∞(X,Z), (g, f) 7→ g ◦ f
• f∗ : C∞(X,Y ) → C∞(X,Z), f∗(g) = f ◦ g, where f ∈ C∞(Y, Z)
• g∗ : C∞(Z, Y ) → C∞(X,Y ), g∗(f) = f ◦ g, where g ∈ C∞(X,Z).
Given Frölicher spaces X , Y , and Z; in view of the cartesian closedness of
the category FRL, the exponential law
C∞(X × Y, Z) ∼= C∞(X,C∞(Y, Z))
holds. Because FX = C∞(X,R), it follows by cartesian closedness of FRL that
the collection FX can be made into a Frölicher space on its own right.
Finally we would like to show how to construct smooth braking functions,
following Hirsch [8]. Smooth braking functions are tools that are behind most
results in this paper. In [11], it is shown that the function ϕ : R → R given by
ϕ(u) =
0 if u ≤ 0
u if u > 0
is smooth. Substituting x2 for u in the above function, one sees that the function
ψ : R → R, given by
ψ(x) =
0 if x ≤ 0
x2 if u > 0
is smooth. Now, let us construct a smooth function α : R → R with the following
properties. Let 0 ≤ a < b. α(t) should satisfy:
• α(t) = 0 for t ≤ a,
• 0 < α(t) < 1 for a < t < b,
• α is strictly increasing for a < t < b,
• α(t) = 1 for t ≥ b.
Define α : R → [0, 1] by
α(t) =
γ(x)dx
γ(x)dx
where γ(x) = ψ(x− a)ψ(b − x).
In the sequel, the notation αǫ, 0 < ǫ <
, will refer to a smooth braking
function with the following properties
• αǫ(t) = 0 for t ≤ ǫ,
• 0 < αǫ(t) < 1 for ǫ < t < 1− ǫ,
• α strictly increasing for ǫ < t < 1− ǫ,
• αǫ(t) = 1 for 1− ǫ ≤ t.
2 Basic Constructions of Homotopy Theory in
In this section, we define the fundamental notions of homotopy theory in the
category FRL, such as the homotopy relation and the mapping cylinder. We
begin with an overview of our approach to homotopy in FRL, and then discuss
alternate Frölicher structures on the unit interval which are used in this and
subsequent sections.
2.1 Our Approach to Homotopy Theory in FRL
One might begin investigating homotopy theory in FRL by simply following
the homotopy theory of topological spaces, replacing continuous functions with
smooth ones. One can certainly define the notion of a homotopy H : I×X → Y
between smooth maps H(0,−) and H(1,−) in this way (which we do). One can
even get as far as the left Puppe sequence (see [4]), but eventually difficulties
begin to arise.
Extending functions defined on a subspace of a Frölicher space tends to be
a little tricky, and so the definition of a cofibration in FRL is one that needs
careful consideration. We envisage to construct the right Puppe sequence in a
future paper. To do this we define a slightly weaker notion of cofibration than
the notion obtained from topological spaces. In addition, we define the mapping
cylinder of a smooth map f : X → Y using not the unit interval, but a modified
version called the weakly flattened unit interval, denoted I, which, as one
can show, is topologically homeomorphic to the unit interval. This modified
structure on the unit interval allows us to show that the inclusion of a space X
into the mapping cylinder of f : X → Y is a cofibration (in our weaker sense ).
The weakly flattened unit interval is useful, but it also has its drawbacks.
It would be ideal to have a single structure on the unit interval that can be
used throughout out homotopy theory, but the weakly flattened unit interval
is not suitable, because it has the rather restrictive property that a smooth
map f : I → I on the usual unit interval often does not define a smooth map
f : I → I unless the endpoints of the interval are mapped to the endpoints. This
restrictive property means that we only use the flattened unit intervals where
they are absolutely necessary.
In our future work, we will investigate whether with our modified notions of
cofibration and mapping cylinder, Baues’ cofibration axioms are satisfied.
2.2 Flattened Structures on the Unit Interval
We define two main Frölicher structures which we call the flattened unit in-
terval and the weakly flattened unit interval . Let (CI ,FI) be the subspace
structure induced on I by the inclusion I →֒ R.
Definition 2.1 The Frölicher space (I, CI,FI), where the structure (CI,FI) is
the structure generated by the set
F = {f ∈ FI| there exists 0 < ǫ < 14 with f(t) = f(0) for t ∈ [0, ǫ) and
f(t) = f(1) for t ∈ (1− ǫ, 1]},
is called the flattened unit interval.
It is easy to see that any continuous map c : R → [0, 1] defines a structure
curve on I if and only if it is smooth at every point t ∈ R, where c(t) ∈ (0, 1), .
We define the left (resp. right) flattened unit interval, denoted by I−
(resp. I+), to be the Frölicher space whose underlying set is the unit interval
[0, 1], and structure is the structure generated by the structure functions in FI
that are constant near 0 (resp. 1).
Definition 2.2 The Frölicher space (I, CI,FI), with the structure defined below
is called the weakly flattened unit interval. The underlying set is the unit
interval; the structure (CI,FI) is generated by the family
F = {f ∈ FI | lim
f(t) = 0, lim
f(t) = 0, n ≥ 1}.
We call the property, for all f ∈ F ,
f(t) = 0, lim
f(t) = 0, n ≥ 1,
the zero derivative property of f .
We shall prove that all structure functions on I have the zero derivative
property, in other words, FI = F . To that effect, we need the following lemma.
Lemma 2.1 Let c : R → R be a smooth real-valued function at t = t0, and let
f : R → R be a smooth real-valued function at t = c(t0). Then,
(f ◦ c)(t0) = f (n)(c(t0))(c′(t0))n + terms of the form
af (k)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(n−1)(t0))
mn−1 ,
where k < n and a ∈ R. In addition, if a 6= 0 then at least one ofm2,m3, . . . ,mn−1
is also non-zero.
Proof. The proof is done by induction. For the sake of brevity, we call the
term f (n)(c(t0))(c
′(t0))
n the primary term for n, and the terms of the form
af (k)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(n−1)(t0))
mn−1 the lower order terms for n.
The statement is true for n = 1 and for n = 2. Suppose the result is true for
n = k. To show that the result holds for n = k + 1, since
dtk+1
(f ◦ c)(t0) =
(f (k)(c(t0))(c
′(t0))
+terms of the form d
(af (j)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(k−1)(t0))
mk−1),
where j < k + 1 and a ∈ R, we need only show that
(af (j)(c(t0))(c
′(t0))
m1(c′′(t0))
m2 . . . (c(k−1)(t0))
mk−1)
gives rise to lower terms for n = k + 1, which is by the way straightforward. �
Theorem 2.1 FI = {f ∈ FI | limt→0+ d
f(t) = 0 = limt→1−
f(t)} =: F
Proof. That F ⊆ FI is evident. We must show the reverse inequality. Let
0 < ǫ < 1
, and 0 < M < 1. Consider the function cM : R → R, given by
cM (t) = (1− αǫ(|t|))βM (t) + αǫ(|t|),
where αǫ : R → R is a smooth braking function as defined in the Preliminaries,
and βM : R → R is given by
βM (t) =
−Mt if t ≤ 0
t if t > 0
It is easily seen that cM is continuous over all R, and smooth over all R except
at t = 0. Also note that 0 < cM (t) < 1 for all t ∈ R, and cM (t) = βM (t) = 0 for
all 0 ≤ t < ǫ. Now,
cM (t) =
βM (t) = −M, for −ǫ < t < 0
cM (t) =
βM (t) = 1, for 0 < t < ǫ
For n > 1, we have
cM (t) =
βM (t) = 0, for t ∈ (−ǫ, 0) ∪ (0, ǫ).
We now show that for cM ∈ ΓF . To this end, let f ∈ F . To show that
f ◦ cM : R → R is smooth, it is obvious that we need only concentrate on the
point t = 0, because f ◦ c is smooth at every t 6= 0. It follows for t 6= 0, and
n ∈ N that Lemma 2.1 applies. But as t → 0, cM (t) → 0+, and so, letting
s = cM (t), we have
f (j)(cM (t)) = lim
f (j)(s) = 0,
for all j ∈ N, by the zero derivative property of f . Thus, as t approaches
the value 0, the primary term and all the lower order terms of d
(f ◦ cM )(t)
vanish, and we have shown that f ◦ cM is smooth at t = 0. This implies that
f ◦ cM ∈ C∞(R,R) for all f ∈ F . It follows that cM ∈ ΓF .
We are now ready to show that FI ⊆ F . To this end, suppose that we
are given a structure function f ∈ FI. We shall show that this f has the zero
derivative property, and is thus an element of F .
Since f ∈ FI, we know that f ◦ c is a smooth real-valued function for every
c ∈ ΓF . In particular, f ◦ cM is smooth for all 0 < M < 1. Thus, for any n ∈ N,
(f ◦ cM )(t) = lim
(f ◦ cM )(t).
As t→ 0−, cM (t) → 0+; let us consider the lower order terms for n. Each term
of the form
af (k)(cM (t))(c
M (t))
m1(c′′M (t))
m2 . . . (c
(n−1)
M (t))
has some term (c
(t))mi , for some i > 1, with mi 6= 0. But limt→0− c
(t) = 0,
if i > 1, and so
af (k)(cM (t))(c
M (t))
m1(c′′M (t))
m2 . . . (c
(n−1)
(t))mn−1 = 0.
So all the lower order terms fall away, therefore
limt→0−
(f ◦ cM )(t) = limt→0− f (n)(cM (t))(c′M (t))n
= limt→0− f
(n)(cM (t))(−M)n
= lims→0+ f
(n)(s)(−M)n,
where s = cM (t). In a similar way one shows that
(f ◦ cM )(t) = lim
f (n)(s).
But f◦cM is smooth, therefore lims→0+ f (n)(s)(−M)n = lims→0+ f (n)(s), which
implies that lims→0+ f
(n)(s) = 0.
We have shown that the zero derivative property of f holds for the left
endpoint of the unit interval. To show that the zero derivative property of f
holds for the right endpoint of f , note that dM : R → R, dM (t) = 1− cM (t), is
a smooth real-valued function with d(0) = 1, and 0 ≤ dM (t) ≤ 1 for all t ∈ R.
One can follow a similar procedure to the above, using dM instead of cM to
show that lims→1− f
(n) = 0. �
2.3 Some Properties of Smooth Functions between the
Flattened Unit Intervals
One has to be careful when dealing with the various flattened unit intervals. A
smooth function f : I → I from the R- Frölicher subspace unit interval I to
itself need not define a smooth function f : I → I, for example. Conversely,
not every smooth function f : I → I defines a smooth function f : I → I. In
particular, we need to be aware of the fact that addition and multiplication of
functions when defined between the various flattened unit intervals does not
preserve smoothness, as is the case with the usual unit interval.
Example 2.1
The function f : I → I, f(t) = 1
t is clearly smooth, but the corresponding
function f : I → I, given by the same formula, is not smooth. To see this, let
α : R → R be a smooth braking function with the properties that
• α(t) = −1, for t < − 3
• α(t) = t, for − 1
< t < 1
• α(t) = 1, for t > 3
Define c : R → I by c(t) = 1− |α(t)|. The curve c is smooth everywhere except
at t = 0, where c(0) = 1. However, every generating function f on I is constant
near 1, and so the composite f ◦ c is smooth. Thus c is a structure curve on I.
Now, f ◦ c : R → I is given by (f ◦ c)(t) = 1
(1 − |α(t)|). Let h : I → R be a
structure function with the properties that
• h(s) = 0, for s < 1
• h(s) = s, for 1
< s < 3
• h(s) = 1, for 7
Then (h ◦ f ◦ c)(t) = 1
(1− |α(t)|) for t near 0, and is not smooth at t = 0. Thus
f does not define a smooth function from I to I.
Example 2.2
The function f : I → I, f(t) =
t, is smooth, but the corresponding f : I → I,
given by the same formula, is not smooth. This follows from the fact that f is
smooth on the open interval (0, 1), and a generating function g on I is constant
near 0 and 1. On the side, f : I → I is not smooth, because if c : R → I is a
structure curve with c(t) = t2 near t = 0, then (f ◦ c)(t) = |t| near t = 0, which
is not smooth on I at t = 0.
Example 2.3
The functions f, g : I− → I−, given by f(t) = 1
t and g(t) = 1
are both
smooth, but the sum f(t) + g(t) = 1
is not smooth.
The following lemma follows from the definition of the Frölicher structures
on the various flattened unit intervals.
Lemma 2.2 Let f : I → I be a smooth function with the properties that f(0) =
0 and f(1) = 1. Then the following maps are smooth:
• f : I → I±,
• f : I → I,
• f : I± → I,
• f : I → I,
• f : I → I.
The function defined in the following example is for later reference.
Example 2.4
Let H : I × I− → I− be given by H(t, s) = (1 − α(t))s, where α : R → R is a
smooth braking function with the properties that
• α(t) = 0 for t < 1
• 0 ≤ α(t) ≤ 1 for all t ∈ R,
• α(t) = 1 for t > 3
We show that H is smooth. To see this, let f : I− → R be a generating function
on I−. So f is constant near 0. Now, let c : R → I × I− be a structure curve,
given by c(v) = (t(v), s(v)). The curve t is a structure curve on I, and so is
a smooth real-valued function for all v ∈ R, except possibly when t(v) = 0 or
t(v) = 1. Similarly, the curve s is a structure curve on I−, and so is smooth
for all v ∈ R except possibly when s(v) = 0. Now consider the composite
H ◦ c : R → I−. Clearly, α(t(v)) is smooth for all v, since the only possible
points for non-smoothness occur when t(v) = 0 or t(v) = 1, and α(t(v)) is
locally constant near these points. Consequently, H ◦ c is smooth everywhere
except possibly when s(v) = 0. Now, let’s consider f ◦H ◦ c : R → R; the only
possible points for non-smoothness are those in which s is 0, i.e. H◦ = 0. But f
is a structure generating function on I−, and so is locally constant near 0. This
shows that f ◦H ◦ c is smooth for all v ∈ R, and thus H is smooth.
2.4 Homotopy in FRL and Related Objects
Definition 2.3 (1) Let X be a Frölicher space, and x0, x1 ∈ X. We say that
x0 is smoothly path-connected to x1 if there is a smooth path c : I → X such
that c(0) = x0 and c(1) = x1. We write x0 ≃ x1. The relation ≃ is called
smooth homotopy when it is applied to hom-sets.
(2) Let f : X → Y be a map of Frölicher spaces. f is called a smooth
homotopy equivalence provided there exists a smooth map g : Y → X such that
f ◦ g ≃ 1Y and g ◦ f ≃ 1X .
One can show that smooth homotopy is a congruence in RFL. In practice, we
say that smooth maps f, g : X → Y are smoothly homotopic if there exists a
smooth map H : I ×X → Y with H(0,−) = f and H(1,−) = g. If A ⊆ X is
subspace of X , then we say that H is a smooth homotopy (rel A) if the map
H has the additional property that H(t, a) = a for each t ∈ I and a ∈ A. See
Cherenack [5] and Dugmore [6] for more detail regarding smooth homotopy.
The notion of deformation retract is fundamental to topological homotopy
theory. The following definitions are adapted for smooth homotopy, and will be
needed at a later stage.
Definition 2.4 Let A ⊆ X be a subspace of a Frölicher space X, and let i :
A →֒ X denote the inclusion map. Then
• We say that A is a retract of X if there exists a smooth map r : X → A
such that ri = 1A. We call r a retraction.
• We call A a weak deformation retract of X if the inclusion i is a smooth
homotopy equivalence.
• The subspace A is called a deformation retract of X if there exists a re-
traction r : X → A such that ir ≃ 1X .
• The subspace A is called a strong deformation retract of X if there exists
a retraction r : X → A such that ir ≃ 1X(relA).
Definition 2.5 The mapping cylinder If of f : X → Y is defined by the fol-
lowing pushout
I ×X // If
where i1 : X → I ×X is given by i1(x) = (1, x), for any x ∈ X. We denote the
elements of If by [t, x] or [y], where (t, x) ∈ I ×X and y ∈ Y .
Replacing I ×X in the above pushout diagram by I×X or I×X, we obtain
the flattened mapping cylinder If and weakly flattened mapping cylinder If of f
respectively. We use the same notation for elements of these flattened mapping
cylinders as described above for the mapping cylinder.
There is also a map i0 : X → I ×X , defined by i0(x) = (0, x) for x ∈ X . This
induces an inclusion map i′0 : X → If , which identifies X with the Frölicher
subspace i′0(X) of If . An inclusion is induced in a similar way for the flattened
mapping cylinders. If one identifies {0}×X to a point in the mapping cylinder
If of a map f : X → Y , then one obtains the mapping cone Tf of the
map f . In a similar fashion, we define the flattened mapping cone Tf and
weakly flattened mapping cone Tf of a smooth map f : X → Y .
2.5 Cofibrations in FRL
A cofibration is a map i : A→ X for which the problem of extending functions
from i(A) to X is a homotopy problem. In other words, if a map f : i(A) → Z
can be extended to a map f∗ : X → Z, then so can any map homotopic to f . For
topological spaces, the usual definition is phrased in a slightly more restrictive
way. The extension of a map g ≃H f , for some homotopy H : I × i(A) → Z, is
required to exist at every level of the homotopy simultaneously. In other words,
one requires each H(t,−) to be extendable in such a way that the resulting
homotopy H∗ : I ×X → Z is continuous.
We weaken this definition somewhat, to enable smooth homotopy extensions
to be more easily constructed using a flattening at the endpoints of the homo-
topy. This enables us to characterize smooth cofibrations in terms of a flattened
unit interval, and then later to relate smooth cofibrations to smooth neigh-
borhood deformation retracts. Our definition of smooth cofibration, though
different from from Cap’s definition, see [1], leads to several classical results as
does Cap’s. As pointed out by Cap, the analogue of the classical definition of
cofibration would not allow even {0} →֒ I to be a smooth cofibration. So, we
have the following
Definition 2.6 A smooth map i : A → X is called a smooth cofibration if,
corresponding to to every commutative diagram of the form
(0,1A)
f // Z
66mmmmmmmmmmmmmm
there exists a commutative diagram in FRL of the form
(0,1X )
::tttttttttt
where G′ : I × A → Z is given by G′(t, a) = G(αǫ(t), a) for some 0 < ǫ < 12 ,
and each t ∈ I, a ∈ A.
The problem of extending a map smoothly from a subspace of a Frölicher
space to the whole space is a more difficult problem than simply extending con-
tinuously. It is mainly for this reason that the definition of smooth cofibration
differs somewhat from the corresponding definition of a topological cofibration.
Lemma 2.3 Let i : A → X be a smooth cofibration, then i is an initial mor-
phism in FRL. In addition, if A is Hausdorff, then i is injective. So in this
case A can be regarded as a subspace of X.
Proof. Let us show that every smooth map f : A→ R factors through i, that
is for every f ∈ FA, there exists f̃ ∈ FX such that f = f̃ ◦ i. To this end,
consider the smooth map G : I × A → R, given by H(t, a) = tf(a). Clearly,
0|A = G(0,−), where 0 : X → R is the constant map 0. It follows that there is
map F : I ×X → R such that F ◦ (1× i) = G′. Then, clearly f̃ := F (1,−) has
the desired property.
The remaining part of the proof of Proposition 3.3, in [1], holds verbatim
here as well. �
In this paper, we are interested only in cofibrations that are injective. Hence-
forth, all cofibrations are assumed to be injective.
All topological cofibrations are inclusions, and this result is true for smooth
cofibrations too. The proof of the following lemma is essentially the same as
the proof given by James [9] for the topological result, although James’s proof
is in some sense dual to ours, using path-spaces in place of cartesian products
and the adjoint versions of our homotopies.
Lemma 2.4 A cofibration
i // X
is a smooth inclusion.
Proof. Let Ii be a mapping cylinder of i, and let j : X → Ii be the standard
inclusion map. Consider the smooth map γ : I → I, γ(t) = 1 − t, for all t ∈ I,
and the quotient map q : (I ×A) ⊔X → Ii; we have the following commutative
diagram
(0,1A)
j // Ii
66mmmmmmmmmmmmmm
where G(t, a) = [(1 − t, a)]. Notice that the map G is smooth. Since i is a
cofibration, we have the commutative diagram
(0,1X )
::uuuuuuuuuu
where G′(t, a) = G(αǫ(t), a) for some 0 < ǫ <
. Define U : X → Ii by
U(x) = F (1, x). We have U ◦ i = G′(1,−), where G′(1, a) = [(0, a)], for every
a ∈ A. Thus the assignment a 7→ G′(1, a) defines the usual inclusion of A into
the mapping cylinder. From this we deduce that U ◦ i is an inclusion, and hence
i is an inclusion. �
There is an equivalent formulation of definition 2.6, given in the following
lemma.
Lemma 2.5 A smooth map
i // X
is a cofibration if and only if, for every smooth map h : (0×X)∪(I−×i(A)) → Z,
the following diagram
(0×X) ∪ (I− × i(A)) h //
I− ×X
77oooooooooooooo
where j is the evident inclusion, exists in FRL.
Proof. Suppose that the inclusion A //
i // X is a smooth cofibration, and
suppose that h : (0 × X) ∪ (I− × i(A)) → Z is a smooth map. We have the
diagram
(0×B) ∪ (I− × i(A)) h //
I− ×X
We need to fill in a smooth map G : I− × X → Z which makes the resulting
diagram commute. To do this, notice that h|I−×i(A) is smooth, and thus the
corresponding map h|I × i(A), using the usual unit interval, is also smooth. We
have the following diagram
(0,1A)
h|0×X // Z
66mmmmmmmmmmmmmm
where h|0×X(0,−) : X → Z is denoted as h|0×X . The fact that i is a smooth
cofibration yields the following FRL-commutative diagram:
h0×X //
(0,1A)
::tttttttttt
where (h|I−×A)′(t, a) = h|I−×A(αǫ(t), a), for some 0 < ǫ < 12 . Now, chose a
smooth braking function β : R → R with the following properties.
• α(t) = 0 for t < ǫ
• α(t) = t for ǫ < t.
F may not be smooth on I− × A due to the flattening requirements of the left
flattened unit interval. To correct this, set G(t, a) = F (β(t), a). Notice that the
insertion of this braking function does not affect the commutativity conditions
of G, since the only adjustments to F occur in the first coordinate where the
map (h|I−×X)′ is constant.
Now, assume the converse, i.e. to every smooth map h : (0 × X) ∪ (I− ×
i(A)) → Z, corresponds a commutative diagram
(0×X) ∪ (I− × i(A)) h //
I− ×X
77oooooooooooooo
We wish to show that the inclusion i : A → X is a cofibration; so assume we
have the following diagram
(0,1A)
f // Z
66mmmmmmmmmmmmmm
There exists the diagram
(0,1A)
f // Z
66mmmmmmmmmmmmmm
where G′(t, a) = G(αǫ(t), a). Our hypothesis allows us to construct the diagram
(0×X) ∪ (I− × i(A))
f∪G′ //
I− ×X
77oooooooooooooo
Note that f ∪ G′ is smooth since αǫ(t) is constant near 0. Since H is smooth
on I− ×X it defines a smooth map on I ×X . One can verify that the diagram
(0,1X)
::tttttttttt
commutes as required. �
3 Smooth Neighborhood Deformation Retracts
This section is concerned with the formulation of a suitable notion of smooth
neighborhood deformation retract. For topological spaces, the statement that a
closed subspace A of X is a neighborhood deformation retract of X is equivalent
to the statement that the inclusion i : A →֒ X is a closed cofibration. We show
that in the category of Frölicher spaces there is a notion of smooth neighborhood
deformation retract that gives rise to an analogous result that a closed Frölicher
subspace A of the Frölicher space X is a smooth neighborhood deformation
retract of X if and only if the inclusion i : A →֒ X comes from a certain subclass
of cofibrations. As an application, we construct the right Puppe sequence.
3.1 SNDR pairs and SDR pairs
The definition of ‘smooth neighborhood deformation retract’ that we adopt in
this paper is similar to the definition of ‘R-SNDR pair’suggested in [6], but we
have modified the definition in order to retain only the essential aspects of ‘first
coordinate independence’ defined in [6].
We begin by defining the ‘first coordinate independence property’ of a func-
tion on a product of a Frölicher space with I (or I−, I+).
Definition 3.1 Let i : A → X be a smooth map, and c : R → X a structure
curve on X. Define
Λ(c, i) = {t∗ ∈ c−1(i(A))| there exists a sequence {tn} of real numbers
with limn→∞ tn = t∗ and each tn ∈ c−1(X − i(A))}.
The points in Λ(c, i) are those values in R where the curve ‘enters’ i(A) from
X − i(A), or ‘touches’ a point in i(A) whilst remaining in X − i(A) nearby.
Now, we are ready to define the ‘first coordinate independence property’ for a
structure function on a product.
Definition 3.2 Let i : A→ X be a smooth map and suppose f : I×X → R is
a structure function on I ×X. Let c : R → I ×X, given by c(s) = (t(s), x(s))
have the following properties
• The map x(s) is a structure curve on X.
• For all ǫ > 0, t(s) is a smooth real-valued function on R−∪s∗∈Λ(x,i)[s∗ −
ǫ, s∗ + ǫ].
If, for every such map c, the composite f ◦ c is a smooth real-valued function,
then we say that f : I×X → R has the first independence property (FCIP) with
respect to i.
Extending the definition, we say that a map g : I × X → Y has the FCIP
with respect to i if the composite h ◦ g : I ×X → R has the FCIP with respect
to i for every h ∈ FY .
Notice that we can formulate a similar definition of the FCIP if we replace
I throughout by I− or I+, leaving the rest of the definition unchanged. We will
have occasion to use this type of first coordinate independence property in the
later part of this work.
Note. Let i : A→ X , and suppose that we are given a map g : I×X → Y . Let
f : Y → R be a structure function on Y , and suppose that f ◦ g : I ×X → R
has the FCIP with respect to i for any such f . Then, given a smooth map
h : Y → Z, the composite f ′ ◦ h ◦ g : I×X → R has the FCIP with respect to
i for any structure function f ′ on Z.
The above note applies equally well if g : I− ×X → Y or g : I+ ×X → Y
has the FCIP with respect to i when composed with a smooth function h on Y .
Example 3.1
1. For any i : A→ X , the projection onto the second coordinate πX : I×X → X
has the FCIP.
2. Let α : R → R be a smooth braking function with the properties that
• α(t) = 0 if t < 1
• 0 < α(t) < 1 if 1
≤ t ≤ 3
• α(t) = 1 if 3
Consider 0 →֒ I−. Let H : I× I− → I− be given by H(t, s) = (1−α(t))s. Then,
f ◦H : I× I− → R has the FCIP with respect to the inclusion 0 →֒ I−, for any
f ∈ FI− .
Definition 3.3 Consider a smooth inclusion i : A →֒ X. Suppose that there
exists a smooth map u : X → I, with u−1(0) = i(A). If there exists a smooth
map H : I×X → X that satisfies the following properties:
• H has the FCIP with respect to i.
• H(0, x) = x for all x ∈ X.
• H(t, x) = x for all (t, x) ∈ I× i(A).
• H(1, x) ∈ i(A) for all x ∈ X with u(x) < 1,
then the pair (X,A) is called a smooth neighborhood deformation retract pair,
or SNDR pair for short.
If, in addition, H is such that H(1 × X) ⊂ i(A), then the pair (X,A) is
called a smooth deformation retract pair, or an SDR pair for short.
The subspace A is called a smooth neighborhood deformation retract or smooth
deformation retract of X if (X,A) is an SNDR pair or SDR pair, respectively.
The pair (u,H) is called a representation for the SNDR (or SDR) pair.
Example 3.2
1. The pair (X, ∅) is an SNDR pair. A representation is u(x) = 1, H(t, x) = x,
for each t ∈ I and x ∈ X .
2. The pair (X,X) is an SNDR pair. A representation is u(X) = 0, H(t, x) = x,
for each t ∈ I and x ∈ X .
Lemma 3.1 The pair (I−, 0) is an SDR pair.
Proof. Let α : R → R be the smooth braking function of Examples 3.1. A
representation for (I−, 0) as an SDR pair is (u,H), where u : I− → I and
H : I× I− → I− are given by u(s) = s, and H(t, s) = (1 − α(t))s. Clearly, the
identity u : I− → I is smooth. And the map H , as shown in Example 2.4, is
smooth and clearly has the FCIP with respect to the inclusion, since whenever
v approaches a value for which s(v) = 0, one has
g((1− α(t(v)))s(v)) = g(0)
for v in a neighborhood of this value and g ∈ FI− . �
Lemma 3.2 The pair (I, {0, 1}) is an SNDR pair.
Proof. A representation (u,H) for the SNDR pair can be given as follows.
Define u : I → I to be a bump function such that
• u(t) = 0 for t = 0 or t = 1,
• u(t) = 1 for t ∈ [ 1
• 0 < u(t) < 1 otherwise,
and let β : I → I be a braking function with the properties that β(s) = 0 for
0 ≤ s ≤ 1
, and β(s) = 1 for 3
≤ s ≤ 1. Let 0 < ǫ 1
, and define H : I× I → I by
H(t, s) = (1− αǫ(t))s+ αǫ(t)β(s). It is clear that H(0, s) = s, H(t, 0) = 0, and
H(t, 1) = 1. Suppose that u(s) < 1. Then, s ∈ [0, 1
) ∪ (3
, 1]. This implies that
β(s) = 0 or β(s) = 1. We then have H(1, s) = 0 or H(1, s) = 1, which means
that H(1, s) ∈ {0, 1} if u(s) < 1.
To see that H is smooth, let f : I → R be a generating function for the
flattened unit interval. The only possible points of non-smoothness are points
where t = 0, 1 and s = 0, 1. The braking function αǫ ensures that H is locally
constant in the tb variable whenever t is near 0 or 1, so no problem arises from
the t component. When s is near s = 0, we have H(t, s) near 0, and so the
generating function f is locally constant. Similarly, when s is near s = 1, we
have H(t, s) near 1, and the generating function f is again locally constant. �
We now show that the product of SNDR pairs is again an SNDR pair.
Theorem 3.1 Let i : A →֒ X and j : B →֒ Y be inclusion mappings. If (X,A)
and (Y,B) are SNDR pairs, then so is
(X × Y, (X ×B) ∪ (A× Y )).
If one of (X,A) or (Y,B) is an SDR pair, then so is the pair
(X × Y, (X ×B) ∪ (A× Y )).
Proof. Let α : R → I be a smooth braking function with the properties that
α(t) = 0 for t ≤ 1
, and α(t) = 1 for t ≥ 3
, and let β : R → R be a smooth
increasing braking function with the properties that β(t) = t for t ≤ 1
, and
β(t) = 1 for t ≥ 3
. Suppose that (u,H) and (v, J) are representations for the
SNDR pairs (X,A) and (Y,B), respectively. Let u : X → I, and v : Y → I be
given by u(x) = β(u(x)) and v(y) = β(v(y)) respectively. Define w : X×Y → I
by w(x, y) = u(x)v(y). The braking function β ensures smoothness of u and
v, and consequently of w. We have w−1(0) = (X × B) ∪ (A × Y ), as required.
Define Q : I×X × Y → X × Y as follows .
Q(t, x, y) =
(H(α(t), x), J(α(t), y)) if u(x) = v(y) = 0
(H(α(t), x), J(α(
)α(t), y)) if v(y) ≥ u(x), v(y) > 0,
(H(α(
)α(t), x), J(α(t), y)) if u(x) ≥ v(y), u(x) > 0.
We must show that Q is a smooth map, with the first coordinate independence
property with respect to the inclusion (X × B) ∪ (A × Y ) →֒ X × Y . We first
consider each part of the definition of Q separately. The first part is clearly
smooth. Let us verify that Q is smooth on the second part of its definition; the
third part is similar.
We need only focus on the component J(α(
)α(t), y). Each function
making up J(α(
)α(t), y) is smooth individually, so we need only pay extra
attention to those parts that involve flattened unit intervals, remembering that
addition and multiplication on the flattened unit interval need not preserve
smoothness, as is the case for the usual unit interval.
So let us consider α(
); it is smooth except possibly when
approaches
0 or 1, since it is here that structure curves on the flattened unit interval need
not be smooth in the usual sense. Clearly, if u(x) approaches 0 and v(y) does
not approach 0, then the braking function α ensures that
= 0 near such
points. If v(y) approaches 0, then u(x) must approach 0 too. This situation is
dealt with later.
Thus, Q, in part two of the definition, is smooth, and one can show similarly
that Q in the third part of the definition is smooth as well.
Let us now consider the overlaps of the three parts of the definition of Q.
Observe that if u(x) is in a sufficiently small neighborhood of v(y), with u(x) 6= 0
and v(y) 6= 0, then we have α(u(x)
) = 1, and so the second and third
parts of the definition of Q coincide here. Thus, it remains only to show that Q
is smooth as u(x) and v(y) both approach 0.
If Q is smooth in each of its coordinates then it is smooth, so consider the
coordinate involving the map J . Let c : R → I×X × Y be a structure that is
given by c(s) = (t(s), x(s), y(s)). Then, the map c1 : R → I× Y , given by
c1(s) =
(α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0
u(x(s))
v(y(s))
)α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0
(α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0
is a map satisfying the conditions of Definition 3.2, since its second coordinate is
smooth, but its first coordinate may be singular as v(y(s)) ( and hence u(x(s)))
approaches 0. Since J has the first coordinate independence property, the map
(Joc1)(s) =
J(α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0
u(x(s))
v(y(s))
)α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0
J(α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0
is smooth. Thus, Q ◦ c is smooth, and since c is arbitrary, Q is smooth. In a
similar way, the coordinate of Q involving H can be shown to be smooth.
We now verify that Q satisfies the required boundary conditions. When t =
0, all three lines defining Q reduce to (H(0, x), J(0, y)) = (x, y). Let x ∈ A and
y ∈ B; then u(x) = v(y) = 0. Therefore, Q reduces to (H(α(t), x), J(α(t), y)) =
(x, y). If x ∈ A and y /∈ B, then Q is given by the second part of its definition,
which reduces to (H(α(t), x), J(0, y)). The case when x /∈ A and y ∈ B is
similar. If t = 1 and 0 < w(x, y) < 1 then either 0 < u(x) < 1 or 0 < v(y) < 1.
Suppose that 0 < u(x) < 1. Then either u(x) ≤ v(y) or v(y) < u(x). If
u(x) ≤ v(y), then Q is given by the second part of its definition, which reduces
to (H(1, x), J(α(
, y)) ∈ i(A)× Y . If v(y) < u(x), then the third part of the
definition of Q applies and Q reduces to (H(α(
), x), J(1, y)) ∈ X × j(B).
Finally, we must show that for any f ∈ FX×Y , f ◦Q has the first coordinate
independence property with respect to the inclusion (X×B)∪(A×Y ) →֒ X×Y .
To this end, consider a map c : R → I×X×Y , given by c(s) = (t(s), x(s), y(s)).
Let {sn} be a sequence of real numbers converging to s∗ with c(sn) ∈ (X×Y )−
((A× Y ) ∪ (X ×B)), and c(s∗) ∈ (A× Y ) ∪ (X ×B). There are three cases to
consider.
• Suppose that c(s∗) ∈ A×B. Then x(s∗) ∈ A and y(s∗) ∈ B. The fact that
H and J have the first coordinate independence property with respect to
i and j respectively means that each coordinate of Q is smooth, and so Q
is smooth.
• Suppose that c(s∗) ∈ A × Y , and that y(s∗) /∈ B. Then at each of the
points c(sn), (Q ◦ c)(sn) is given by the second part of the definition of
Q, for n large enough. Since x(s∗) ∈ A, the component of Q involving H
is smooth, since H has the first coordinate independence property. For
any s in a neighborhood of s∗, α(
u(x(s))
v(y(s))
) = 0. Thus, the component of
Q involving J is constant for s in a neighborhood of s∗, and so is smooth
there.
• The case with c(s∗) ∈ X ×B, and x(s∗) /∈ A is similar to the second case
above.
For the last part of the theorem, suppose that (u,H) represent (X,A) as an
SDR pair. If we replace u by u′ = 1
u, then (u′, H) also represent (X,A) as an
SDR pair. Making the above constructions now with u′ in place of u, it follows
that w(x, y) < 1 for all (x, y) and so Q(1, x, y) ∈ (X × B) ∪ (A × Y ). This
completes the proof. �
4 Cofibrations
In this section, we show that for a subspace A ⊆ X that is closed in the under-
lying topology, the inclusion i : A → X is a cofibration if and only if (X,A) is
an SNDR pair.
Definition 4.1 Let i : A→ X be a cofibration. We call i a cofibration with
FCIP if any homotopy extension can be chosen to have the FCIP with respect
to i.
Using the equivalent formulation of the notion of cofibration, given by Lemma
2.5, we may restate Definition 4.1 as follows: A cofibration i : A → X is a
cofibration with the FCIP if and only if the map G that we may fill in to
complete the commutative diagram
(0×X) ∪ (I− ×A) h //
I− ×X
may be chosen to have the FCIP with respect to the inclusion i.
We have the following result, which corresponds to a similar topological
result.
Lemma 4.1 A smooth map i : A → X is a cofibration (with the FCIP) if
and only if (0 × X) ∪ (I− × A) is a retract of I− × X, (where the retraction
r : I− ×X → (0×X) ∪ (I− ×A) has the FCIP ).
Proof. In the one direction, suppose that (0 × X) ∪ (I− × A) is a retract of
I− ×X . We wish to complete the following diagram:
(0×X) ∪ (I− ×A) h //
I− ×X
By hypothesis, there exists r : I−×X → (0×X)∪ (I− ×A) such that r ◦ j = 1.
Define G = h ◦ r. If r has the FCIP, then so does h ◦ r.
Conversely, suppose that i : A → X is a cofibration (with the FCIP). We
may find a map r such that the diagram
(0×X) ∪ (I− ×A) 1//
(0 ×X) ∪ (I− ×A)
I− ×X
commutes. Thus, r ◦ j = 1. If i is cofibration with the FCIP with respect to i,
then r can be chosen to have the FCIP. �
The next theorem shows the relationship between cofibrations, retracts and
SNDR pairs.
Theorem 4.1 Let i : A → X be an inclusion, with A closed in the underlying
topology of X. Then the following are equivalent.
(1) The pair (X,A) is an SNDR pair.
(2) There is a smooth retraction r : I− × X → (0 ×X) ∪ (I− × A) with the
FCIP.
(3) The map i : A→ X is a cofibration with the FCIP.
Proof. To show that (1) and (2) are equivalent, note that the pair (I−×X, (0×
X) ∪ (I− × A)) is an SDR pair, as a consequence of Lemma 3.1 and Theorem
3.1. Let (w,Q) be a representation for the pair (I− ×X, (0×X)∪ (I− ×A)) as
an SDR pair, and let Q be constructed as in Theorem 3.1. Define
r : I− ×X → (0×X) ∪ (I− ×A)
by r(t, x) = Q(1, t, x), where (t, x) ∈ I− ×X . We observe that r has the FCIP,
since Q has this property, and Q has this property since each of its components
has this property.
The equivalence of (2) and (3) is Lemma 4.1.
We need only show that (2) implies (1). Let r : I−×X → (0×X)∪ (I−×A)
be a retraction with the FCIP with respect to i. Define H : I × X → X
by H(t, x) = (πX ◦ r)(α(t), x), where πX is the projection onto the second
coordinate, and α : R → R is a braking function with the following properties:
α(t) = 0 for t ≤ 0, α(t) = 1 for t ≥ 3
, and 0 < α(t) < 1 for 0 < t < 3
. This
braking function is necessary to ensure smoothness at the right endpoint of the
flattened unit interval I. Smoothness at the left endpoint is already taken care
of by the fact that r is defined in terms of the left flattened unit interval. The
map H satisfies the following properties:
• H has the FCIP since r has this property.
• H(0, x) = (πX ◦ r)(0, x) = x, for x ∈ X .
• H(t, x) = (πX ◦ r)(α(t), x) = x, for x ∈ A.
We now construct u : X → I. Let πI : I×X → I denote the projection onto I.
Define a smooth function β : R → R by
β(t) =
0 if t ≤ 0
t2 if t > 0.
Now, define u : X → I by
u(x) =
β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt
β(α(t))dt
It is clear that u is a smooth mapping.
We now verify that (u,H) represents (X,A) as an SNDR pair.
(1) Let x ∈ A. Clearly, (πI ◦ r)(1, x) = 1 and πI ◦ r)(α(t), x) = α(t), and so
β(α(t)− (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt = 0. Thus, u(x) = 0, for all x ∈ A.
(2) Suppose that x ∈ X−A. Since 0×(X−A) is open in the underlying topology
on (0×X)∪ (I− ×A), we may choose an open neighborhood W ⊆ 0× (X −A)
of (0, x). Since r is continuous, there is a neighborhood V ⊆ I− ×X such that
r(V ) ⊆W ⊆ 0× (X −A). Now, consider the mapping qx : I → I×X , given by
qx(t) = (α(t), x), for each x ∈ X . This is clearly smooth. Thus, there exists a
neighborhood U ⊆ I− such that qx(U) ⊆ V . In other words, U × {x} ⊆ V . So,
we have (πI ◦ r)(α(t), x) = 0, for all t ∈ U . Thus, we have
u(x) =
β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt +
β(α(t))dt
β(α(t))dt
Combining this with part (1), we deduce that u−1(0) = A.
(3) Suppose that x is such that u(x) < 1. There must be a neighborhood U of I
such that (πI◦r)(1, x)(πI ◦r)(α(t), x) > 0, for t ∈ U . Thus (πI◦r)(1, x) > 0, but
this implies that r(1, x) ∈ I×A, and hence H(1, x) ∈ A. The proof is complete.
5 The Mapping Cylinder
In this section we show that the inclusion of X into the flattened mapping
cylinder If of a map f : X → Y is a cofibration with the FCIP.
Theorem 5.1 Let f : X → Y be a smooth map. Then, the pair (If , X) is an
SNDR pair.
Proof. Let α : I → R be a smooth braking function with the following proper-
ties: α(t) = 0 if 0 ≤ t ≤ 1
, α(t) = 1 if 3
≤ t ≤ 1, 0 < α(t) < 1, otherwise. Define
two more braking functions α1, α2 : I → R as follows: α1(0) = 0, 0 < α1(t) < 1
if 0 < t < 3
, α1(t) = 1 if
≤ t ≤ 1, and α2(t) = 0 if 0 ≤ t ≤ 34 , α2(t) = 1
≤ t ≤ 1. Now, define u : If → I by u([t, x]) = α1(t) and u([y]) = 1, for
(t, x) ∈ I×X and y ∈ Y . Define H : I× If → If by
H(s, [t, x]) = [(1 − α(s))t+ α(s)α2(t), x] if (t, x) ∈ I×X
H(s, [y]) = [y] if y ∈ Y .
That u is smooth comes from the fact that it is smooth when restricted to
each component of the coproduct (I×X)⊔Y ; it is thus smooth on the quotient
To see that the map H : I × If → If is smooth, note that since we are
working in a cartesian closed category, products commute with quotients, i.e. if
q is quotient, then so is 1× q, where 1 is an identity map. Thus, we may think
of H as being defined on the space
(I× I×X) ⊔ (I× Y )
where ∼ is the identification (t, 1, x) = (t, f(x)) for t ∈ I, and x ∈ X . Since H is
smooth when restricted to each component of the coproduct (I×I×X)⊔(I×Y ),
H is smooth on the quotient I× If .
We now verify that (u,H) is a representation for (If , X) as an SNDR pair.
• u−1(0) = [0, x] = i0(X).
• H(0, [t, x]) = [t, x] and H(0, [y]) = [y].
• H(s, [0, x]) = [0, x].
• If u[t, x] < 1, then t < 3
and so α2(t) = 0. Thus, H(1, [t, x]) = [0, x].
This completes the proof. �
Finally, we have the following important corollary.
Corollary 5.1 Given any smooth map f : X → Y , the inclusion X →֒ If is a
cofibration with the FCIP.
6 The Exact Sequence of a Cofibration
Our aim in this section is to show how one can use SNDR pairs to prove the
existence of the right exact Puppe sequence. We state the result in Theorem
6.1 and break the proof of the result up into a number of lemmas. We follow
the method used by Whitehead [12] for the topological case.
Throughout this section we work in the category FRL∗ of pointed Fr—’olicher
spaces, and basepoint preserving smooth maps.
Theorem 6.1 Let W be an object in FRL∗, and suppose that i : A →֒ X is a
cofibration in FRL∗. For any basepoint x0 ∈ A ⊆ X there is a sequence
. . . // [
A,W ]
Ti,W ]
X,W ]
A,W ] // . . .
. . . // [
A,W ]
// [Ti,W ]
// [X,W ]
// [A,W ]
which is an exact sequence in SETS∗, where j : X → Ti is the inclusion dis-
cussed in Paragraphe 2.4 and k : Ti →
A is the quotient map defined below.
It is, in fact, possible to prove that the sequence above is an exact sequence
of groups as far as
A,W ] and that the morphisms to this point are group
homomorphisms, but we shall not do so here.
The reduced(flattened)suspension of a pointed Frölicher space X is de-
fined as
X = (I/{0, 1}) ∧X,
where the reduced join is defined as for topological spaces with the identified
set taken as basepoint, and with 0 the basepoint of I.
In this section, whenever we refer to the suspension of a space , we mean
the reduced flattened suspension defined above.
Lemma 6.1 If (x,A) is an SNDR pair and p : X → X/A the quotient map,
then the sequence
i // X
p // X/A
is right exact.
Proof. To show that the given sequence is right exact we must show that for
any Frölicher space W the following sequence is exact in SETS:
[X/A,W ]
// [X,W ]
// [A,W ] .
It is easy to see that im p∗ ⊆ ker i∗. To see the reverse inclusion, let g : X →
W be an element of [X,W ], with g|A ≃ w0 (rel w0), where w0 ∈ W . Since
i // X is an SNDR pair, the map i is a cofibration, and so we may extend
w0 to a smooth map g
′ : X → W such that g′ ≃ g. But g′ is constant on A,
and so there exists a smooth map g1 : X/A → W such that p∗(g1) = g′. This
shows that ker i∗ ⊂ im p∗. �
Lemma 6.2 For any smooth map f : X → Y , the sequence
f // Y
l // Tf
is right exact, where l is the usual inclusion of Y into the mapping cone; i.e.
y 7→ [y] ∈ Tf .
Proof. One can show that there is a homotopy commutative diagram
i ��@
// Tf
where i, j, and l are the usual inclusions, and p is the quotient map that collapses
away {0} ×X to a point. Since, by Theorem 5.1, (If , X) is an SNDR pair, it
follows from Lemma 6.1 that the sequence
i // If
p // Tf
is right exact. It is fairly easy to show that j : Y → If is a homotopy equivalence.
Therefore, the sequence
f // Y
l // Tf
is right exact. �
Lemma 6.3 For any smooth map i : A → X, there is an infinite right exact
sequence
i // X
// Ti
// . . . i
// Tin−2
// Tin−1
// . . .
where in, n ≥ 1, are inclusion maps.
Proof. The pair (Ti, X) is an SNDR pair. The representation for the pair
(If , X) in Theorem 5.1 can be adapted to show this. One iterates the procedure
of Lemmas 6.1 and 6.2. �
One can easily see that there is an isomorphism between Ti/X and
Define q : Ti →
A to be the map which identifies X ⊂ Ti to a point, followed
by the isomorphism Ti/X →
Lemma 6.4 The sequence
// Ti
is right exact.
Proof. As noted above the pair (Ti, X) is an SNDR pair. We have the com-
mutative diagram
// Ti
where p : Ti → Ti/X is the identification map, and q0 : Ti/X →
A is an
isomorphism. The top line of the diagram is right exact, by Lemma 6.1, and so
the sequence
// Ti
is right exact. �
There is a commutative diagram
// Ti
where q1 is a homotopy equivalence. ( See Whitehead [12] for more details of
this map. ) Using commutative diagrams of this form, one can now proceed
almost exactly as one does in the topological situation, as in Whitehead [12] for
example, to get the following infinite right exact sequence:
i // X
// Ti
// . . .
. . . //
// . . .
The definition of right exactness now gives us the exact sequence of Theorem
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	Preliminaries
	Basic Constructions of Homotopy Theory in FRL
	Our Approach to Homotopy Theory in FRL
	Flattened Structures on the Unit Interval
	Some Properties of Smooth Functions between the Flattened Unit Intervals
	Homotopy in FRL and Related Objects
	Cofibrations in FRL
	Smooth Neighborhood Deformation Retracts
	SNDR pairs and SDR pairs
	Cofibrations
	The Mapping Cylinder
	The Exact Sequence of a Cofibration
