topological cofiber sequence

under construction



topology (point-set topology, point-free topology)

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Basic facts




We discuss in detail the realization of the abstract concept of cofiber sequences in its explicit incarnation in point-set topology, the way it is traditionally presented in topology textbooks.

Hence we use the concepts of homotopy equivalence instead of weak homotopy equivalence. For discussion using the latter in the context of the classical model structure on topological spaces see instead at Introduction to Homotopy Theory the section Homotopy fiber sequences.


Throughout, write I[0,1]I \coloneqq [0,1] \subset \mathbb{R} for the closed interval equipped with its Euclidean metric topology.

Homotopy equivalences


For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a homotopy

η:f Lg \eta \colon f \,\Rightarrow_L\, g

is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the standard cylinder object over XX: the product space of XX with the Euclidean closed interval, such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.

(graphics grabbed from J. Tauber here)


(homotopy equivalence)

A continuous function f:XYf \colon X \to Y is called a homotopy equivalence if there exist

  1. a continuous function g:YXg \colon Y \to X;

  2. homotopies (def. 1) of the form

    gfid XAAAAfgid Yg \circ f \Rightarrow id_X \phantom{AAAA} f \circ g \Rightarrow id_Y


(contractible topological space)

A topological space XX is called contractible if the unique map to the point space X*X \to \ast is a homotopy equivalence (def. 2).



(Hurewicz cofibration)

A continuous function i:AXi \colon A \longrightarrow X is called a Hurewicz cofibration or just cofibration if it satisfies the homotopy extension property in that:

  • for any topological space YY,

  • for all continuous functions f:AY f \colon A\to Y, f˜:XY\tilde{f}:X\to Y such that f˜i=f\tilde{f}\circ i=f

    A f Y i f˜ X \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }
  • and any left homotopy F:A×IYF \colon A\times I\to Y such that F(,0)=fF(-,0)=f

there is a homotopy F˜:X×IY\tilde{F} \colon X\times I\to Y such that

  • F˜(i×id I)=F\tilde{F}\circ(i\times id_I)=F

    A×I F Y i×id I F˜ X×I \array{ A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X \times I }
  • and F˜(,0)=f˜\tilde{F}(-,0)=\tilde{f}

    A id A×const 0 A×I F Y i×id I i×id I F˜ X id X×const 0 X×IAAA=AAAA f Y i f˜ Xcco \array{ A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X &\underset{id_X \times const_0}{\longrightarrow}& X \times I } \phantom{AAA} = \phantom{AAA} \array{ \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }cco }

If the image i(A)Xi(A) \subset X is a closed subset, then ii is called a closed Hurewicz cofibration.


(retracts of relative cell complex inclusions are closed Hurewicz cofibrations)

If AA is a topological space and i:AXi \colon A \hookrightarrow X is a relative cell complex inclusion, then ii is a closed Hurewicz cofibration (def. 3).

Also every retract of such ii (this sense) is a closed Hurewicz cofibration.

Pushouts / space attachments

Cofiber sequences (below) are constructed by iterated pushouts in the category Top of topological spaces with continuous functions between them (space attachments), see at Top – Universal constructions.


(quotient space by a subspace)

Let XX be a topological space and AXA \subset X a non-empty subset. Consider the equivalence relation on XX which identifies all points in AA with each other. The resulting quotient space is denoted X/AX/A.

Notice that X/AX/A is canonically a pointed topological space, with base point the equivalence class A/AX/AA/A \subset X/A of AA.

If A=A = \emptyset is the empty space, then one defines

X/X +X* X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast

to be the disjoint union space of XX with the point space. This is no longer a quotient space, but both constructions are unified by the pushout i:AXi \colon A \to X along the map A*A \to \ast, equivalently the cokernel of the inclusion:

A i X (po) * X/A. \array{ A &\overset{i}{\hookrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A } \,.

(union of two open or two closed subspaces is pushout)

Let XX be a topological space and let A,BXA,B \subset X be subspaces such that

  1. A,BXA,B \subset X are both open subsets or are both closed subsets;

  2. they constitute a cover: X=ABX = A \cup B

Write i A:AXi_A \colon A \to X and i B:BXi_B \colon B \to X for the corresponding inclusion continuous functions.

Then the commuting square

AB A (po) i A B i B X \array{ A \cap B &\longrightarrow& A \\ \downarrow &(po)& \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X }

is a pushout square in Top (see there).

By the universal property of the pushout this means in particular that for YY any topological space then a function of underlying sets

f:XY f \;\colon\; X \longrightarrow Y

is a continuous function as soon as its two restrictions

f| A:AYAAAAf| A:BY f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y

are continuous.


Clearly the underlying diagram of underlying sets is a pushout in Set. Therefore (by this prop.) we need to show that the topology on XX is the final topology induced by the set of functions {i A,i B}\{i_A, i_B\}, hence that a subset SXS \subset X is an open subset precisely if the pre-images (restrictions)

i A 1(S)=SAAAAandAAAi B 1(S)=SB i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B

are open subsets of AA and BB, respectively.

Now by definition of the subspace topology, if SXS \subset X is open, then the intersections ASAA \cap S \subset A and BSBB \cap S \subset B are open in these subspaces.

Conversely, assume that ASAA \cap S \subset A and BSBB \cap S \subset B are open. We need to show that then SXS \subset X is open.

Consider now first the case that A;BXA;B \subset X are both open open. Then by the nature of the subspace topology, that ASA \cap S is open in AA means that there is an open subset S AXS_A \subset X such that AS=AS AA \cap S = A \cap S_A. Since the intersection of two open subsets is open, this implies that AS AA \cap S_A and hence ASA \cap S is open. Similarly BSB \cap S. Therefore

S =SX =S(AB) =(SA)(SB) \begin{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned}

is the union of two open subsets and therefore open.

Now consider the case that A,BXA,B \subset X are both closed subsets.

Again by the nature of the subspace topology, that ASAA \cap S \subset A and BSBB \cap S \subset B are open means that there exist open subsets S A,S BXS_A, S_B \subset X such that AS=AS AA \cap S = A \cap S_A and BS=BS BB \cap S = B \cap S_B. Since A,BXA,B \subset X are closed by assumption, this means that AS,BSXA \setminus S, B \setminus S \subset X are still closed, hence that X(AS),X(BS)XX \setminus (A \setminus S), X \setminus (B \setminus S) \subset X are open.

Now observe that (by de Morgan duality)

S =X(XS) =X((AB)S) =X((AS)(BS)) =(X(AS))(X(BS)). \begin{aligned} S & = X \setminus (X \setminus S) \\ & = X \setminus ( (A \cup B) \setminus S ) \\ & = X \setminus ( (A \setminus S) \cup (B \setminus S) ) \\ & = (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S)) \,. \end{aligned}

This exhibits SS as the intersection of two open subsets, hence as open.

A general abstract fact about pushouts which we will use repeatedly in the proofs below is the following:


(pasting law)

Consider a diagram in Top (or in any other category) of the following form:

(po) , \array{ &\longrightarrow& &\longrightarrow& \\ \downarrow &(po)& \downarrow && \downarrow \\ &\longrightarrow& &\longrightarrow& } \,,

where the left square is a pushout.

Then: The total rectangle is a pushout precisely if the right square is.

We need the following two facts (prop. 2 prop. 3) regarding the stability of cofibrations under pushout.


(pushout of cofibrations)

Let AA be a topological space and let AXA \hookrightarrow X be a closed Hurewicz cofibration.

Then for every continuous function f:AYf \colon A \to Y, the pushout f *if_\ast i in

A f Y i (po) f *i X XAY \array{ A &\overset{f}{\longrightarrow}& Y \\ {}^{\mathllap{i}}\downarrow &(po)& \downarrow^{\mathrlap{f_\ast i}} \\ X &\longrightarrow& X \underset{A}{\sqcup} Y }

is also a closed Hurewicz cofibration. Moreover, if in addition

  1. ii is a retract of a relative cell complex inclusion, then so is f *if_\ast i,

  2. ii is a homotopy equivalence, then so is f *if_\ast i.

Prop. 2 is a consequence of the existence of the Strøm model structure Top StromTop_{Strom} and the classical model structure on topological spaces Top QuillenTop_{Quillen} and of the fact that the identity functors Top StromididTop QuillenTop_{Strom} \underoverset{\underset{id}{\longrightarrow}}{\overset{id}{\longleftarrow}}{\bot} Top_{Quillen} constitute a Quillen adjunction.


(quotient by contractible closed subspace)

Let i:AXi \colon A \longrightarrow X be a closed Hurewicz cofibration.

If f:AYf \colon A \to Y is a homotopy equivalence (def. 2) then its pushout i *fi_\ast f in

A i X f (po) i *(f) Y YAX \array{ A &\overset{i}{\longrightarrow}& X \\ {}^{\mathllap{f}}\downarrow &(po)& \downarrow^{\mathrlap{i_\ast(f)}} \\ Y &\longrightarrow& Y \underset{A}{\sqcup} X }

is also a homotopy equivalence.

In particular if AA is a contractible topological space (example 1), then the coprojection

XX/A X \longrightarrow X/A

to the quotient space (example 3) is a homotopy equivalence.

Prop. 3 is again a consequence of the existence of the Strøm model structure Top StromTop_{Strom}: It is the statement that Top StromTop_{Strom} is a left proper model category which follows (this cor.) since all its objects are evidently cofibrant. An elementary proof in point-set topology is offered in (Hatcher, prop. 0.17)

Mapping cones


(topological cylinder and cone)

Let XX be a topological space. Then

  1. the standard cylinder on XX is the product topological space

    Cyl(X)X×[0,1] Cyl(X) \coloneqq X \times [0,1]
  2. the standard cone on XX is the quotient space (example 3)

    Cone(X)Cyl(X)/(X×{0}) Cone(X) \coloneqq Cyl(X) / (X \times \{0\})

    of the standard cylinder by the subspace X×{0}X×[0,1]X \times \{0\} \subset X \times [0,1].

    Equivalently this is the following pushout in Top

    X×{0} Cyl(X) (po) * Cone(X) \array{ X \times \{0\} &\hookrightarrow& Cyl(X) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(X) }

Let XX be a topological space. Then the canonical inclusions

XCyl(X)AAAA*Cone(X) X \longrightarrow Cyl(X) \phantom{AAAA} \ast \longrightarrow Cone(X)

are homotopy equivalences. Hence every cone is a [contractible topological space]] (example 1).


(mapping cylinder and mapping cone)

Let f:XYf \colon X \to Y be a continuous function between topological spaces. Then

  1. the mapping cylinder of ff is the space attachment

    Cyl(f)Y fCyl(X) Cyl(f) \coloneqq Y \cup_f Cyl(X)

    of YY with the cylinder on XX, according to def. 4, hence the following pushout in Top

    X f Y (po) Cyl(X) Cyl(f) \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cyl(X) &\longrightarrow& Cyl(f) }
  2. the mapping cone of ff is the space attachment

    Cone(f)Y fCone(X) Cone(f) \coloneqq Y \cup_f Cone(X)

    hence the following pushout in Top:

    X f Y (po) Cone(X) Cone(f) \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cone(X) &\longrightarrow& Cone(f) }

In summary, def. 4 and def. 5 say that for f:XYf \colon X \to Y a continuous function then we have a pasting of pushout diagrams in Top of the following form:

X f Y i 1 (po) X i 0 Cyl(X) Cyl(f) (po) (po) * Cone(X) Cone(f). \array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow \\ X &\stackrel{i_0}{\to}& Cyl(X) &\to & Cyl(f) \\ \downarrow &(po)& \downarrow &(po)& \downarrow \\ {*} &\to& Cone(X) &\to& Cone(f) } \,.

Since XCone(X)X \to Cone(X) is a closed Hurewicz cofibration , the pasting law together with prop. 2 therefore implies that also YCone(f)Y \to Cone(f) is a closed Hurwicz cofibration.



For XX a topological space, then the mapping cone (def. 5) of the unique function X*X \to \ast to the point space is

SXCone(X*)Cone(X)XCone(X). S X \coloneqq Cone(X \to \ast) \simeq Cone(X) \underset{X}{\sqcup} Cone(X) \,.

This is called the suspension of XX.

The mapping cone of a map XYX \to Y is to be thought of as the homotopy-quotient of the YY by XX, as opposed to be the naive quotient. This is made precise by the following two statements, lemma 2 and lemma 3:


If f:XYf \colon X \to Y is a closed Hurewicz cofibration (def. 3), then the coprojection

Cone(f)Y/f(X) Cone(f) \longrightarrow Y/f(X)

from its mapping cone (def. 5) to the naive quotient space (example 3) is a homotopy equivalence.


Consider the diagram

X f Y (po) Cone(X) Cone(f) (po) * Cone(f)/Cone(X) \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cone(X) &\longrightarrow& Cone(f) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(f)/Cone(X) }

Since ff is a closed Hurewicz cofibration, so is Cone(X)Cone(f)Cone(X) \to Cone(f), by prop. 2. Since Cone(X)*Cone(X) \to \ast is a homotopy equivalence (lemma 1), so is Cone(f)Cone(f)/Cone(X)Cone(f) \to Cone(f)/Cone(X), by prop. 3. But by lemma 3 there is also a homeomorphism Cone(f)/Cone(X)Y/f(X)Cone(f)/Cone(X) \simeq Y/f(X).


Let f:XYf \colon X \to Y be a continuous function such that the image f(X)Yf(X) \subset Y is a closed subset.

Then there is a homeomorphism

Cone(f)/Cone(X)Y/f(X) Cone(f)/Cone(X) \simeq Y / f(X)

between the quotient space (example 3) of the mapping cone of ff (def. 5) by the cone of XX (def. 4) and the quotient space of YY be the image of XX.


Consider the following diagram in Top:

f(X) Y (po) f(Cone(X)) Cone(f) (po) * Cone(f)/Cone(X) \array{ f(X) &\longrightarrow& Y \\ \downarrow &(po)& \downarrow \\ f(Cone(X)) &\longrightarrow& Cone(f) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(f)/Cone(X) }

Here the top square is a pushout by example 4, while the bottom square is a pushout by def. 5. Hence the total rectangle is also a pushout, by the pasting law (prop. 1). But that total rectangle is the defining pushout for Y/f(X)Y/f(X) (by example 3). Hence the statement follows by the universal property of the pushout.

Cofiber sequences


Let f:XYf \colon X \to Y be a continuous function with closed image f(X)Yf(X) \subset Y. Write Cone(g)Cone(g) in

X f Y g Cone(f) Cone(g) \array{ X &\overset{f}{\longrightarrow}& Y &\overset{g}{\longrightarrow}& Cone(f) &\overset{}{\longrightarrow}& Cone(g) }

for the mapping cone (spring) of the inclusion gg of YY into the mapping cone of ff.

Then the canonical quotient coprojection

Cone(g)ΣX Cone(g) \to \Sigma X

to the suspension of XX (example 5) is a homotopy equivalence (def. 2).


Since g:YCone(f)g \colon Y \to Cone(f) is a closed Hurewicz cofibration (by remark 1), lemma 2 gives that

Cone(g)Cone(f)/g(Y) Cone(g) \to Cone(f)/g(Y)

is a homotopy equivalence. But then there is the following evident homeomorphism

Cone(f)/g(Y)=(Y fCone(X))/YCone(X)/XSX. Cone(f)/g(Y) = (Y \cup_f Cone(X))/Y \simeq Cone(X)/X \simeq S X \,.

(graphics taken from Muro 10)

Hence from every f:XYf \colon X \to Y with closed image, we get long sequences

X f Y g Cone(f) Cone(g) homotopyequivalence SX Sf SY \array{ X &\overset{f}{\longrightarrow}& Y &\overset{g}{\longrightarrow}& Cone(f) &\longrightarrow& Cone(g) \\ && && && \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}} \\ && && && S X &\overset{S f}{\longrightarrow}& S Y &\longrightarrow& \cdots }

Pointed cofiber sequence


Revised on July 4, 2017 14:22:39 by Urs Schreiber (