CW-pair (changes) in nLab

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Idea

In algebraic topology, by a CW-pair (X,A)(X,A) is meant a CW-complex XX equipped with a sub-complex inclusion A↪XA \hookrightarrow X.

The concept appears prominently in the discussion of ordinary relative homology and generally in the Eilenberg-Steenrod axioms for generalized homology/generalized cohomology.

Properties

General

For instance (Hatcher 2002, prop. A.5).

For instance (Hatcher 2002, prop. 2.22).

Collapsing contractible subcomplexes

\begin{proposition}\label{Pinching} If the sub-complex AA is contractible, then the quotient coprojection is a homotopy equivalence.

*⟶∼A−−−⇒−−−X⟶∼X/A. \ast \overset{\sim}{\longrightarrow} A \phantom{---} \Rightarrow \phantom{---} X \overset{\sim}{\longrightarrow} X/A \,.

\end{proposition} (e.g. Hatcher p 11).

\begin{example}\label{SphereWithArcAttached} Let XX be the pushout (in Top) which attaches an arc — identified as A⊂XA \subset X — to the 2-sphere, connecting a pair of distinct points:

\begin{imagefromfile} “file_name”: “PinchingSphereAtTwoPoints.png”, “float”: “right”, “width”: 300, “unit”: “px”, “margin”: { “top”: -30, “bottom”: 20, “right”: 0, “left”: 10 }, “caption”: “(from Hatcher 2002)” \end{imagefromfile}

\begin{tikzcd} & S^0 \ar[r, hook] \ar[d, hook] \ar[dr, phantom, { \mbox{ (po) } }{pos=.8,gray, scale=.7}] & S^2 \ar[d, hook] \ & \mathllap{ A := \, } D^1 \ar[r] & X \end{tikzcd}

and let BB be any arc inside S 2S^2 connecting these two distinct points. Then Prop. \ref{Pinching} gives homotopy equivalences

S 2/S 0=X/A⟶∼X⟶∼X/B=S 2∨S 1. S^2 / S^0 \,=\, X/A \overset{\sim}{\longrightarrow} X \overset{\sim}{\longrightarrow} X/B \,=\, S^2 \vee S^1 \,.

\end{example} (e.g. Hatcher 2002 p 11).

In generalization of this example:

\begin{example} For Σ g,n 2≔Σ g 2∖{s 1,⋯,s n}\Sigma^2_{g,n} \,\coloneqq\, \Sigma^2_g \setminus \{s_1, \cdots, s_n\} the result of subjecting a closed surface Σ g 2\Sigma^2_g to n≥2n \geq 2 punctures, the one-point compactification (Σ g,n 2) *\big(\Sigma^2_{g,n}\big)^\ast is homotopy equivalent to the wedge sum of the original surface with n−1n-1 circles:

(Σ g,n 2) *≃Σ g 2∨⋁ n−1S 1. \big(\Sigma^2_{g,n}\big)^\ast \;\simeq\; \Sigma^2_g \,\vee\, \textstyle{\bigvee_{n-1}} S^1 \,.

This follows from Prop. \ref{Pinching} by generalizing Ex. \ref{SphereWithArcAttached} as follows: Instead of a single arc, take AA and BB there to each be, in themselves, the linear graph consisting of n−1n-1 arcs (edges ) and observe that that after identifying thevertices all with each other, this gives the wedge sum of (n−1)(n-1) circles.

Then take X≔Σ g 2∪AX \coloneqq \Sigma^2_g \cup A to be the result of attaching this graph to the closed surface by gluing the vertices to the marked points (the would-be punctures) and observe that (Σ g,n 2) *=X/A(\Sigma^2_{g,n})^\ast = X/A. With BB similarly identified as the corresponding graph but now embedded inside Σ g 2\Sigma^2_g, we have X/B=Σ g 2∨⋁ n−1S 1X/B = \Sigma^2_g \,\vee\, \textstyle{\bigvee_{n-1}} S^1.

\end{example}

References

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), webpage]

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 5.1 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)