Contents

Contents

Idea

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

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

Properties

Proposition

For $X$ a CW complex, the inclusion $A \hookrightarrow X$ of any subcomplex has an open neighbourhood in $X$ which is a deformation retract of $A$. In particular such an inclusion is a good pair in the sense of relative homology.

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

Proposition

For $(X,A)$ a CW-pair, then the $A$-relative singular homology of $X$ coincides with the reduced singular homology of the quotient space $X/A$:

$H_n(X , A) \simeq \tilde H_n(X/A) \,.$

For instance (Hatcher, prop. 2.22).

Proof

By assumption we can find a neighbourhood $A \stackrel{j}{\to} U \hookrightarrow X$ such that $A \hookrightarrow U$ has a deformation retract and hence in particular is a homotopy equivalence and so induces also isomorphisms on all singular homology groups.

It follows in particular that for all $n \in \mathbb{N}$ the canonical morphism $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an isomorphism, by homotopy invariance of relative singular homology.

Given such $U$ we have an evident commuting diagram of pairs of topological spaces

$\array{ (X,A) &\stackrel{(id,j)}{\to}& (X,U) &\leftarrow& (X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ (X/A, A/A) &\stackrel{(id,j/A)}{\to}& (X/A, U/A) &\leftarrow& (X/A - A/A, U/A - A/A) } \,.$

Here the right vertical morphism is in fact a homeomorphism.

Applying relative singular homology to this diagram yields for each $n \in \mathbb{N}$ the commuting diagram of abelian groups

$\array{ H_n(X,A) &\underoverset{\simeq}{H_n(id,j)}{\to}& H_n(X,U) &\stackrel{\simeq}{\leftarrow}& H_n(X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H_n(X/A, A/A) &\underoverset{\simeq}{H_n(id,j/A)}{\to}& H_n(X/A, U/A) &\stackrel{\simeq}{\leftarrow}& H_n(X/A - A/A, U/A - A/A) } \,.$

Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by excision and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).

References

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

Last revised on March 7, 2016 at 10:30:22. See the history of this page for a list of all contributions to it.