nLab
CW-pair

Contents

Context

Topology

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

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

In algebraic topology by a CW-pair (X,A)(X,A) is meant a CW-complex XX equipped with a sub-complex inclusion AXA \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 XX a CW complex, the inclusion AXA \hookrightarrow X of any subcomplex has an open neighbourhood in XX which is a deformation retract of AA. 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)(X,A) a CW-pair, then the AA-relative singular homology of XX coincides with the reduced singular homology of the quotient space X/AX/A:

H n(X,A)H˜ n(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 AjUXA \stackrel{j}{\to} U \hookrightarrow X such that AUA \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 nn \in \mathbb{N} the canonical morphism H n(X,A)H n(id,j)H n(X,U)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 UU we have an evident commuting diagram of pairs of topological spaces

(X,A) (id,j) (X,U) (XA,UA) (X/A,A/A) (id,j/A) (X/A,U/A) (X/AA/A,U/AA/A). \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 nn \in \mathbb{N} the commuting diagram of abelian groups

H n(X,A) H n(id,j) H n(X,U) H n(XA,UA) H n(X/A,A/A) H n(id,j/A) H n(X/A,U/A) H n(X/AA/A,U/AA/A). \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

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