Show that Proposition 4G.2 and its corollary hold also for CW complexes and covers by families of subcomplexes.
Solution
1. Proposition 4G.2 (the nerve lemma) states: if \(X\) is covered by a family of open sets \(\{U_\alpha\}\) such that every nonempty finite intersection \(U_{\alpha_1} \cap \cdots \cap U_{\alpha_k}\) is contractible, then \(X\) is homotopy equivalent to the nerve \(N\) of the cover.
statement of the proposition.
2. The same conclusion holds when \(X\) is a CW complex covered by subcomplexes \(\{X_\alpha\}\) with every nonempty finite intersection contractible.
2.1. Each subcomplex \(X_\alpha\) has an open neighborhood \(U_\alpha\) that deformation retracts onto \(X_\alpha\).
a subcomplex of a CW complex has a regular neighborhood that deformation retracts onto it.
2.2. The open sets \(U_\alpha\) form an open cover of \(X\), and every nonempty finite intersection \(U_{\alpha_1} \cap \cdots \cap U_{\alpha_k}\) deformation retracts onto \(X_{\alpha_1} \cap \cdots \cap X_{\alpha_k}\), which is contractible.
the regular neighborhoods can be chosen so that intersections of neighborhoods retract onto intersections of subcomplexes.
2.3. Hence every nonempty finite intersection of the \(U_\alpha\) is contractible.
2.
2.4. By the nerve lemma (<1>1), \(X\) is homotopy equivalent to the nerve of the cover \(\{U_\alpha\}\).
3.
2.5. The nerve of \(\{U_\alpha\}\) equals the nerve of \(\{X_\alpha\}\) (the same intersection pattern).
\(U_{\alpha_1} \cap \cdots \cap U_{\alpha_k} \neq \varnothing\) iff \(X_{\alpha_1} \cap \cdots \cap X_{\alpha_k} \neq \varnothing\).
3. Hence \(X\) is homotopy equivalent to the nerve of the subcomplex cover \(\{X_\alpha\}\).
2.
4. The corollary (that a CW complex covered by contractible subcomplexes with contractible intersections is homotopy equivalent to the nerve) follows immediately.
3 is exactly the corollary’s statement.
5. Q.E.D.
3 and 4.