For a space \(X\), let \(X^+\) be the one-point compactification. If the added point, denoted \(\infty\), has a neighborhood in \(X^+\) that is a cone with \(\infty\) the cone point, show that the evident map \(H_c^n(X; G) \to H^n(X^+, \infty; G)\) is an isomorphism for all \(n\).
Solution
1. \(H_c^n(X; G) = \varinjlim_K H^n(X, X - K; G)\), the direct limit over compact subsets \(K \subseteq X\).
definition of compactly supported cohomology.
2. \(H^n(X^+, \infty; G) \cong H^n(X^+, X^+ - U; G)\) for any neighborhood \(U\) of \(\infty\) (by excision).
excision (removing the complement of a neighborhood of \(\infty\)).
3. The neighborhoods \(U\) of \(\infty\) correspond to complements of compact sets \(K \subseteq X\) (since \(X^+\) is the one-point compactification).
a neighborhood of \(\infty\) is the complement of a compact set in \(X\).
4. Hence \(H^n(X^+, \infty; G) \cong \varinjlim_K H^n(X^+, X^+ - K; G) \cong \varinjlim_K H^n(X, X - K; G)\).
2 and 3 (the direct limit over neighborhoods of \(\infty\) is the direct limit over compact \(K\)).
5. The cone condition ensures the direct limit is well-behaved (the map from the direct limit to \(H^n(X^+, \infty)\) is an isomorphism).
4 and the hypothesis (the cone neighborhood of \(\infty\) makes the direct limit stabilize).
6. Hence \(H_c^n(X; G) \cong H^n(X^+, \infty; G)\).
1 and 4.
7. Q.E.D.
6.