One-point compactification and the existence of a contractible open neighborhood of infinity

Question

My question is related to this one. I would like to know what extra conditions a locally compact (but not compact), Hausdorff space should satisfy such that in its one point compactification the point at infinity has a contractible open neighborhood?

Answer 1

I doubt that there exists a reasonable condition on $X$ assuring that $\infty$ has some contractible neighborhood in $X^+$. Hatcher's requirement is that $\infty$ has a neighborhood in $X^+$ that is a cone with $\infty$ the cone point. This is a very special neighborhood as it assures the existence of arbitrary small contractible neighborhoods of $\infty$. If the latter holds, let us say that $X$ satisfies the ASCN-condition (this is only an adhoc notation).

In fact, it is precisely the ASCN-condition which is needed to prove $H^n_c(X)\cong H^n(X^+,\infty)$ where $X^+$ is the one-point compactification (see the comments to this question).

The existence of a "cone neighborhood" is just a convenient sufficient (but not necessary) criterion to enforce the ASCN-condition. Note that if there exists a compact cone neigborhood $U \approx CZ$ (which implies that the base $Z$ is compact), then $X$ must be $\sigma$-compact. That is, $X$ is the union of countably many compact subsets. To see this, observe that $U \backslash \{ \infty \} \approx Z \times [0,1)$ is the union of countably many compact sets.

Here is a criterion on $X$ which implies that the ASCN-condition is satisfied:

There exists a sequence of $K_n \subset X$ such that the closure $\overline{K}_n$ is compact with $\overline{K}_n \subset int(K_{n+1})$ and $\bigcup_{n=1}^\infty K_n = X$ and such that for each $n$ there exists a homotopy $H^n : (X \backslash K_n) \times [1-\frac{1}{n},1-\frac{1}{n+1}] \to X \backslash K_n$ with the following properties:

a) $H^n$ is stationary on $X \backslash K_{n+2}$

b) $H^n_t = id$ for $t = 1-\frac{1}{n}$

c) $H^n_1t(X \backslash K_n) \subset X \backslash K_{n+1}$ for $t = 1-\frac{1}{n+1}$

These conditions allow us to paste the $H^n$ to a homotopy $H : (X^+ \backslash K_1) \times I \to X^+ \backslash K_1$ with the following properties:

a) $H((X^+ \backslash K_n) \times [1-\frac{1}{n},1]) \subset X^+ \backslash K_n$

b) $H_1(X^+ \backslash K_1) = \{ \infty \}$.

This shows that the ASCN-condition is satisfied. Note that our criterion is more general than the existence of a cone neighborhood. However, it also applies only to $\sigma$-compact $X$.