Vrftsjtryk

If $X$ is a space, then let us say that $pi_0^infty(X)$ is the set of equivalence classes of proper maps $[0,infty) to X$, modulo proper homotopy (the map $[0,1] times [0,infty) to X$ should be proper). Call an element of this set an "end of X".

If $X$ is compact, $X$ has no ends. If $X = Bbb R$, then $X$ has two ends, corresponding to the identity and negation maps $[0, infty) to Bbb R$. If $X = Bbb R^n$ for $n > 1$, then $X$ has one end.

In fact, generalizing this, if $M^circ$ is the interior of some compact manifold with boundary $M$, then we may identify $pi_0^infty(X) = pi_0(partial M)$.

Suppose $M$ is connected and thus, if $M^circ$ has more than one end, then $H_0(partial M;Bbb Z/2)$ is larger than 1-dimensional (all (co)homology groups will have $Bbb Z/2$ coefficients from now on); the relative long exact sequence then implies that $H_1(M, partial M)$ is non-trivial ; applying Poincare-Lefschetz duality we find that $H^{n-1}(M)$ is nonzero.

So we conclude: if $M$ is a connected compact manifold with boundary whose interior has more than on end, then $H^{n-1}(M) neq 0$. This fits with situations we see in practice: the easiest way to construct $M$ with two ends is to take the interior of $N times [0,1]$ for $N$ a closed connected manifold.

After all this, a new definition. An $n$-dimensional homology manifold is a locally compact separable Hausdorff space so that at each point $x in M$, we have $H_k(M, M -x) = H_k(Bbb R^n, Bbb R^n - 0)$. Theorems like Alexander duality and Poincare duality continue to hold in this context.

Is it still true that a connected $n$-dimensional homology manifold with more than one end has $H^{n-1}(M) neq 0$?

The name of the game seems to be to figure out if one can find a proof just using homological duality theorems, and not using some sort of compactification to a manifold with boundary.

This question came.up in the course of answering this question, where I needed to show that certain contractible homology manifolds have 1 end. I ended up restricting to the 2-dimensional case, where a contractible homology manifold must be $Bbb R^2$.

First of all, let $X$ be a reasonably nice space, say, metrizable and locally compact. Define
$$
H^i(Ends(X))=lim_K H^i(X-K),
$$
where the direct limit is taken over compact subsets $K$ in $X$. (Similarly, one defines $H_i(Ends(X))$ by taking the inverse limit.)
In fact, these groups are the Chech cohomology groups of the space of ends of $X$ but I will not need this.

The space $X$ has more than one end if and only if
$$
tilde{H}^0(Ends(X))ne 0,
$$
where I am using the reduced cohomology. On the other hand, the cohomology with local support of $X$ satisfies
$$
H^1_c(X)cong lim_K H^1(X, X-K).
$$
Assume now that $X$ is acyclic as in your case. Then by the long exact sequence of a pair,
$$
lim_K H^1(X, X-K) cong lim_K tilde{H}^0(X-K)cong tilde{H}^0(Ends(X)).
$$
By the Alexander duality, assuming that $X$ is an $n$-dimensional homology manifold,
$$
H^1_c(X)cong H_{n-1}(X).
$$
Hence, since $X$ is acyclic, $$tilde{H}^0(Ends(X))cong H^1_c(X)cong H_{n-1}(X)=0,$$ i.e. $X$ has exactly one end.