So I was watching this talk by Peter Scholze, where at around 9:50 he says that the homology of the quotient $X := H^3/SL_2(\mathbb{Z}[i])$ only has homology in degrees 0, 1, and 2, because it's noncompact. Here $H^3$ is hyperbolic upper 3-space. (I understand that homology must vanish beyond degree 3, so the question is why the top dimensional homology vanishes)
I'm trying to understand why the noncompactness makes the 3-dimensional homology vanish. In particular I'd be interested in the relevant facts/theorems, and if possible, references would be awesome.
Other simpler but enlightening examples would be appreciated as well.
As a side question - this space $X$ is triangularizable right?
This is a consequence of Poincare duality. Let $X$ be a connected $n$-manifold (without boundary). Then by Poincare duality, there is an isomorphism $H_n(X;\mathbb{Z})\cong H^0_c(X;\omega)$ where $\omega$ is the orientation local system of $X$. Now if $X$ is not compact, $H^0_c(X;A)=0$ for any local system $A$. Indeed, if $\alpha$ is any nonzero compactly supported $0$-cochain on $X$, let $\gamma$ be a singular $1$-simplex (i.e., a path) in $X$ which starts at a point in the support of $\alpha$ and ends at a point not in the support of $\alpha$. Then $\alpha(\partial\gamma)$ is nonzero. Thus $\alpha$ is not a cocycle. So there are no nonzero compactly supported $0$-cocycles, and so $H^0_c(X;A)=0$ trivially.
Thus if $X$ is a connected non-compact $n$-manifold, $H_n(X;\mathbb{Z})=0$.