I am trying to solve the following problem.
Two properly homotopic maps induce the same map on end-cohomology.
Let $X$ be a locally compact Hausdorff space and $R$ be a commutative ring. Note that end-cohomology is defined as follows: Consider the cochain complex whose $q$-th cochain is $\frac{C^q(X;R)}{C^q_{\textbf{c}}(X;R)}$. Here, $\textbf{c}$ stands for compactly supported. The Cohomology groups of this complex will be denoted by $H_\textbf e^q(X;R)$. Here, $\textbf e$ stands for the end.
Now, let Let $f,g:X\to Y$ be two proper maps between two locally compact Hausdorff space, and $\mathscr H:f\simeq g$ be a proper homotopy. I want to show, $f_*=g_*:H^q_\textbf e(X;R)\to H^q_\textbf e(Y;R)$.
My Thoughts: I am trying to using the long exact sequence $$\cdots \to H_\textbf c^q(X;R)\xrightarrow{i_X}H^q(X;R)\to H_\textbf e^q(X;R)\to H_\textbf c^{q+1}(X;R)\to \cdots$$ obtained from the short exact sequnce $$0\to C_\textbf c^q(X;R)\to C^q(X;R)\to \frac{C^q(X;R)}{C^q_\textbf c(X;R)}\to 0.$$ Since, $\mathscr H$ is a proper homotopy we have $f_*=g_*:H^q_\textbf c(X;R)\to H^q_\textbf c(Y;R)$. Also, $\mathscr H$ is a homotopy, then $f_*=g_*:H^q(X;R)\to H^q(Y;R)$. Now, I am trying to use some algebra stuff like, Snake Lemma or $5$-lemma to prove the problem. But cannot proceed further.
Any help will be appreciated. Thanks in advance.
Best Answer
This strategy (using maps you know and the five lemma) will NEVER work to show that some pair of maps are the same. Indeed consider the pair of maps of short exact sequences from $V \to V^2 \to V$ to itself, which is zero on the outer factors and either zero in the middle or $\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}.$ As you can see, only the maps in the middle differ. Another way to say this is "maps which between filtered vector spaces whose associated graded map is zero need not themselves be zero maps." Think about this long enough and you will see that in fact this is geometrically obvious.
Now for your question show that a proper homotopy induces a chain homotopy on $C^*$ which preserves the subcomplex of compactly supported things (this is where you use properness), so that it induces a chain homotopy on the quotient.