# Two proper maps inducing same map at the level of fundamental groups are properly homotopic

I am dealing with the following problem:

Let $$X$$ and $$Y$$ be two connected non-compact surfaces, possibly with boundary. Let $$f_0,f_1:(X,x_0)\to (Y,y_0)$$ be two proper maps with $$f_{0*}=f_{1*}:\pi_1(X,x_0)\to \pi_1(Y,y_0)$$, then $$f_0$$ is properly homotopic to $$f_1$$.

Thoughts: Since $$Y$$ is a non-compact surface, its universal cover is a convex subset of $$\Bbb R^2$$, i.e., $$Y=K(G,1)$$ for some group $$G$$. If one goes through the proof the Hatcher's proposition 1B.9. on page 90, then the construction of the homotopy is dependent on $$f_0$$ and $$f_1$$, i.e., homotopy itself is proper if we are given $$f_0$$ and $$f_1$$ as proper maps.

Any help will be appreciated. Thanks in advance.

This is (a) false, and (b) obviously false.

I will explain (b) by showing that your question fails in the first non-trivial example. (You should therefore try to think of examples...)

Take $$X = Y = S^1 \times \Bbb R$$, with $$x_0 = y_0 = (1, 0)$$. Set $$f_0$$ to be the identity map, and $$f_1(x,t) = (x,-t)$$. Then these clearly induce the same map on the fundamental group --- they are the identity on the central circle, and $$X$$ deformation retracts to its central circle --- but they are also obviously not properly homotopic, as $$\text{End}(X)$$ is a two-point set, and proper maps induce maps on $$\text{End}(X)$$ which are equal if the original maps are properly homotopic. But $$f_0$$ is the identity on $$\text{End}(X)$$ and $$f_1$$ swaps the two ends.

If you want a proper map of a surfaces to be properly homotopic to the identity, you should (a) assume it is the identity on $$\pi_1$$, (b) assume it preserves ends, (c) assume it induces the identity on the fundamental group at infinity of each end. (For instance, this last condition distinguishes between the identity on $$\Bbb R^2$$ and the reflection map on $$\Bbb R^2$$.)

• Could you tell me any source of the portion "If you want a proper map of a surfaces to be properly homotopic to the identity....." Commented Dec 23, 2020 at 12:41
• I'd rather not. Try proving it first for closed surfaces with finitely many points removed. Then try doing the general case using what you learned.
– j.q
Commented Dec 23, 2020 at 13:32
• Got it. So, you are telling me to go through the following proof: Let $K$ and $L$ be finite dimensional CW-complexes. Let $f : K \to L$ be a proper map. Then $f$ is a proper homotopy equivalence if and only if: $(1)$ $\underline\pi_0(f) :\underline\pi_0(K) \to \underline\pi_0(L)$ is a homeomorphism. $(2)$ For each $n$, $\pi_n(f):\pi_n(K,*)\to \pi_n(L,f*)$ is an isomorphism. $(3)$ For each $n$ and each end $[a]$ of $K$, $\underline\pi_n(f) : \pi_n(K,\underline a) \to \underline\pi_n(L,\underline{fa})$ is an isomorphism. Commented Dec 23, 2020 at 14:24