拓扑学作业 11

[Arm.C5.T15]

[Arm.C10.T23] If $X$ is a connected, locally path-connected, Hausdorff space, and if a finite group $G$ of order $n$ acts freely on $X$, show that $X$ is an $n$-sheeted covering of $X/G$.

[Hat.C2S1.T11] Show that if $A$ is a retract of $X$ then the map $H_n(A)\to H_n(X)$ induced by the inclusion $A\subseteq X$ is injective.

[Hat.C2S1.T13] Verify that $f\sim g$ implies $f_* = g_*$ for induced homomorphisms of reduced homology groups.

[Hat.C2S1.T30] In each of the following commutative diagrams assume that all maps but one are isomorphisms. Show that the remaining map must be an isomorphism as well.