TITLE: Hatcher Theorem 3.26 - orientability
QUESTION [5 upvotes]: I am reading Hatcher, the beginning of the chapter on Poincare duality. I am trying to understand how theorem 3.26 is deduced from lemma 3.27 and I must admit I find Hatcher's proof very esoteric. 
Theorem: let $M$ be a closed connected $n$-manifold. Then: 
(a) If $M$ is $R$-orientable, then the map $H_n(M;R) \to H_n(M \mid x;R) \cong R$ is an isomorphism for all $x \in M$. 
(b) If $M$ is not $R$-orientable, then the map $H_n(M;R) \to H_n(M \mid x;R) \cong R$ is injective with image $\{ r \in R \mid 2r =0\}$ for all $x \in M$. 
Lemma: under the same assumptions on $M$ as above, if $x \mapsto \alpha_x$ is a section of the covering space $M_R \to M$ then there is a unique class $\alpha \in H_n(M;R)$ whose image in $H_n(M \mid x;R)$ is $\alpha_x$ for all $x \in M$. 
Hatcher defines the $R$-module $\Gamma_R(M)$ of sections of $M_R \to M$ and infers from the lemma that the map sending a class $\alpha \in H_n(M;R)$ to the section $x \mapsto \alpha_x$, where $\alpha_x$ is the image of $\alpha$ under the map $H_n(M;R) \to H_n(M \mid x;R)$, is an isomorphism. He then says that if $M$ is connected then each section is uniquely determined by its value at one point. He then says that the theorem "is apparent" from the earlier discussion of the structure of $M_R$. 
I would be grateful if someone could spell out why it is so "apparent" that the theorem follows, and what features of the structure of $M_R$ are relevant. The main problem for me is that the lemma doesn't say anything about orientability, so I'm not sure how one can use it to prove the theorem.

REPLY [4 votes]: We know that $M_R$ is a disjoint union of the covering spaces $M_r$, which are isomorphic to the trivial cover $M$ if $r=-r$ and to $\widetilde{M}$ otherwise. Furthermore, $\widetilde{M}$ is homeomorphic to two copies of $M$ iff $M$ is orientable.
So, for (a) when $M$ is orientable, $M_R$ is a union of copies of $M$. Hence any section $\alpha_x$ is determined by its value at a single point and is just a homeomorphism from $M$ to the relevant component of $M_R$. The $R$-module of sections is thus isomorphic to $R$, so the isomorphism in 3.27 gives the result.
For (b) when $M$ is not orientable, $M_R$ is a union of copies of $M$ corresponding to the elements of $R$ with $r=-r$ along with copies of $\widetilde{M}$ corresponding to the other elements of $R$. By the lifting criterion, there are no sections $M\to\widetilde{M}$, so the $R$-module of sections is isomorphic to $\{r\in R: r=-r\}$, and again 3.27 gives the result.