Let $M$ be an orientable closed manifold of dimension $n$ covered by coordinate discs $ \{ U_i : 1 \le i \le k\} $ such that for each $i$, $\bar{U_i}-U_i$ is homeomorphic to $S^{n-1}$, and suppose for each $i$, $\alpha_i$ is a given generator of $H_n (M, M-U_i)$ (thus each $\alpha_i$ forms a local $R$-orientation of $M$ at a point in $U_i$). ($R$ is given commutative unitary)
The problem is to construct a generator $\zeta \in H_n(M) $ mapping to $\alpha_i$ for all $i$.
At first I used Mayer-Vietoris sequence to constructed an injective map $H_n(M) \rightarrow \oplus_{i=1}^k H_n (M, M-U_i)$. I think this injective map should give the answer but I don't know how to finish the proof. Here is one my thought: We know that $H_n(M,M-U_i) = H_{n-1}(S^{n-1}) = R$. The generator $\alpha_i$ is a eq class of a chain. Then the simplexes of which the chain consist are actually simplexes of $M$, that is, the chain is actually an $n$-chain of $M$. Clearly the injective map above sends this chain of $M$ to $\alpha_i$. Doing this several times we will obtain a $n$-chain $\alpha$ of $M$ which is sent to ($\alpha_1, \alpha_2, \cdots, \alpha_k$) (after removing some 'overlapping' chains). But is this $\alpha$ really a generator of $H_n(M)$? How can I verify that? If I am on a wrong way, how could I do this problem?
Thank you for your help! I am sorry if there is any mistake in my English. I am not a fluent English speaker.