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This occurs to me when considering the homomorphism $p_*$ induced by the covering map/attaching map: $$p:S^n\to \mathbb{R}P^n$$ on its homology group: $H_n(S^n)\to H_n(\mathbb{R}P^n)$ which sends the generator to twice the generator if $n$ is odd, with integral coefficient implicit.

Meanwhile, since the covering map gives a very fine conclusion on homotopy groups, and comparatively rather little information on homology/cohomology groups, I was wondering what we can say for a general covering map: $$p:X\to Y$$ of the relation between their homology groups, or the order of $p$ if we consider $X, Y$ be both manifolds.

Any contribution is more than welcomed.

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A covering map $p : X \to Y$ of degree $n$ corresponds to a local system of finite sets of size $n$ on $Y$. Taking the free abelian group on this gives a local system of abelian groups on $Y$ with fiber $\mathbb{Z}^n$; call this $L$. Then the cohomology of $X$ is the cohomology $H^{\bullet}(Y, L)$ of $Y$ with local coefficients in $L$.

You can say more with more hypotheses. For example, if $p$ is a Galois cover with Galois group $G$ and $k$ is a field with characteristic not dividing $|G|$, then it turns out that

$$H^{\bullet}(X, k)^G \cong H^{\bullet}(Y, k).$$

This lets you compute the cohomology of $\mathbb{RP}^n$ from the cohomology of $S^n$ over any field of characteristic not equal to $2$.

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