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Suppose we have a sufficiently nice (example below) pathwise connected topological space $X$ with a double cover $\pi : \widetilde X \to X$. There is an associated free $\mathbb Z_2$-action on $\widetilde X$ interchanging the sheets of the cover; denote by $\iota : \widetilde X \to \widetilde X$ the map corresponding to the nonzero element of $\mathbb Z_2$. The question is: under which conditions is $\iota$ detectable by its action on the homology of $\widetilde X$, $\iota_* : H_*(\widetilde X) \to H_*(\widetilde X)$? In other words, when is $\iota_*$ not the identity? And when it is not, how can one compute the $\pm 1$-eigenspaces of $\iota_*$?

(The coefficients may be freely chosen to give the "best" answer possible.)

Examples:

1) If $\pi$ is the trivial covering, $\widetilde X$ is the disjoint union of two copies of $X$ and therefore $H_*(\widetilde X) = H_*(X) \oplus H_*(X)$. The action of $\iota_*$ is by interchanging the summands and therefore it is detectable.

2) If $X = \widetilde X = S^1$ and $\pi$ is the unique nontrivial double covering, then $\iota$ is isotopic to the identity on $S^1$ and therefore acts trivially on $H_*(S^1)$.

3) If $X = \mathbb R P^2$, $\widetilde X = S^2$ and $\pi : \widetilde X \to X$ is the orientation covering, then $\iota$ acts as the antipodal map on $S^2$. The map $\iota_*$ acts trivially on $H_0(S^2)$ but on $H_2(S^2;\mathbb Z) = \mathbb Z$, for example, it acts as the multiplication by $-1$.

4) The real example I have in mind is as follows. Let $M$ be a smooth manifold, $L \subset M$ a smooth submanifold, and assume for simplicity both are closed and connected. Fix a point $x_0 \in L$, and let $X$ be the space of smooth maps $(D^2, \partial D^2, 1) \to (M,L,x_0)$, representing $0 \in \pi_2(M,L,x_0)$. Here $D^2 \subset \mathbb C$ is the closed unit disk. There is an isomorphism $\pi_1(X,x_0) \simeq \pi_3(M,L,x_0)$ ($x_0$ being the constant map, of course). Suppose $\widetilde X$ is the double cover corresponding to a real line bundle whose first Stiefel-Whitney class is given by a homomorphism $w : \pi_3(M,L,x_0) \to \mathbb Z_2$ (in view of the above isomorphism this is equivalent to specifying a class in $H^1(X;\mathbb Z_2)$). What can be said about this example?

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3  
As a generalization of (3), homology detects the involution on the orientation double cover of any closed manifold. –  Eric Wofsey Sep 9 '13 at 0:57

2 Answers 2

Here's something you can say. Recall (Hatcher 3G.1) that if $G$ is a finite group and $\pi : X \to X/G$ a covering map associated to an action of $G$ on a space $X$, then over a field $F$ of characteristic $0$ or not dividing $|G|$ the transfer map $H^k(X/G, F) \to H^k(X, F)$ (given by sending a singular simplex in $X/G$ the sum of its $|G|$ distinct lifts in $X$) is injective and has image precisely the fixed subgroup $H^k(X, F)^G$.

In particular, letting $F = \mathbb{Q}$, if $X$ has finitely generated homology and $G$ acts trivially on the homology of $X$ then $X$ and $X/G$ have the same Euler characteristic, as in your example 2). Taking the contrapositive, if $X$ has finitely generated homology and $\chi(X) \neq \chi(X/G)$ then $G$ can't act trivially on the homology of $X$, as in your example 3). Unfortunately I have no idea if your spaces have finitely generated homology.

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Like Qiaochu says, if you use a field of characteristic $\not= 2$ as coefficients then the cohomology of the base is simply isomorphic to $+1$ eigenspace of the cohomology the double cover. I don't know about the specific application you have in mind in 4), but generally you can also get some useful information about the cohomology with $\mathbb{Z}_2$ coefficients by applying the Gysin sequence for a 0-dimensional sphere bundle (unoriented, unless the double cover is trivial). This gives a long exact sequence

$$ H^0(X; \mathbb{Z}_2) \stackrel{\pi^*}{\to} H^0(\widetilde X; \mathbb{Z}_2) \stackrel{I}{\to} H^0(X; \mathbb{Z}_2) \stackrel{\cup\, w_1}{\to} H^1(X; \mathbb{Z}_2) \cdots ,$$

where $I$ is fibrewise integration, and the snake maps are cup product with the Stiefel-Whitney class of the line bundle corresponding to the cover. Note that $\pi^* \circ I$ equals $1 + \iota^*$ on $H^k(\widetilde X; \mathbb{Z}_2)$. This can be useful for instance for understanding whether the integral cohomology $H^k(\widetilde X; \mathbb{Z})$ is spanned by the $\pm 1$ eigenspaces of $\iota^*$ or whether their direct sum has some coquotient $\mathbb{Z}_2^r$.

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