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Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commentator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint $\mathbb{Q}$-cohomology class on $X$ is a pull-back from $B$.

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commutator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint $\mathbb{Q}$-cohomology class on $X$ is a pull-back from $B$.

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commentator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint cohomology class on $X$ is a pull-back from $B$.

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commentator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint $\mathbb{Q}$-cohomology class on $X$ is a pull-back from $B$.

Let $X\to B$ be a connected (un-ramified) covering map with group of deck transformations $G$; e.g. $\mathbb{C}\to T= \mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$. The covering may be finite or infinite.

I am looking for a reference for the following statement (finite case is easy; thus, mainly for the infinite case):

Every $G$-invariant cohomology class on $X$ is pull-back of a cohomology class from the base.

RMK: I am not sure this is going to be true always; I just need it for quotients of maximal abelian covering.

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commentator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint cohomology class on $X$ is a pull-back from $B$.