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Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map $f: E \rightarrow E$ that covers a map $g: B \rightarrow B$. If the maps $g_*: H_*(B, \mathbb{Q}) \rightarrow H_*(B, \mathbb{Q})$ and $(f_F)_*: H_*(F,\mathbb{Q}) \rightarrow H_*(F,\mathbb{Q}) $ are the identity map where $f_F$ denotes the restriction to the fiber, does it follows then that $f_*$ is the identity in the homology of $E$?

If I look at the Serre spectral sequence, this information tells me that $f_*$ is the identity just up to a filtration of $H_*(E)$, but it do not how to conclude that $f_*$ is the identity map or not

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