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Homotopy theory, homological algebra, algebraic treatments of manifolds.
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Accepted
question about the induced homomorphism of etale fundamental groups
(This was meant to be a comment elaborating on Dan Petersen's answer, but got too long.) The error is that you cooked up an isomorphism between $F_x \circ f^{\ast}$ and $F_x$ and by means of that simp …