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This question is related to my previous question Cylindrical structure and homology.

Let $S$ be a oriented compact (topological) 2-manifold. We consider a cylinder $M=S\times I$ over $S$, here $I=[0, 1]$.

By pushing a loop in the bottom boundary $S \times 0$ to a loop in in the top boundary $S \times I$ using a cylindrical structure of $S\times I$, we have an isomorphism $f: H_1(S\times 0, \mathbb{Z}) \to H_1(S \times 1, \mathbb{Z})$.

Is $f$ identity isomorphism? (if we assume the identification of $S \times 0$ with $S\times 1$?)

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