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Let us consider a cobordism $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is homeomorphic to $T \times I$, here $T$ is a torus $S^1 \times S^1$ and $I=[0, 1]$.

I encountered the statement "isomorphism $f:H_1(\partial_{-}M, \mathbb{Z}) \to H_1(\partial_{+}M, \mathbb{Z})$ is obtained by pushing loops in the bottom base of $M$ to the top base using the cylindrical structure on $M$.

So the questions are;

  1. What does "cylindrical structure" really mean here?

  2. Does it make sence only when we can "see" $M$ as a subset of $R^3$?

  3. A guess is that a loop $\alpha$ in the bottom is mapped a loop $\beta$ in the top by $f$ if and only if $\alpha=\beta$ in $H_1(M, \mathbb{Z})$. Is this ok?

  4. If #3 is correct (or not), how can we check it without having $M$ in $R^3$? Does a cylindrical structure depend on the isomorphism $M \to T\times I$?

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