Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $M \to T\times I$?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.