Isomorphism of cobordisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:17:20Z http://mathoverflow.net/feeds/question/88334 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88334/isomorphism-of-cobordisms Isomorphism of cobordisms knot 2012-02-13T08:01:37Z 2012-02-13T16:16:10Z <p>Let $(M, \partial_{-}M, \partial_{+}M)$ be a decorated 3-cobordism, where $M$ be a (topological) 3-manifold and $\partial M=-\partial_{-}M \cup \partial_{+}M$. (decorated in a sense of Turaev, Quantum invariants of knots and 3-anifold, on page 159.) Suppose $M$ is homeomorphic to a cylinder over a torus $\Sigma \times I$, where $\Sigma$ is a torus $S^1 \times S^1$.</p> <p>Let $f_{-}: \Sigma \to \partial_{-}M$ and $f_{+}: \Sigma \to \partial_{+}M$ be parametrizations of bottom and top boundaries respectively.</p> <p>Consider the composition</p> <p>$H_1(\Sigma; \mathbb{Z}) \to H_1(\partial_{-}M; \mathbb{Z}) \to H_1(\partial_{+}M; \mathbb{Z}) \to H_1(\Sigma; \mathbb{Z})$.</p> <p>Here the first and the third ismorphism are induced by the parametrizations $f_{\pm}$ respectively and the second isomorphism (let's say $h$) is obtained by pushing loops in the bottom base of $M$ to the top base using the cylindrical structure on $M$.</p> <p><strong>Question</strong></p> <p>Is this cobordism $(M, \partial_{-}M, \partial_{+}M)$ (d-)homeomorphic to a cobordims $(\Sigma \times I, \Sigma \times 0, \Sigma \times 1)$, where the parametrization on the top is given by the identity of $\Sigma$ and the bottom is given by $f_{+}^{-1}hf_{-}$? </p> <p>Please give me a proof. Thanks.</p> <p><strong>Edit</strong></p> <p>In this context, "decorated" can be almost ignored. What important here is that the homeomorphism of two cobordism is a homeomorphism of the two manifolds and if it is restricted on boundaries, it should commute with the given parametrizations.</p> <p>The parametrization in this context means a degree 1 homeomorphism from a fixed torus to $\partial_{-}M$ and $\partial_{+}M$</p>