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 $S^1 \times S^1$.

Let $f_{-}: \Sigma \to \partial_{-}M$ and $f_{+}: \Sigma \to \partial_{+}M$ be parametrizations of bottom and top boundaries respectively.

Consider the composition

$H_1(\Sigma; \mathbb{Z}) \to H_1(\partial_{-}M; \mathbb{Z}) \to H_1(\partial_{+}M; \mathbb{Z}) \to H_1(\Sigma; \mathbb{Z})$.

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 $M$.

Question

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 $f_{+}^{-1}hf_{-}$?

Please give me a proof. Thanks.

Edit

In this contest, "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.

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 $S^1 \times S^1$.

Let $f_{-}: \Sigma \to \partial_{-}M$ and $f_{+}: \Sigma \to \partial_{+}M$ be parametrizations of bottom and top boundaries respectively.

Consider the composition

$H_1(\Sigma; \mathbb{Z}) \to H_1(\partial_{-}M; \mathbb{Z}) \to H_1(\partial_{+}M; \mathbb{Z}) \to H_1(\Sigma; \mathbb{Z})$.

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 $M$.

Question

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 $f_{+}^{-1}hf_{-}$?

Please give me a proof. Thanks.

Edit

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.

The parametrization in this context means a degree 1 homeomorphism from a fixed torus to $\partial_{-}M$ and $\partial_{+}M$

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 $S^1 \times S^1$.

Let $f_{-}: \Sigma \to \partial_{-}M$ and $f_{+}: \Sigma \to \partial_{+}M$ be parametrizations of bottom and top boundaries respectively.

Consider the composition

$H_1(\Sigma; \mathbb{Z}) \to H_1(\partial_{-}M; \mathbb{Z}) \to H_1(\partial_{+}M; \mathbb{Z}) \to H_1(\Sigma; \mathbb{Z})$.

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 $M$.

Question

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 $f_{+}^{-1}hf_{-}$?

Please give me a proof. Thanks.

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 $S^1 \times S^1$.

Let $f_{-}: \Sigma \to \partial_{-}M$ and $f_{+}: \Sigma \to \partial_{+}M$ be parametrizations of bottom and top boundaries respectively.

Consider the composition

$H_1(\Sigma; \mathbb{Z}) \to H_1(\partial_{-}M; \mathbb{Z}) \to H_1(\partial_{+}M; \mathbb{Z}) \to H_1(\Sigma; \mathbb{Z})$.

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 $M$.

Question

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 $f_{+}^{-1}hf_{-}$?

Please give me a proof. Thanks.

Edit

In this contest, "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.