Topological Gluing

Regular Interval Theorem

Let $M:{\Sigma }_0\to {\Sigma }_1$ be a cobordism, and let $f:M\to [0,1]$ be a Morse function without any critical points, then there is a diffeomorphism from the cylinder ${\Sigma }_0\times [0,1]$ to $M$ that is compatible with the projection to $[0,1]$:

And similarly there is another diffeomorphism ${\Sigma }_1\times [0,1]\to M$ compatible with the projection.

Corollary

Let $M:{\Sigma }_0\to {\Sigma }_1$ be a cobordism. Then there is a decomposition $M=M_{[0,\epsilon ]}{M}_{[\epsilon ,1]}$ such that $M_{[0,\epsilon ]}$ is diffeomorphic to the cylinder ${\Sigma }_0\times [0,\epsilon ]$.

Proof Take a Morse function $f:M\to [0,1]$, then there is some $\epsilon >0$ such that $f$ has no critical points in $M_{[0,\epsilon ]}:=f^{-1}([0,\epsilon ])$. By the regular interval theorem, there is a diffeomorphism ${\Sigma }_0\times [0,\epsilon ]\to M_{[0,\epsilon ]}$. $\square$

Gluing of General Cobordisms

Theorem

Suppose $M:{\Sigma }_0\to {\Sigma }_1$ and $N:{\Sigma }_1\to {\Sigma }_2$ are cobordisms, then there always exists a smooth manifold $NM$ which is homeomorphic to $M{\bigsqcup }_{{\Sigma }_1}N$ and whose smooth structure agrees with both $M$ and $N$.