Let $M^n$ be a smooth, connected, closed $n$ dimensional manifold. A handle is $H_{\lambda}^n = B^{\lambda} \times B^{n-\lambda}$ where $B^{\lambda}$ is a $\lambda$ ball.

Any such manifold $M$ is diffeomorphic to the union of finitely many such handles where each handle $H_{\lambda}^n$ is in a one-to-one correspondence with the critical points of index $\lambda$ of a Morse function on $M$