Mathematics Colloquia and Seminars

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The idea of Morse theory is to use nice real valued functions on a space to study the topology of the space itself. Suppose $M$ is a closed, smooth $n$ dimensional manifold and $f$ a smooth function on $M$ with critical point $c$. Morse proved that if $c$ is non-degenerate (in a certain precise sense) then the behavior of $f$ in a neighborhood of $c$ depends only on the index of $f$ at $c$, an invariant of the matrix of second derivatives of $f$ at $c$.

For a regular value $a$ of $f$, let $M(f, a)$ denote the submanifold ${x | f(x) /leq a}$. I will present the following two results:
(i) If $f^{-1}[a, b]$ is without critical points then $M(f,a)$ is diffeomorphic to $M(f,a)$.
(ii) If $f^{-1}[a, b]$ contains exactly one critical point, $c$, then $M(f, b)$ is obtained from $M(f, a)$ by attaching a handle of index equal to the index of $f$ at $c$, or homotopically by attaching a cell of dimension equal to the index of $f$ at $c$.

As a nearly immediate corollary we see that if $f$ has only non-degenerate critical points then $M$ is homotopy equivalent to a CW-complex with one cell of dimension $k$ for each critical point of $f$ with index $k$.

Other interesting consequences will be discussed as time permits.