Editing Morse theory (section)

==Morse–Bott theory==
The notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A '''{{visible anchor|Morse–Bott function|Morse–Bott function}}''' is a smooth function on a manifold whose [[critical set]] is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the Hessian at critical points is non-degenerate in every direction, that is, has no kernel).

The index is most naturally thought of as a pair
<math display="block">\left(i_-, i_+\right),</math>
where <math>i_-</math> is the dimension of the unstable manifold at a given point of the critical manifold, and <math>i_+</math> is equal to <math>i_-</math> plus the dimension of the critical manifold. If the Morse–Bott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between <math>i_-</math> and <math>i_+.</math>

Morse–Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. [[Raoul Bott]] used Morse–Bott theory in his original proof of the [[Bott periodicity theorem]].

[[Round function]]s are examples of Morse–Bott functions, where the critical sets are (disjoint unions of) circles.

[[Morse homology]] can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a [[spectral sequence]].  Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.