nLab framed generalized Morse function

Idea

A variant of a Morse function that yields a contractible space of such functions.

Definition

A generalized Morse function $f$ on a smooth manifold is a smooth real-valued function $f$ whose critical points either have a nondegenerate Hessian or a Hessian with a 1-dimensional kernel $K$ such that the third derivative of $f$ along $K$ is nonzero.

The Morse lemma shows that in a neighborhood of such a critical point we can pick a coordinate system in which $f$ has the form

$$f(x_1,\ldots,x_n)=x_1^2+\cdots+x_k^2-x_{k+1}^2-\cdots-x_n^2$$

or

$$f(x_1,\ldots,x_n)=x_1^2+\cdots+x_{k-1}^2+x_k^3-x_{k+1}^2-\cdots-x_n^2,$$

respectively.

Properties

The space of framed generalized Morse functions is contractible. For a proof, see Eliashberg–Mishachev or Kupers.

This property distinguishes framed generalized Morse functions from ordinary Morse functions, whose space is not contractible.

Related concepts

• Morse function

References

• Kiyoshi Igusa, Higher Singularities of Smooth Functions are Unnecessary, Annals of Mathematics 119:1 (1984), 1–58. DOI

• Kiyoshi Igusa, On the homotopy type of the space of generalized Morse functions, Topology 23:2 (1984), 245–256. DOI

• Kiyoshi Igusa, The space of framed functions, Transactions of the American Mathematical Society 301:2 (1987), 431–477. DOI

• Y. M. Eliashberg, N. M. Mishachev, The space of framed functions is contractible, Essays in Mathematics and its Applications. In Honor of Stephen Smale’s 80th Birthday (2012), 81–109. arXiv, DOI.

• Alexander Kupers, Three applications of delooping to h-principles, Geometriae Dedicata 202:1 (2019), 103–151. arXiv, DOI.