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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 12th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: \tableofcontents ## 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][Igusa1984aom] [Igusa1984aom]: https://doi.org/10.2307/2006962 * [[Kiyoshi Igusa]], _On the homotopy type of the space of generalized Morse functions_, Topology 23:2 (1984), 245–256. [DOI][Igusa1984top] [Igusa1984top]: https://doi.org/10.1016/0040-9383(84)90043-0 * [[Kiyoshi Igusa]], _The space of framed functions_, Transactions of the American Mathematical Society 301:2 (1987), 431–477. [DOI][Igusa1987] [Igusa1987]: https://doi.org/10.1090/s0002-9947-1987-0882699-7 * [[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](https://arxiv.org/abs/1108.1000), [DOI](https://doi.org/10.1007/978-3-642-28821-0_5). * [[Alexander Kupers]], _Three applications of delooping to h-principles_, Geometriae Dedicata 202:1 (2019), 103–151. [arXiv](https://arxiv.org/abs/1701.06788), [DOI](https://doi.org/10.1007/s10711-018-0405-7). <a href="https://ncatlab.org/nlab/revision/framed+generalized+Morse+function/1">v1</a>, <a href="https://ncatlab.org/nlab/show/framed+generalized+Morse+function">current</a>

   Created:

   \tableofcontents

   Idea

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

   Definition

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

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

   or

   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.

   v1, current