The space of framed functions is contractible

TL;DR

This paper provides a geometric proof that the space of framed functions on an n-dimensional manifold is contractible, confirming a conjecture by Igusa and Lurie through a new approach.

Contribution

It offers a geometric proof of the contractibility of the space of framed functions, complementing previous algebraic topological proofs.

Findings

The space of framed functions is contractible.

Provides a new geometric proof of Igusa-Lurie's theorem.

Confirms the conjecture about the topological structure of framed functions.

Abstract

According to Kiyoshi Igusa a generalized Morse function on an n-dimensional manifold M is a smooth function with only Morse and birth-death singularities and a framed function is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of the function f. In his paper "The space of framed functions" (Trans. of Amer. Math. Soc., 301(1987), 431-477) Igusa proved that the space of framed generalized Morse functions is (n-1)-connected. In the paper "On the Classification of Topological Field Theories" (arXiv:0905.0465) Jacob Lurie gave an algebraic topological proof that the space of framed functions is contractible. In this paper we give a geometric proof of Igusa-Lurie's theorem in the spirit of our paper "Wrinkling of smooth mappings - II. Wrinkling of embeddings and K.Igusa's theorem" (Topology, 39(2000), 711-732.