The Space of Framed Functions is Contractible

Y. M. Eliashberg () and N. M. Mishachev ()
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Y. M. Eliashberg: Stanford University
N. M. Mishachev: Lipetsk Technical University

A chapter in Essays in Mathematics and its Applications, 2012, pp 81-109 from Springer

Abstract: Abstract According to Igusa (Ann Math 119:1–58, 1984) a generalized Morse function on M is a smooth function $$M \rightarrow \mathbb{R}$$ with only Morse and birth-death singularities and a framed function on M is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of f. In (Igusa, Trans Am Math Soc 301(2):431–477, 1987) Igusa proved that the space of framed generalized Morse functions is $$(\dim \,M - 1)$$ -connected. Lurie gave in (arXiv:0905.0465) 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 using methods of our paper (Eliashberg and Mishachev, Topology 39:711–732, 2000).

Keywords: Vector Field; Line Field; Formal Extension; Manifold Versus; Injective Homomorphism (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-3-642-28821-0_5

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