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).
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Acknowledgements
We are grateful to D. Kazhdan and V. Hinich for their encouragement to write this paper and to S. Galatius for enlightening discussions.
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Dedicated to the 80th Anniversary of Professor Stephen Smale
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Eliashberg, Y.M., Mishachev, N.M. (2012). The Space of Framed Functions is Contractible. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_5
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DOI: https://doi.org/10.1007/978-3-642-28821-0_5
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