# Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

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Mathematische Annalen

, Volume 368, Issue 3–4, pp 945–970

First Online: 03 August 2016Received: 26 May 2015Revised: 27 June 2016

Abstract

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base a similar result was also obtained by Nadler. The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.

Mathematics Subject Classification53D12 53D40 Communicated by Denis Auroux.

Autor: Thomas Kragh