The exactness theorem for Floer homology
type: Item
instance of: scholarly article
title: The exactness theorem for Floer homology
zbMATH Open document ID: 0911.57013
author: Hongyu Wang
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1999-05-02T00:00:00Z
review text: The author illustrates some aspects of the work of A. Floer on the surgery formula for instanton Floer homology, known as the exact triangle. A detailed overview of Floer's work can be found in [\textit{P. J. Braam} and \textit{S. K. Donaldson}, Floer's work on instanton homology, knots and surgery, in `The Floer Memorial Volume', Progr. Math. 133 (1995)]. In this paper the author concentrates on the part concerning the proof of the exactness of the sequence determined by the surgery cobordisms. A suitable perturbation of the Chern-Simons functional on a homology sphere \(Y\) provides a decomposition of the moduli space of irreducible critical points (perturbed flat connections) on \(Y\) as \(R_Y = R_{Y'} \cup R_{Y''}\), where \(Y'\) is the homology sphere obtained by 1-surgery on a knot \(K\) in \(Y\) and \(Y''\) is the 3-manifold, with the homology of \(S^1\times S^2\), obtained by 0-surgery on \(K\). This induces exact sequences of abelian groups relating the Floer complexes of \(Y\), \(Y'\), and \(Y''\). Using the surgery cobordisms, it is possible to define chain homomorphisms between the same Floer complexes. Considering the natural filtration of the Floer complexes, it is possible to relate these chain homomorphisms to the exact sequence of groups induced by the identification of the flat connections. The difference between the maps induced by the cobordism and the exact sequence of groups is given by maps that are strictly monotonic with respect to the filtration. This is the essential ingredient in the proof of the exactness of the cobordism sequence. The author then proves that the connecting homomorphism in the exact triangle is also a map induced geometrically by counting solutions on a cobordism: in fact, the homomorphism is induced by the three-dimensional moduli space of solutions of the anti-self-dual equation on the composite cobordism, connecting \(Y''\) to \(Y'\). The composite cobordism can be written as a connected sum \(W_3 \# CP^2\), where \(W…
reviewed by: Matilde Marcolli
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 1195594
zbMATH Keywords: Floer homology
zbMATH Keywords: surgery
zbMATH Keywords: exact triangle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195145014
OpenAlex ID: W2079127232
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DOI: 10.2977/PRIMS/1195145014
description: scientific article; zbMATH DE number 1195594