Coherent orientation of mixed moduli spaces in Morse-Floer theory
type: Item
instance of: scholarly article
title: Coherent orientation of mixed moduli spaces in Morse-Floer theory
zbMATH Open document ID: 1216.53079
author: Jelena Katić
author: Darko Milinković
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2009-10-15T00:00:00Z
review text: The goal of the paper under review is to provide a coherent orientation for the spaces of intersections of gradient trajectories and holomorphic disks in the cotangent bundle case along the line first introduced by \textit{S. Donaldson} [``The orientation of Yang-Mills moduli spaces and 4-manifold topology'', J. Differ. Geom. 26, 397--428 (1987; Zbl 0683.57005)] for moduli spaces of (anti)-self-dual connections and by \textit{A. Floer} and \textit{H. Hofer} [``Coherent orientations for periodic orbit problems in symplectic geometry'', Math. Z. 212, No.~1, 13--38 (1993; Zbl 0789.58022)] for the symplectic moduli spaces. The coherent orientation for Lagrangian intersections was given by Fukaya, Oh, Ohta and Ono and for Morse theory by Schwarz. Consider the cotangent bundle \(P=T^*M\) of a smooth compact closed \(n\)-dimensional manifold and a compactly supported Hamiltonian \(H: P \times [0, 1]\to\mathbb{R}\). Define \(L_0 = 0_M\) to be the zero section of \(P\) and \(L_1 = \phi_1^H(L_0)\) to be a Hamiltonian deformation of \(L_0\) by the symplectic diffeomorphism \(\phi_1\) generated by \(H\). Let \(CF_*(H)\) be the Floer chain groups generated by the transversal intersection points of \(L_0\) and \(L_1\) with \({\mathbb{Z}}_2\)-coefficients. The boundary map \(\partial_F: CF_*(H) \to CF_*(H)\) measures the number of solutions \(u: \mathbb R\times [0, 1]\to T^*M\) of \[ \frac{\partial u}{\partial s} + J\left (\frac{\partial u}{\partial t} - X_H(u)\right) = 0,\;u(s, i)\in L_i,\;i=0, 1,\;u(\pm \infty, t)\in L_0 \cap L_1. \] The authors constructed in [``Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections'', Differ. Geom. Appl. 22, No.~2, 215--227 (2005; Zbl 1064.37041)] an isomorphism between the Floer homology \(HF_*(H)\) of \(CF_*(H)\) and the Morse homology \(HM_*(f)\) of a Morse function \(f\) by connecting the gradient flow on \((-\infty, 0]\) with a \(J\)-holomorphic curve \(u\) to form a moduli space which gives rise to a chain homomorphism from …
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 37J05
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 5615325
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: Floer homology
zbMATH Keywords: Morse theory
zbMATH Keywords: coherent orientation
zbMATH Keywords: cotangent bundle
MaRDI profile type: Publication
full work available at URL: s00574-009-0013-0
OpenAlex ID: W1994202466
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DOI: 10.1007/S00574-009-0013-0
description: scientific article; zbMATH DE number 5615325