Combinatorial proofs in bordered Heegaard Floer homology
type: Item
instance of: scholarly article
title: Combinatorial proofs in bordered Heegaard Floer homology
zbMATH Open document ID: 1365.57038
author: Bohua Zhan
published in: Algebraic \& Geometric Topology
publication date: 2016-11-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.3004
reviewed by: Marco Marengon
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57R56
zbMATH DE Number: 6653751
zbMATH Keywords: bordered Heegaard Floer homology
zbMATH Keywords: strongly based mapping class groupoid
Wikidata QID: Q113727211
describes a project that uses: bfh_python
MaRDI profile type: Publication
OpenAlex ID: W3105419823
arXiv ID: 1405.3004
DOI: 10.2140/AGT.2016.16.2571
review text: Bordered Heegaard Floer homology is an invariant of a 3-manifold with parameterised boundary, defined by \textit{R.~Lipshitz}, \textit{P.~S.~Ozsváth}, and \textit{D.~P.~Thurston} [``Bordered Heegaard Floer homology: invariance and pairing'', Preprint, \url{arXiv:0810.0687}], which behaves nicely under gluing. For instance, from a splitting of a closed 3-manifold \(Y\) into two pieces \(Y_1\) and \(Y_2\) with common boundary, one can recover \(\widehat{\text{HF}}(Y)\) as a `box tensor product' of bordered invariants of \(Y_1\) and \(Y_2\). By breaking a closed 3-manifold into simpler pieces, and by using the gluing theorem in bordered Floer homology, Lipshitz, Ozsváth, and Thurston were able to give a combinatorial construction of \(\widehat{\text{HF}}(Y)\) alternative to [\textit{S.~Sucharit} and \textit{J.~Wang}, Ann.~Math.~(2) 171, No.~2, 1213--1236 (2010, Zbl 1228.57017)].NEWLINENEWLINEThe paper under review gives a combinatorial proof of the fact that \(\widehat{\text{HF}}(Y)\) is a topological invariant of \(Y\). Given a closed 3-manifold \(Y\), one can construct it (up to diffeomorphism) as \(Y = \boldsymbol{H}^g \cup_\psi {-\boldsymbol{H}^g}\), where \(\boldsymbol{H}^g\) is a standard genus-\(g\) handlebody with parameterised boundary, and \(\psi\) is an orientation-preserving diffeomorphism of \(\partial\boldsymbol{H}^g\). The Heegaard Floer homology of \(Y\) can be defined as NEWLINE\[NEWLINE {\widehat{\text{HF}}(Y) := (\widehat{\text{CFAA}}(\psi, \boldsymbol{\tau}) \boxtimes \widehat{\text{CFD}}(\boldsymbol{H}^g)) \boxtimes \widehat{\text{CFD}}(-\boldsymbol{H}^g).} NEWLINE\]NEWLINENEWLINENEWLINEHere \(\boldsymbol{\tau}\) is a factorisation of \(\psi\) into a set of generators (called \textit{arcslides}) of the strongly based mapping class group of \(\partial\boldsymbol{H}^g\). The latter can be thought of as the group of self-diffeomorphisms of \(\partial\boldsymbol{H}^g\) fixing the neighbourhood of a basepoint, up to isotopies also fixing the same neighb…
description: scientific article; zbMATH DE number 6653751