Combinatorial proofs in bordered Heegaard Floer homology
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 neighbourhood. The author proves combinatorially that the homotopy type of \(\widehat{\text{HF}}(Y)\) as defined above is independent of the chosen factorisation of \(\psi\) into arcslides, and of the chosen Heegaard splitting of \(Y\). This is done by constructing the chain maps in a combinatorial way.NEWLINENEWLINEIn the same paper, the author also gives a full combinatorial description of the representation of the strongly based mapping class groupoid of a based closed genus-\(g\) surface (or equivalently a genus-\(g\) surface with one boundary component) \(F_{g,1}\), which associates to any element \(\psi\) the bimodule \(\widehat{\text{CFDA}}(\psi)\). The use of a groupoid rather than a group is because \(F_{g,1}\) can be parameterised in different ways.
scientific article; zbMATH DE number 6653751