Three-manifold mutations detected by Heegaard Floer homology

Clarkson, Corrin

doi:10.2140/agt.2017.17.1

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Three-manifold mutations detected by Heegaard Floer homology

Corrin Clarkson

Algebraic & Geometric Topology 17 (2017) 1–16

DOI: 10.2140/agt.2017.17.1

Abstract

Given an orientation-preserving self-diffeomorphism $φ$ of a closed, orientable surface $S$ with genus at least two and an embedding $f$ of $S$ into a three-manifold $M$ , we construct a mutant manifold by cutting $M$ along $f (S)$ and regluing by $f φ f^{- 1}$ . We will consider whether there exist nontrivial gluings such that for any embedding, the manifold $M$ and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if $φ$ is not isotopic to the identity map, then there exists an embedding of $S$ into a three-manifold $M$ such that the rank of the nontorsion summands of $\hat{HF}$ of $M$ differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of $S$ into a three-manifold $M$ such that the total rank of $\hat{HF}$ of $M$ differs from that of its mutant.

Keywords

Heegaard Floer homology, mapping class group, Thurston norm, Fukaya category, three-manifolds, mutation

Mathematical Subject Classification 2010

Primary: 57M27, 57M60

References

Publication

Received: 16 October 2013

Revised: 8 June 2016

Accepted: 4 July 2016

Published: 26 January 2017

Authors

Corrin Clarkson
Department of Mathematics Indiana University 831 E Third St Bloomington, IN 47405 United States
http://pages.iu.edu/~cjclarks/

Volume 17, issue 1 (2017)