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Three-manifold mutations detected by Heegaard Floer homology

Corrin Clarkson

Algebraic & Geometric Topology 17 (2017) 1–16

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 fφf1. 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 HF̂ 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 HF̂ of M differs from that of its mutant.

Heegaard Floer homology, mapping class group, Thurston norm, Fukaya category, three-manifolds, mutation
Mathematical Subject Classification 2010
Primary: 57M27, 57M60
Received: 16 October 2013
Revised: 8 June 2016
Accepted: 4 July 2016
Published: 26 January 2017
Corrin Clarkson
Department of Mathematics
Indiana University
831 E Third St
Bloomington, IN 47405
United States