Three-manifold mutations detected by Heegaard Floer homology

Clarkson, Corrin¹

¹Department of Mathematics, Indiana University, 831 E Third St, Bloomington, IN 47405, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 1-16

Résumé

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 fφf−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 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.

Détail

DOI : 10.2140/agt.2017.17.1

Classification : 57M27, 57M60
Keywords: Heegaard Floer homology, mapping class group, Thurston norm, Fukaya category, three-manifolds, mutation

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