Naturality issues in involutive Heegaard Floer homology
Heegaard Floer homology is an invariant of 3-manifolds, and knots and links within them, introduced by P. Oszváth and Z. Szabó in the early 2000s. Because of its relative computability by the standards of gauge and Floer theoretic invariants, it has enjoyed considerably popularity. However, it is not immediately obvious from the construction that Heegaard Floer homology is natural, that is, that it assigns to a basepointed 3-manifold a well-defined module over an appropriate base ring rather than an isomorphism class of modules, and well-defined cobordism maps to 4-manifolds with boundary. This situation was improved in the 2010s when A. Juhász, D. Thurston, and I. Zemke showed naturality of the various versions of Heegaard Floer homology. In this talk we consider involutive Heegaard Floer homology, a refinement of the theory introduced by C. Manolescu and I in 2015, whose definition relies on Juhász-Thurston-Zemke naturality but which is itself not obviously natural even given their results. We prove that involutive Heegaard Floer homology is a natural invariant of basepointed 3-manifolds together with a framing of the basepoint, and has well-defined maps associated to cobordisms, and discuss some consequences and implications. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.