Locally equivalent Floer complexes and unoriented link cobordisms

TL;DR

This paper introduces new link invariants derived from Floer complexes that serve as concordance invariants and provide bounds for slice genus and crosscap number, with a focus on unoriented link cobordisms.

Contribution

It defines new invariants for links from Floer complexes and proves they are concordance invariants, also relating them to slice genus and crosscap number bounds.

Findings

Invariants from Floer complexes are concordance invariants.

New bounds for slice genus and crosscap number are established.

Behavior of homology groups under unoriented cobordisms is analyzed.

Abstract

We show that the local equivalence class of the collapsed link Floer complex $cCFL^\infty(L)$, together with many $\Upsilon$-type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants $\Upsilon_L(t)$ and $\nu^+(L)$ when $L$ is a link and we prove that they give a lower bound for the slice genus $g_4(L)$. Furthermore, in the last section of the paper we study the homology group $HFL'(L)$ and its behaviour under unoriented cobordisms. We obtain that a normalized version of the $\upsilon$-set, introduced by Ozsv\'ath, Stipsicz and Szab\'o, produces a lower bound for the 4-dimensional smooth crosscap number $\gamma_4(L)$.