Locally equivalent Floer complexes and unoriented link cobordisms

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 $\gamma_4(L)$.