Locally equivalent Floer complexes and unoriented link cobordisms

We show that the local equivalence class of the collapsed link Floer complex $cCF{L}^{\mathrm{\infty }}(L)$, together with many $\mathrm{{\rm Y}}$-type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants ${\mathrm{{\rm Y}}}_L(t)$ 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 $HF{L}^{\mathrm{\prime }}(L)$ 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 ${\gamma }_4(L)$.