We show that the local equivalence class of the collapsed link Floer complex
cCFL∞(L), together with many Υ-type invariants extracted from
this group, is a concordance invariant of links. In particular, we define a
version of the invariants ΥL​(t) and ν+(L) when L is a link and
we prove that they give a lower bound for the slice genus g4​(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 Ï…-set, introduced by Ozsv\'ath, Stipsicz and
Szab\'o, produces a lower bound for the 4-dimensional smooth crosscap number
γ4​(L).Comment: Remark 1.3 and acknowledgements were change