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We study the problem of defining maps on link Floer homology induced byunoriented link cobordisms. We provide a natural notion of link cobordism,disoriented link cobordism, which tracks the motion of index zero and indexthree critical points. Then we construct a map on unoriented link Floerhomology associated to a disoriented link cobordism. Furthermore, we give acomparison with Oszv\'ath-Stipsicz-Szab\'o's and Manolescu's constructions oflink cobordism maps for an unoriented band move.

Unoriented Cobordism Maps on Link Floer Homology