An odd Khovanov homotopy type

Abstract

For each link L⊂S3 and every quantum grading j, we construct a stable homotopy type Xoj(L) whose cohomology recovers Ozsváth-Rasmussen-Szabó's odd Khovanov homology, H˜i(Xoj(L))=Khoi,j(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. We also construct a potentially new even Khovanov homotopy type with a Z/2 action, with fixed point set a desuspension of Xoj(L).