An odd Khovanov homotopy type

TL;DR

This paper constructs a stable homotopy type for odd Khovanov homology of links, revealing a Z/2 symmetry relating it to the even Khovanov homotopy type, extending previous homotopy constructions.

Contribution

It introduces a stable homotopy type for odd Khovanov homology and establishes a Z/2 action linking it to the even Khovanov homotopy type, expanding the homotopy-theoretic framework.

Findings

The odd Khovanov homotopy type recovers odd Khovanov homology via cohomology.

A Z/2 action on the odd homotopy type has a fixed point set related to the even homotopy type.

A Z/2 action on the even homotopy type has a fixed point set related to the odd homotopy type.

Abstract

For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(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 Z/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of X^j_o(L).