An odd Khovanov homotopy type

For each link L subset of S-3 and every quantum grading j, we construct a stable homotopy type X-o(j)(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, (H) over tilde (i)(X-o(j)(L)) = Kh(o)(i, 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 X-o(j)(L). (C) 2020 Elsevier Inc. All rights reserved.