An odd Khovanov 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).

Figures (2)

• Figure 1: The Burnside categories and some functors between them. We have the relations $\mathcal{Q}\circ \mathcal{D} = \mathcal{F}$ and $\mathcal{F}\circ\mathcal{I} = \mathrm{Id}$. The $\mathcal{F}$'s are forgetful functors, $\mathcal{I}$ is a subcategory inclusion, $\mathcal{Q}$ is a quotient functor, and $\mathcal{D}$ stands for doubling.
• Figure 2: The odd functor for the type-X assignment recovers the even functor for the right ladybug matching. Consider the two types of ladybugs, X and Y, and name the circles appearing in the various resolutions $a,b_1,b_2,c_1,c_2,d$ as shown (their ordering does not matter). The coefficients of the relevant portion of the functor $\mathfrak{F}_o$ (as well as those of the assignment $\mathfrak{F}'_o$ in parentheses) are shown. Since we are considering a type-X assignment, $\mathfrak{F}_o$ is chosen to differ from $\mathfrak{F}'_o$ in one edge for the X-ladybug, and is chosen to agree with $\mathfrak{F}'_o$ for the Y-ladybug. In either case, the unique sign-respecting $2$-isomorphism is the matching $(a,b_1,d)\leftrightarrow (a,c_1,d), (a,b_2,d)\leftrightarrow (a,c_2,d)$, which is the right ladybug matching from lshomotopytype.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.3
• Definition 2.5