Equivariant Khovanov Homotopy Type and Periodic Links

Abstract

In this thesis, we give two equivalent definitions for a group $G$ acting on a strictly-unitary-lax-2-functor $D:\CC\rightarrow\mathscr{B}$ from the cube category to the Burnside category. We then show that the natural $\mathbb{Z}/p\mathbb{Z}$ action on a $p$-periodic link $L$ induces such an action on Lipshitz and Sarkar's Khovanov functor $F_{Kh}(L): \CC \rightarrow \mathscr{B}$ which makes the Khovanov homotopy type $\mathcal{X}(L)$ into an equivariant knot invariant. That is, if a link $L'$ is equivariantly isotopic to $L$, then $\mathcal{X}(L')$ is Borel homotopy equivalent to $\mathcal{X}(L)$.