A Khovanov stable homotopy type (Q2922790)

Khovanov homology of an oriented link is a link invariant which takes the form of a bigraded abelian group \(\mathrm{Kh}^{i,j}(L)\), and the Euler characteristic of which is the Jones polynomial. In this paper, the authors construct a family of spectra \(\mathcal{X}_{\mathrm{Kh}}^{j}(L)\) with the properties that the reduced cohomology is the Khovanov homology, namely \(\widetilde{H}^{i}(\mathcal{X}_{\mathrm{Kh}}^{j}(L))=\mathrm{Kh}^{i,j}(L)\), and moreover the homotopy type of \(\mathcal{X}_{\mathrm{Kh}}^{j}(L)\) is a link invariant.NEWLINENEWLINEThe Khovanov spectrum \(\mathcal{X}_{\mathrm{Kh}}(L)\) is the suspension spectrum of the Khovanov space \(|\mathcal{C}_{K}(L)|\), which is a CW complex constructed from a framed flow category \(\mathcal{C}_{K}(L)\) (which is called the Khovanov flow category in this paper). The construction from \(\mathcal{C}_{K}(L)\) to \(|\mathcal{C}_{K}(L)|\) follows the method in Section 3.3, originally owed to Cohen-Jones-Segal. The Khovanov flow category \(\mathcal{C}_{K}(L)\) has one object for each generator of the Khovanov complex, and the set of morphisms between two objects (called moduli space) is a smooth manifold with corners which is constructed inductively.NEWLINENEWLINEThe invariance of \(\mathcal{X}_{\mathrm{Kh}}(L)\) is discussed in Section 6. The authors show that the homotopy type does not depend on the choice of the link diagram and other various choices during the construction. Some properties and calculations are provided in the end of the paper.