The homotopy theory of Khovanov homology
The purpose of this paper is to establish an interesting interpretation of Khovanov homology in homotopy-theoretic terms. More precisely, the authors start by regarding the cube of abelian groups based on resolutions of a link projections (this is the usual cube used for constructing the Khovanov chain complex) as a functor/presheaf \(F_{KH}: \mathbf Q^{op}\to \mathbf{Ab}\) where \(\mathbf Q\) is the poset of subsets of a finite set with inclusions as morphisms and with an extra final object added in (this is so that the homotopy limit of \(F_{KH}\) comes out to be interesting). This functor also depends on the link diagram \(D\). The main theorem is that the derived functors of the (inverse) limit of \(F_{KH}\) are precisely the unnormalized Khovanov homology: NEWLINE\[NEWLINE \overline{KH}^i(D)\cong \lim_{\mathbf Q^{op}}\,\! ^i F_{KH}, NEWLINE\]NEWLINE where \(D\) is a link diagram.NEWLINENEWLINESince the derived functors can be interpreted as the homotopy limits of diagrams of Eilenberg-MacLane spaces, one gets the following space-level interpretation of the above result: NEWLINE\[NEWLINE \overline{KH}^{n-i}(D)\cong \pi_i(\text{holim}_{\mathbf Q^{op}}(K(-,n)\circ F_{KH}), NEWLINE\]NEWLINE for \(0\leq i\leq n\). Varying \(n\) gives a collection of spaces on the right side of the above which form a spectrum; delooping this spectrum gives the normalized Khovanov homology.NEWLINENEWLINEThe homotopy limit on the right of the above formula is a product of Eilenberg-MacLane spaces and is hence determined by the Khovanov homology. The authors therefore obtain a homotopy type for Khovanov homology, although, in their own admission, in a seemingly uninteresting way. Nevertheless, this results places Khovanov homology into a homotopy-theoretic context and opens various avenues for examining Khovanov homology with techniques of algebraic topology.NEWLINENEWLINEThe paper is very clear and well-written. In order to appeal to both knot theorists and homotopy theorists, the authors give helpful background sections on the construction of Khovanov homology, derived functors, and homotopy limits. They also work out an illustrative example using the machinery of homotopy limits. The hope is that this paper will incite a deeper study and understanding of a space-level version of Khovanov homology.
scientific article; zbMATH DE number 6369097
scientific article