The homotopy theory of Khovanov homology
type: Item
instance of: scholarly article
title: The homotopy theory of Khovanov homology
zbMATH Open document ID: 1312.57014
published in: Algebraic \& Geometric Topology
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.3460
reviewed by: Ismar Volic
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 6369097
zbMATH Keywords: Khovanov homology
zbMATH Keywords: homotopy limits
zbMATH Keywords: higher inverse limits
MaRDI profile type: Publication
OpenAlex ID: W3104156427
arXiv ID: 1112.3460
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DOI: 10.2140/AGT.2014.14.2747
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author: Brent Everitt
author: Paul Turner
review text: 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 auth…
description: scientific article; zbMATH DE number 6369097