Becker, Hanno: Homotopy-Theoretic Studies of Khovanov-Rozansky Homology. - Bonn, 2015. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
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author = {{Hanno Becker}},
title = {Homotopy-Theoretic Studies of Khovanov-Rozansky Homology},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2015,
month = jul,

note = {This dissertation applies homotopy-theoretic methods to the study of Khovanov-Rozansky homology, a generalization of Khovanov homology (which in turn categorifies the Jones polynomial) that is constructed using categories of matrix factorizations: these are variants of chain complexes for which we have δ2 ≠ 0 and for which therefore no homotopy theory in terms of ordinary derived categories is available.
In this work I study alternative approaches to the construction of homotopy theories for matrix factorizations, and more generally categories of curved modules and singularity categories, and describe the use of these homotopy-theoretic considerations to the understanding of Khovanov-Rozansky homology. The dissertation consists of a knot-theoretic part focusing on concrete applications of homotopy-theoretic techniques to Khovanov-Rozansky homology, and a homotopy-theoretic part, in which these techniques are developed independently and with the aim of large generality in the context of the theory of abelian model structures.
The central results of the knot-theoretic part are the following: Firstly, the development of a conceptual definition of stable Hochschild homology and the description of Khovanov-Rozansky homology as stable Hochschild homology of Rouquier complexes of Soergel bimodules. These are well-known from representation theory and also play an important role in the construction of other knot invariants. Further, the classical knowledge about the combinatorics of Rouquier complexes leads to a direct proof of the fact that Khovanov-Rozansky homology is indeed a knot invariant. Afterwards, the introduction of a combinatorial approximation to Khovanov-Rozansky homology through a diagrammatic calculus similar to ones that can be used for the definition of the Jones polynomial and Khovanov homology.
The central results of the homotopy-theoretic part are the following: Firstly, the construction of abelian model structures for categories of curved modules and singularity categories, on the basis of general techniques for the localization and the proof of cofibrant generation of abelian model structures. Afterwards, the discussion of numerous examples and enrichments of classical equivalences and recollements between triangulated categories to the level of model categories. Finally, the construction of a realization functor from the derived category of an Grothendieck abelian category A to the homotopy category of any reasonable abelian model structure on A, by means of constructing a Quillen equivalent abelian model structure on the category of chain complexes over A.},

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