Representation homology and knot contact homology

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This thesis has four parts. In the first part, we introduce and study representation homology of topological spaces, which is a higher homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to Pirashvili's construction of higher Hochschild homology; in fact, we establish a direct relation between the two theories by proving that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other known homology theories associated with spaces (such as the Pontryagin algebra $ H_*(\Omega X) $, the S1-equivariant homology of the free loop space and the stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly in a number of interesting cases, including the spheres Sn, the complex projective spaces $CP^r $, closed surfaces of arbitrary genus and some 3-dimensional manifolds, such as link complements in R3 and lens space L(p,q). One of our main results, which we call Comparison Theorem, expresses the representation homology of a simply-connected topological space of finite rational type in terms of its Quillen and Sullivan models. The second part is a compendium of the first part. We prove some technical results required to establish some basic properties of representation homology. A result that might be of independent interest is Theorem 10.3.2, which says that if k is a field of characteristic zero, then for any k-linear operad P, the model category of simplicial P algebras is Quillen equivalent to the model category of non-negatively graded DG P algebras. This is the result that allows us to transition between simplicial commutative algebras and commuatative DG algebras, thereby proving some results (see, e.g., Proposition 11.3.13 and 11.3.14) about smooth extensions of simplicial commutative algebras.These results would find applications in derived representations schemes, which are simplicial commutative algebras that give rise to representation homology. In the third part, we give a new algebraic construction of knot contact homology in the sense of Ng [Ng05a]. For a link L in R3, we define a differential graded (DG) k-category $ \tilde{\mathscr{A}}_L $ with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of a distinguished object of $\tilde{\mathscr{A}}_L $ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng and Sullivan in [EENS13a]. The input of our construction is a natural action of the braid group Bn on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson and Vilonen in [GMV96]. As an application, we show that the category of finite-dimensional representations of the link k-category $ \tilde{A}_L = H_0(\tilde{\mathscr{A}}_L) $ defined as the 0-th homology of $ \tilde{\mathscr{A}}_L $ is equivalent to the category of perverse sheaves on R3 that are singular along the link $ L $. We also obtain several generalizations of the category $ \tilde{\mathscr{A}}_L $ by extending the Gelfand-MacPherson-Vilonen braid group action. In the forth part, we generalize Keller's construction [Kel11] of deformed n-Calabi-Yau completions to the relative contexts. This gives a universal construction that extends any given DG functor F:A→B to a DG functor F~:A~→B~, together with a family of deformations of F~ parametrized by relative negative cyclic homology classes [η]∈\HCn−2−(B,A). We show that, under a finiteness condition, these extensions have canonical relative n-Calabi-Yau structures in the sense of [BD]. This is applied to give a construction that associates a DG category A(N,M;Φ) to a pair (N,M) consisting of a manifold N and an embedded submanifold M of codimension ≥2, together with a trivialization Φ of the unit normal bundle of M in N. In the case when (N,M) is the pair consisting of a set of n points in the interior of the 2-dimensional disk, A(N,M;Φ) is the multiplicative preprojective algebra [CBS06] with non-central parameters. In the case when (N,M) is the pair consisting of a link L in \bR3, A(N,M;Φ) is the link DG category [BEY] that extends the Lengendrian DG algebra [Ng05a, Ng05b, Ng08, EENS13a] of the unit conormal bundle STL∗(\bR3)⊂ST∗(\bR3). We show that, when M⊂N has codimension 2, then the category of finite dimensional modules over the 0-th homology H0(A(N,M;Φ)) of this DG category is equivalent to the category of perverse sheaves on N with singularities at most along M. Our main references are the papers [BRY], [BEY16a] and [Yeu], which form Part I, III and IV of this thesis respectively.

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2017-08-30
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Mathematics
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Berest, Yuri
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Sjamaar, Reyer
Knutson, Allen
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Mathematics
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Ph. D., Mathematics
Degree Level
Doctor of Philosophy
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