Representation homology of topological spaces
| Autor: | Ajay C. Ramadoss, Yuri Berest, Wai-kit Yeung |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General Mathematics Homology (mathematics) Topological space 01 natural sciences Mathematics::Algebraic Topology symbols.namesake Mathematics - Geometric Topology Mathematics::K-Theory and Homology 0103 physical sciences FOS: Mathematics Algebraic Topology (math.AT) Free loop Mathematics - Algebraic Topology 0101 mathematics Invariant (mathematics) Representation Theory (math.RT) Mathematics::Symplectic Geometry Mathematics Hochschild homology Riemann surface 010102 general mathematics Geometric Topology (math.GT) K-Theory and Homology (math.KT) Automorphism Mathematics::Geometric Topology Spectral sequence Mathematics - K-Theory and Homology symbols 010307 mathematical physics Mathematics - Representation Theory |
| Popis: | In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the 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 homology theories associated with spaces (such as Pontryagin algebras, $S^1$-equivariant homology of the free loop space and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in $\R^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $\R^3$. A substantially revised version. New results are added, including the existence of the derived representation adjunction and the commutativity of the derived representation functor with arbitrary homotopy colimits. We deduce these results from a version of (derived) adjunction theorem for categories with weak equivalences that extends formally Quillen's classical theorem for model categories |
| Databáze: | OpenAIRE |
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