Rational homotopy theory of function spaces and Hochschild cohomology

2014/06/24 by Amrani, Ilias
#Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1406.6269

Abstract

Given a map f: X→ Y of simply connected spaces of finite type such. The space of based loops at f of the space of maps between X and Y is denoted by Ω_f Map(X,Y). For n> 0, we give a model categorical interpretation of the existence (in functorial way) of an injective map of ℚ-vector spaces π_n Ω_fMap(X,Y_ℚ) → HH^-n(C^∗(Y),C^∗(X)_f), where HH^∗ is the (negative) Hochschild cohomology and C^∗(X)_f is the rational cochain complex associated to X equipped with a structure of C^∗(Y)-differential graded bimodule via the induced map of differential graded algebras f^∗: C^∗(Y)→ C^∗(X). Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when X=Y, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space X i.e., π₁ Aut(X_ℚ)_id.

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