Hochschild cohomology of Sullivan algebras and mapping spaces
type: Item
instance of: scholarly article
title: Hochschild cohomology of Sullivan algebras and mapping spaces
zbMATH Open document ID: 1413.55006
published in: Arab Journal of Mathematical Sciences
publication date: 2019-01-30T00:00:00Z
review text: Let \(X\) and \(Y\) be simply connected spaces having the homotopy types of finite CW complexes, and let \(f:X\to Y\) be a continuous map. Let \(\phi :(\wedge V,d)\to (B,d)\) be a Sullivan model of \(f\) and let \(\mathrm{map}(X,Y;f)\) denote the path component of \(f\) in the space of continuous maps from \(X\) to \(Y\). For a graded algebra \(A\) and a differential \((A,d)\)-bimodule \((M,d)\), let \(HH^*(A;M)\) denote the Hochschild cohomology of \(A\) with coefficients in \(M\) defined by \(HH^*(A;M)=\mathrm{Ext}_{A^e}(A,M)\), where \(A^e=A\otimes A^{op}\) and the \(A^e\)-module structure of \(A\) is given by the action \((a_1\otimes a_2)a=(-1)^{|a||a_2|} a_1aa_2\). By the work due to \textit{M. Gerstenhaber} and \textit{S. D. Schack} [J. Pure Appl. Algebra 48, 229--247 (1987; Zbl 0671.13007)], it is known that there is a Hodge type decomposition of the Hochschild cohomology \(HH^*(\wedge V;M)\cong\bigoplus_{n\geq 0}HH^*_{(n)}(\wedge V;M)\). In this paper, the author proves that there is a canonical injection \[ \pi_*(\Omega \mathrm{map}(X,Y;f))\otimes \mathbb{Q}\to HH^*(\wedge V;B) \] and he also obtains the isomorphism \(\pi_*(\Omega \mathrm{map}(X,Y;f))\otimes \mathbb{Q}\cong HH^*_{(1)}(\wedge V;B)\). The proof is based on the careful analysis of the \(L_{\infty}\)-models of mapping spaces due to the works of Buijs-Félix-Murillo, cf. [\textit{U. Buijs} et al., J. Math. Soc. Japan 63, No. 2, 503--524 (2011; Zbl 1252.55006) and Rev. Mat. Complut. 26, No. 2, 573--588 (2013; Zbl 1334.55003)].
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 7008436
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: mapping space
zbMATH Keywords: \(L_{\infty}\) algebra
author: Jean Baptiste Gatsinzi
MaRDI profile type: Publication
full work available at URL: j.ajmsc.2018.07.002
OpenAlex ID: W2886196782
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DOI: 10.1016/J.AJMSC.2018.07.002
description: scientific article; zbMATH DE number 7008436