Hochschild cohomology of Sullivan algebras and mapping spaces
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)].
scientific article; zbMATH DE number 7008436