Rational homotopy theory of mapping spaces via Lie theory for \(L_\infty\)-algebras (Q5963055)

The author calculates the higher homotopy groups of the Deligne-Getzler \(\infty\)-groupoid \(\gamma_\bullet(\mathfrak g)\) associated to a nilpotent \(L_\infty\)-algebra \(\mathfrak g\) [\textit{E. Getzler}, Ann. Math. (2) 170, No. 1, 271--301 (2009; Zbl 1246.17025)]. The main result gives natural group isomorphisms: NEWLINE\[NEWLINE \pi_{n+1}( \gamma_\bullet(\mathfrak g), \tau) \cong H_n(\mathfrak g^\tau) NEWLINE\]NEWLINE for \(n \geq 0\). Here \(\tau\) is a Maurer-Cartan element and \(\mathfrak g^\tau\) has the twisted bracket induced by \(\tau\). The group structure on \(H_0(\mathfrak g^\tau)\) is given by the Campbell-Hausdorff formula. This result is extended to complete \(L_\infty\)-algebras.NEWLINENEWLINEAs a consequence, the author obtains a model for the space \(\mathrm{Map}(X, Y_\mathbb{Q})\) of continuous maps from a connected space \(X\) into the rationalization \(Y_\mathbb{Q}\) of a nilpotent space \(Y\) in terms of the Deligne-Getzler \(\infty\)-groupoid. Specifically, the author deduces a homotopy equivalence of the form: NEWLINE\[NEWLINE\mathrm{Map}(X, Y_\mathbb{Q}) \simeq \gamma_\bullet(A \hat{\otimes} L),NEWLINE\]NEWLINE where \(A\) is a commutative differential graded algebra model for \(X\), \(L\) is an \(L_\infty\) model for \(Y\) and \(A \hat{\otimes} L\) is the inverse limit of the nilpotent \(L_\infty\)-algebras \(A \otimes L/L_{\geq r}\). In particular, the higher homotopy groups of the function space at a basepoint \(f : X \to Y_\mathbb{Q}\) are given by \(\pi_{n+1}(\mathrm{Map}(X, Y_\mathbb{Q}), f) \cong H_n(A \hat{\otimes} L^\tau)\) for \(n \geq 0\) and \(\tau\) the Maurer-Cartan element corresponding to the map \(f\). Several applications of these results are provided, including a formula for the rational homotopy groups of the space of self-equivalences of a Koszul space [\textit{A. Berglund}, Trans. Am. Math. Soc. 366, No. 9, 4551--4569 (2014; Zbl 1301.55006)].