The structuring effect of a Gottlieb element on the Sullivan model of a space

Gregory Lupton ; Samuel Bruce Smith

Homology, homotopy, and applications, Tome 25 (2023) no. 2, pp. 275-296.

Résumé

We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the rational cohomology of a formal space of finite type. We apply the odd-degree result to affirm a special case of the $2N$-conjecture on Gottlieb elements of a finite complex. We combine our results to make a contribution to the realization problem for the classifying space $\operatorname{\mathit{B}aut}_1 (X)$. We prove a simply connected space $X$ satisfying $\operatorname{\mathit{B}aut}_1 (X_\mathbb{Q}) \simeq S^{2n}_\mathbb{Q}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n=1,2,3$.

Détail

DOI : 10.4310/HHA.2023.v25.n2.a12

Classification : 55P62, 55R35, 55R15
Keywords: Gottlieb element, Sullivan minimal model, classifying space for a fibration, derivation