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

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 $n=1,2,3$.

Keywords

Gottlieb element, Sullivan minimal model, classifying space for a fibration, derivation

2010 Mathematics Subject Classification

Primary 55P62, 55R35. Secondary 55R15.

Full Text (PDF format)

Received 5 January 2022

Received revised 29 June 2022

Accepted 29 September 2022

Published 1 November 2023