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

TL;DR

This paper investigates how Gottlieb elements influence the structure of Sullivan minimal models in rational homotopy theory, revealing parity-dependent results and applications to conjectures and classifying spaces.

Contribution

It establishes new structural results for Sullivan minimal models based on Gottlieb elements, including parity-specific properties and implications for the realization problem.

Findings

In even-degree cases, rational Gottlieb elements are terminal homotopy elements.

An even-degree Gottlieb element induces a free factor in the rational cohomology of formal spaces.

The results support a case of the $2N$-conjecture and inform the structure of classifying spaces.

Abstract

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 $B\mathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying $B\mathrm{aut}_1(X_{\mathbb{Q}}) \simeq S_{\mathbb{Q}}^{2n}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n= 1, 2, 3.$