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

2022/06/29 by Lupton, Gregory, Smith, Samuel Bruce
#55P62 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.14622

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 Baut₁(X). We prove a simply connected space X satisfying Baut₁(X_ℚ) ≃ S_ℚ²ⁿ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree 2n-1 for n= 1, 2, 3.

Related

• A rational realization problem in Gottlieb group
• A rational obstruction to be a Gottlieb map
• Rational homotopy theory: a brief introduction
• Rational homotopy theory via Sullivan models: a survey
• Rational Homotopy Equivalence