Weight decompositions of Thom spaces of vector bundles in rational homotopy theory

TL;DR

This paper advances the rational homotopy theory of Thom spaces of vector bundles by establishing a Thom isomorphism without certain restrictions, and explores applications to submanifold representability, formality, Massey products, and mixed Hodge theory.

Contribution

It extends the Thom isomorphism in rational homotopy to non-nilpotent, non-orientable cases and links weight decompositions with mixed Hodge theory for motivic Thom spaces.

Findings

Established a Thom isomorphism at the rational homotopy level without nilpotency or orientability assumptions.

Provided a criterion for class representability by submanifolds using weight decompositions.

Derived formulas for Massey products of Thom spaces.

Abstract

Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Felix-Oprea-Tanr\'e by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Lastly, we link the theory of weight decompositions with mixed Hodge theory and apply our results to motivic Thom spaces.