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 Félix-Oprea-Tanré 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.

中文翻译：

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有理同伦中向量束的Thom空间的权重分解

基于子流形的可表示性类的理论，我们研究了矢量束Thom空间的有理同伦理论。我们首先给出有理同伦水平的Thom同构，通过消除碱基无能和束可定向性的假设来扩展Félix-Oprea-Tanré的工作。然后，我们使用有理同伦中的权重分解理论，给出了子流形表示类的可表示性的标准，从而将Papadima的结果进行了推广。在此过程中，我们研究形式问题并给出Thom空间的Massey乘积的公式。最后，我们将权重分解理论与混合Hodge理论联系起来，并将我们的结果应用于动机Thom空间。