Weight decompositions of Thom spaces of vector bundles in rational homotopy

  • Urtzi Buijs
  • Federico Cantero MoránEmail author
  • Joana Cirici


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.


Thom spaces Rational homotopy theory Smoothing theory Thom isomorphism theorem Mixed Hodge structures Weight decompositions Motivic Thom spaces 



We thank Yves Félix for answering our many questions and for telling us about spaces with weight decompositions. Thanks also to Vicente Navarro for his ideas on mixed Hodge theory and useful comments. We are also grateful to Andrew Baker, Pascal Lambrechts, Luc Menichi and Jean-Claude Thomas for their feedback at the early stages of this project. Thanks also to Nero Budur for spotting an imprecision in our exposition of mixed Hodge theory.


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© Tbilisi Centre for Mathematical Sciences 2019

Authors and Affiliations

  • Urtzi Buijs
    • 1
  • Federico Cantero Morán
    • 2
    Email author
  • Joana Cirici
    • 2
  1. 1.Department of Algebra, Geometry and TopologyUniversidad de MálagaMálagaSpain
  2. 2.Mathematics and Computer ScienceUniversitat de BarcelonaBarcelonaSpain

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