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Tautological Rings of Fibrations

Nils Prigge

Abstract

We study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincar\' e fibre, i.e. the ring obtained by fibre integrating powers of the fibrewise Euler class. We discuss how to compute the Euler ring with tools from rational homotopy theory and completely determine the tautological ring for even spheres, complex projective spaces and some products of odd spheres.

Tautological Rings of Fibrations

Abstract

We study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincar\' e fibre, i.e. the ring obtained by fibre integrating powers of the fibrewise Euler class. We discuss how to compute the Euler ring with tools from rational homotopy theory and completely determine the tautological ring for even spheres, complex projective spaces and some products of odd spheres.

Paper Structure

This paper contains 14 sections, 19 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Rational homotopy theory of fibrations
  4. Rational models of fibrations
  5. Fibre integration in rational homotopy theory
  6. Chain level fibre integration
  7. The fibrewise Euler class
  8. A cocycle representative via rational homotopy theory
  9. The fibrewise Euler class of Leray--Hirsch fibrations
  10. Fibrations with positively rationally elliptic fibre
  11. Computations
  12. The Euler ring of even spheres
  13. The Euler ring of complex projective space
  14. The Euler ring of products of odd spheres

Key Result

Theorem 1

The Euler ring of complex projective space is $E^*(\mathbb{C} P^n)\cong \mathbb{Q}[\kappa_1,\hdots,\kappa_{n-1},\kappa_{n+1}]$.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 20 more