Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

Johannes Ebert, Jens Reinhold

Abstract

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller--Morita--Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups.

Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

Abstract

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds , for large and , up to approximately degree . The answer is that it is a free graded commutative algebra on an appropriate set of Miller--Morita--Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, (c) use pseudoisotopy theory and algebraic -theory to get at actual diffeomorphism groups.
Paper Structure (31 sections, 60 theorems, 314 equations)

This paper contains 31 sections, 60 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Context: Madsen--Weiss type theorems
  3. Main result
  4. Relation to Hebestreit--Perlmutter's work
  5. Method of proof and outline of the paper
  6. Acknowledgements
  7. Characteristic classes of smooth and block bundles
  8. Automorphism groups
  9. Tautological classes
  10. Some vanishing theorems for tautological classes
  11. Borel classes
  12. Rational cohomology of block diffeomorphism spaces: general theory
  13. Some words about rational homotopy theory
  14. Block diffeomorphisms versus tangential homotopy automorphisms
  15. Cohomology of mapping spaces
  16. ...and 16 more sections

Key Result

Theorem A

The map is surjective in degrees $* \leq \min (\frac{g-4}{2}, n-3)$, and in that range of degrees, the kernel is the ideal generated by the classes $\mu_{L_m}$ (all $m$) and by the linear subspace $H^1(\Omega^\infty_0 \mathrm{MT} \theta_{2n+1}^n;\mathbb{Q})$.

Theorems & Definitions (125)

  • Theorem A
  • Remark 1.2
  • Theorem 1.4
  • Theorem B
  • Remark 1.9
  • Remark 1.13
  • Lemma 2.3
  • proof
  • Lemma 2.8
  • proof
  • ...and 115 more