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Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

Anthony Conway, Diarmuid Crowley, Mark Powell, Joerg Sixt

Abstract

For every $k \geq 2$ we construct infinitely many $4k$-dimensional manifolds that are all stably diffeomorphic but pairwise not homotopy equivalent. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In fact we construct infinitely many such infinite sets. To achieve this we prove a realisation result for appropriate subsets of Kreck's modified surgery monoid $\ell_{2q+1}(\mathbb{Z}[π])$, analogous to Wall's realisation of the odd-dimensional surgery obstruction $L$-group $L_{2q+1}^s(\mathbb{Z}[π])$.

Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

Abstract

For every we construct infinitely many -dimensional manifolds that are all stably diffeomorphic but pairwise not homotopy equivalent. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In fact we construct infinitely many such infinite sets. To achieve this we prove a realisation result for appropriate subsets of Kreck's modified surgery monoid , analogous to Wall's realisation of the odd-dimensional surgery obstruction -group .

Paper Structure

This paper contains 32 sections, 47 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Manifolds with infinite homotopy stable class
  3. Realising modified surgery obstructions
  4. Boundary isomorphisms and primitive embeddings
  5. Extended symmetric forms
  6. Forms and formations
  7. Forms and formations
  8. Quasi-formations and the l-monoid.
  9. $0$-stabilised quadratic forms and the structure of $\ell_{2q+1}(R)$
  10. Realising odd dimensional modified surgery obstructions
  11. Normal smoothings, normal bordisms and Wall forms
  12. The modified surgery obstruction
  13. Realising modified surgery obstructions
  14. Boundary isomorphisms
  15. Stable isomorphisms of split formations
  16. ...and 17 more sections

Key Result

Theorem 1.1

Fix an integer $k \geq 2$. There is a closed, connected oriented, stably parallelisable, smooth $4k$-manifold $M$ with $\pi_1(M) \cong \mathbb{Z}$ and $|\mathcal{S}^{\rm st}_h(M)| = \infty$. Indeed, there are infinitely many stable diffeomorphism classes of such manifolds, $\{[M_i]_{\rm st} \}_{i=1}

Theorems & Definitions (133)

  • Theorem 1.1
  • Theorem 1.2: Wall Realisation
  • Theorem 1.3: Realisation of modified surgery obstructions
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.9
  • Theorem 1.10
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 123 more