Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extended surgery theory for simply-connected $4k$-manifolds

Csaba Nagy

Abstract

Kreck proved that two $2q$-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q-1)$-smoothings over the same normal $(q-1)$-type $(B,ξ)$. We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k-1)$-connected $4k$-manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected $4k$-manifold $M$ with normal $(q-1)$-type $(B,ξ)$ such that $H_q(B)$ is free, an algebraic description of the stable class of $M$, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to $M$. Using this description, we explicitly compute the stable class of manifolds $M$ with rank-$2$ hyperbolic intersection form.

Extended surgery theory for simply-connected $4k$-manifolds

Abstract

Kreck proved that two -manifolds are stably diffeomorphic if and only if they admit normally bordant normal -smoothings over the same normal -type . We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on ), when the manifolds are simply-connected, is even and is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a -connected -manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For , the combination of these results determines the inertia groups. We also obtain, for a simply-connected -manifold with normal -type such that is free, an algebraic description of the stable class of , that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to . Using this description, we explicitly compute the stable class of manifolds with rank- hyperbolic intersection form.
Paper Structure (25 sections, 86 theorems, 50 equations)

This paper contains 25 sections, 86 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Extended quadratic forms
  3. Preliminaries
  4. Basic constructions
  5. Full geometric metabolic forms
  6. The group $\mathop{\mathrm{RU}}\nolimits_{\mathop{\mathrm{st}}\nolimits}(\underline{M},L)$
  7. Extended $\ell$-monoids
  8. Definitions
  9. The monoid $\ell_{2q+1}(Q,v)$
  10. The group $L_{2q+1}(Q,v)$
  11. The $\ell$-monoid associated to a space
  12. Normal maps and Q-forms
  13. Normal maps
  14. The Q-form of a map
  15. The extended surgery obstruction $\theta_{W,F}$
  16. ...and 10 more sections

Key Result

Theorem 1

A pair of $2q$-manifolds are stably diffeomorphic if and only if they have the same normal $(q{-}1)$-type and they admit bordant normal $(q{-}1)$-smoothings over it.

Theorems & Definitions (212)

  • Theorem : Kreck kreck99
  • Theorem 1.1
  • Theorem 1.2: Q-form conjecture for simply-connected $B$ and even $q$, with free $H_q(B)$
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5: $q=4$: Senger-Zhang senger-zhang22, Crowley-Nagy ann-3c8m, $q=8$: Senger-Zhang senger-zhang22, Crowley-Olbermann
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Definition 1.9
  • ...and 202 more