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On four-manifolds without 1- and 3-handles

R. Inanc Baykur

Abstract

We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i.e. they can be given handle decompositions without 1- and 3-handles. In particular, there are many such 4-manifolds homeomorphic but not diffeomorphic to the standard 4-manifolds # m (S^2 x S^2) and # n (CP^2 # -CP^2), respectively, which answers Problem 4.91 on Kirby's 1997 list.

On four-manifolds without 1- and 3-handles

Abstract

We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i.e. they can be given handle decompositions without 1- and 3-handles. In particular, there are many such 4-manifolds homeomorphic but not diffeomorphic to the standard 4-manifolds # m (S^2 x S^2) and # n (CP^2 # -CP^2), respectively, which answers Problem 4.91 on Kirby's 1997 list.
Paper Structure (2 sections, 1 theorem, 6 equations)

This paper contains 2 sections, 1 theorem, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem and examples

Key Result

Theorem 1

There exist irreducible, closed, simply connected $4$--manifolds that admit perfect Morse functions, where the $4$--manifolds have prescribed signature $\sigma$ and prescribed spin type when $\sigma$ is divisible by $16$, and arbitrarily large Euler characteristic.

Theorems & Definitions (1)

  • Theorem 1