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A cut-and-paste mechanism to introduce fundamental group and construct new four-manifolds

Valentina Bais, Rafael Torres

Abstract

We introduce a simple cut-and-paste mechanism to construct both orientable and nonorientable four-manifolds from a given initial one. This mechanism alters the fundamental group while preserving other essential topological invariants. It avoids codimension two cut-and-paste fundamental group computations and fast tracks the search for fixed-point free involutions. The mechanism proves useful to unveil novel exotic irreducible smooth structures on closed four-manifolds with finite cyclic fundamental group, which include $\Q$-homology real projective four-spaces.

A cut-and-paste mechanism to introduce fundamental group and construct new four-manifolds

Abstract

We introduce a simple cut-and-paste mechanism to construct both orientable and nonorientable four-manifolds from a given initial one. This mechanism alters the fundamental group while preserving other essential topological invariants. It avoids codimension two cut-and-paste fundamental group computations and fast tracks the search for fixed-point free involutions. The mechanism proves useful to unveil novel exotic irreducible smooth structures on closed four-manifolds with finite cyclic fundamental group, which include -homology real projective four-spaces.

Paper Structure

This paper contains 18 sections, 30 theorems, 108 equations.

Table of Contents

  1. Introduction.
  2. Acknowledgements
  3. Constructions of oriented four-manifolds.
  4. Cut-and-paste procedure: oriented setting
  5. Examples of Theorem \ref{['Theorem A']}.
  6. Raw materials for Theorem \ref{['Theorem A']}
  7. Diffeomorphism obstruction
  8. Proof of Theorem \ref{['Theorem A']}
  9. Mechanism in Theorem \ref{['Theorem Main']}.
  10. Seiberg-Witten invariants of smoothly reducible double covers
  11. Proof of Theorem \ref{['Theorem Main']}
  12. Proof of Theorem \ref{['Theorem Geo']}: new irreducible examples with cyclic fundamental group.
  13. Constructions of nonorientable four-manifolds and the examples of Theorem \ref{['Theorem D']}.
  14. Cut-and-paste procedure: nonorientable setting
  15. Nonorientable raw material and its properties
  16. ...and 3 more sections

Key Result

Theorem 3

For every odd $p \in \mathbb{Z}_{> 0}$, there is a closed smooth oriented four-manifold $\mathcal{R}_{2p}$ that is homeomorphic but not diffeomorphic to the connected sum $\Sigma_{2p}\mathbin{\#} 4\overline{{\mathbb C\mkern-0.5mu\mathrm P}^2}$, where $\Sigma_{2p}$ is a $\mathbb{Q}$-homology four-sph

Theorems & Definitions (39)

  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Example 1
  • Lemma 2
  • Remark 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • ...and 29 more