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Some examples of small irreducible exotic 4-manifolds with free abelian fundamental group

Valentina Bais, Rafael Torres, Daniele Zuddas

Abstract

We produce examples of pairwise non-diffeomorphic closed irreducible 4-manifolds with non-trivial free abelian fundamental group of rank less than three and small Euler characteristic. These exotic smooth structures become standard after taking a connected sum with a single copy of $S^2\times S^2$. The contributions of this paper include an explicit mechanism to computate the equivariant intersection form of 4-manifolds that are obtained via torus surgeries and a new stabilization result concerning exotic smooth structures with arbitrary fundamental group.

Some examples of small irreducible exotic 4-manifolds with free abelian fundamental group

Abstract

We produce examples of pairwise non-diffeomorphic closed irreducible 4-manifolds with non-trivial free abelian fundamental group of rank less than three and small Euler characteristic. These exotic smooth structures become standard after taking a connected sum with a single copy of . The contributions of this paper include an explicit mechanism to computate the equivariant intersection form of 4-manifolds that are obtained via torus surgeries and a new stabilization result concerning exotic smooth structures with arbitrary fundamental group.

Paper Structure

This paper contains 16 sections, 16 theorems, 68 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction.
  2. Acknowledgements
  3. Preliminary results and construction tools.
  4. Equivariant intersection form
  5. Classification up to homeomorphism
  6. Torus surgeries and framings
  7. Luttinger surgery, irreducibility, and the symplectic Kodaira dimension
  8. Infinitely many diffeomorphism types and irreducibility
  9. The second homotopy group of a closed orientable 4-manifold with free abelian fundamental group of small rank.
  10. A construction of irreducible 4-manifolds.
  11. An irreducible symplectic 4-manifold with free abelian fundamental group
  12. Some 4-manifolds obtained from torus surgeries on $M$ and their fundamental groups
  13. A surgical procedure to represent homology classes by surfaces with trivial $\pi_1$-inclusion.
  14. Topological prototypes.
  15. Stabilization of $p$-torus surgeries.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $k\in \{2, 3, 4, 5\}$. There are infinite collections of closed smooth oriented pairwise non-diffeomorphic 4-manifolds that satisfy the following properties. Moreover, the 4-manifolds $M_{1, k}$ and $N_{1, k}$ admit a symplectic structure of symplectic Kodaira dimension two.

Figures (3)

  • Figure 1: Reciprocal positions of $T$, $S$ and $\Sigma$ under the hypothesis of Section \ref{['construction']}.
  • Figure 2: Curves in the 2-tori $T_1$ and $T_2$.
  • Figure 3: Band desingularization.

Theorems & Definitions (40)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Theorem 2.7
  • ...and 30 more