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Exotic structures on 4-manifolds with infinite dihedral fundamental group

Simone Tagliente

Abstract

The aim of this paper is to produce infinite exotic structures on smooth closed oriented $4-$manifolds with fundamental group isomorphic to the infinite dihedral group, assuming that $b_2^+$ and $b_2^-$ are at least $12$.

Exotic structures on 4-manifolds with infinite dihedral fundamental group

Abstract

The aim of this paper is to produce infinite exotic structures on smooth closed oriented manifolds with fundamental group isomorphic to the infinite dihedral group, assuming that and are at least .
Paper Structure (17 sections, 16 theorems, 20 equations, 4 figures)

This paper contains 17 sections, 16 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basics
  3. Seiberg-Witten invariant
  4. Torus surgery
  5. Luttinger surgery
  6. Knot-surgery
  7. Subgroups of $D_\infty$
  8. Homeomorphism classification
  9. Building blocks
  10. The fibration $\textgoth{Z}_m$
  11. From $\textgoth{Z}_m$ to $Z_m$ by surgery
  12. Properties of the manifold $Z_m$
  13. Exotica
  14. Other Lemmas
  15. A more general Theorem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix positive integers $a,b$ with $a,b\geq 12$. Then there exists an infinite family of homeomorphic but pairwise not diffeomorphic closed manifolds with fundamental group isomorphic to the infinite dihedral group, $b_2^+=a$, and $b_2^-=b$. When $a=b$, the families consist of irreducible manifolds.

Figures (4)

  • Figure 1: The fibration $\textgoth{Z}_m$ and its decomposition into blocks $A$, $B_0$, $C_0$. The red asterisk indicates the chosen base-point (the intersection between the three blocks is diffeomorphic to $S^1$, and we choose the base-point there). Surgeries on the link of Lagrangian tori displayed in $C_0 \cup B_0$ in dark blue yield the simply connected manifold $Z_m$, with a similar decomposition $Z_m=A\cup B \cup C$. We also define $\overline{Z}_m$ to be $K\cup B \cup (C - \Sigma)$.
  • Figure 2: When gluing two manifolds with boundary parametrized by $S\times S^1$, where $S$ is a surface, we will always use the map given by the composition between the reflection along the red plane and the $\pi$-degree rotation along the blue axis on $S$, and the identity on $S^1$.
  • Figure 3: A schematic representation of the manifolds $Z$, $X$, $W$, $Y$, and $\tilde{Y}$.
  • Figure 4: The universal cover of $Y$. We remove neighborhoods of surfaces in each block, and identify the boundaries according to the red and blue lines. Each group element sends a copy of $Z-(\nu S_1\cup \nu S_2)$ to another one via the identity map, and permutes them according to the $D_\infty$ action. So, order $2$ elements act as 'reflections' across the dotted lines on the schematic, while the $\mathbb{Z}$ subgroup acts via translations.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 3.1
  • Proposition 3.2: bayku, Proposition 2
  • Definition 3.3
  • Definition 3.4
  • ...and 24 more