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On smooth structures over $4$-manifolds with fundamental group of even order

Roberto Ladu, Simone Tagliente

Abstract

We show that any topological, closed, oriented, non-spin $4$-manifold with fundamental group $\mathbb{Z}_{4k}$ and $\min(b_2^+, b_2^-)\geq 15$, has either none or infinitely many distinct smooth structures. Furthermore, we construct infinitely many non-diffeomorphic, irreducible, smooth structures on manifolds with signature zero, $b_2^+$ even and fundamental group $\mathbb{Z}_2\times G$, for any finite group $G$. This extends the results of Baykur-Stipsicz-Szabó.

On smooth structures over $4$-manifolds with fundamental group of even order

Abstract

We show that any topological, closed, oriented, non-spin -manifold with fundamental group and , has either none or infinitely many distinct smooth structures. Furthermore, we construct infinitely many non-diffeomorphic, irreducible, smooth structures on manifolds with signature zero, even and fundamental group , for any finite group . This extends the results of Baykur-Stipsicz-Szabó.

Paper Structure

This paper contains 24 sections, 13 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Organization.
  3. Conventions and background
  4. Conventions and notation
  5. Remarks on surgery along tori
  6. Review of the Baykur-Hamada manifold.
  7. Manifolds with fundamental group $\mathbb{Z}_{4}$
  8. The manifold $X$
  9. The manifolds $\mathcal{X}_p$
  10. The tori $T_{\gamma_i}$
  11. Proof of \ref{['Thm1']} case $k=1$
  12. Fundamental group $\mathbb{Z}_{4k}$
  13. The construction.
  14. Proof of \ref{['Thm1']}, case $k>1$
  15. Fundamental group $\mathbb{Z}_{2}\times G$ with $G$ finite
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $a,b\in \mathbb{N}$ with $a,b\geq 15$. Then the smoothable closed, non-spin, topological, oriented $4$-manifold $Q$ with $\pi_1(Q) \simeq \mathbb{Z}_{4k}$, $b_2^+(Q)=a, b_2^-(Q)=b$, admits infinitely many non-diffeomorphic smooth structures.

Figures (4)

  • Figure 2: Left: arcs $\alpha_i = \gamma_{2-i} \cap B^{(1)}$ lying on the fiber of $B^{(1)}\setminus \nu^\circ (S^{(1)}\cup T_1^{(1)}\cup T_2^{(1)})$. Right: Schematic picture of $X$ and the loops $\gamma_1,\gamma_2$. Internal edges represent identification of sections, external edges identification of tori.
  • Figure 3: Left: the arcs $\alpha_i = \gamma_{2-i} \cap B^{(1)}$ lying on the fiber of $B^{(1)}\setminus \nu (S^{(1)}\cup T_1^{(1)}\cup T_2^{(1)})$. Note that $a_7 = \partial D_2$ (this is $a_7$ not $\alpha_7$). Right: schematic depiction of $X$ for $k=2$. Each vertex is a block $B^{(i)}$, the internal edges represent identification of sections whilst the external edges are due to the identification of tori. The picture also shows how the paths $\gamma_1$ and $\gamma_2$ cross the blocks.
  • Figure 4: Surface $\Sigma_4$ with the curves $x_i, y_i$, $i=1,\dots,4$. The involution $r$ is given by a $\pi$-rotation in the axis $A$ followed by a reflection in the plane $P$.
  • Figure 5: Left: a neighbourhood of $p$ in $S_G$. Right: addition of genus and modification of $\gamma_i$ with consequent elimination of the intersection point. Notice that the whole orbit of $\gamma_i$ is modified by this operation.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 20 more