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Exotic diffeomorphisms of 4-manifolds with b_+ = 2

David Baraglia, Joshua Tomlin

TL;DR

The paper addresses the structure of mapping class groups and Torelli groups for simply-connected 4-manifolds with $b_+ = 2$ by developing $1$-parameter Seiberg--Witten invariants in the presence of chamber structures. It defines constant and zero chambers for families over mapping cylinders and establishes gluing formulas for connected sums and blowups, enabling the transfer of invariants from simpler manifolds to more complex ones. The main results show that for all $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}}^2$ surjects to $\\mathbb{Z}^\\infty$, and that the mapping class group of $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}}^2$ is not finitely generated; the authors construct exotic diffeomorphisms via nontrivial $1$-parameter SW invariants with chambers. These results extend previous work at higher $b_+$ to the delicate $b_+=2$ regime and reveal a profusion of exotic diffeomorphisms in low-dimensional gauge-theoretic settings, with further implications for the global structure of mapping class groups in 4-manifolds.

Abstract

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The mapping class group of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The Torelli group of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg-Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg-Witten invariants plays an important role.

Exotic diffeomorphisms of 4-manifolds with b_+ = 2

TL;DR

The paper addresses the structure of mapping class groups and Torelli groups for simply-connected 4-manifolds with by developing -parameter Seiberg--Witten invariants in the presence of chamber structures. It defines constant and zero chambers for families over mapping cylinders and establishes gluing formulas for connected sums and blowups, enabling the transfer of invariants from simpler manifolds to more complex ones. The main results show that for all , the Torelli group of surjects to , and that the mapping class group of is not finitely generated; the authors construct exotic diffeomorphisms via nontrivial -parameter SW invariants with chambers. These results extend previous work at higher to the delicate regime and reveal a profusion of exotic diffeomorphisms in low-dimensional gauge-theoretic settings, with further implications for the global structure of mapping class groups in 4-manifolds.

Abstract

Let be a compact, oriented, smooth, simply-connected -manifold. The mapping class group of is defined as the group of smooth isotopy classes of diffeomorphisms of . The Torelli group of is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each , the Torelli group of surjects to . We also prove that the mapping class group of is not finitely generated. Our proofs of these results makes use of Seiberg-Witten invariants for -parameter familes of -manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have , the chamber structure of the -parameter Seiberg-Witten invariants plays an important role.

Paper Structure

This paper contains 9 sections, 14 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Outline of the proofs of the main results
  3. $1$-parameter Seiberg--Witten invariants for $b_+ = 2$
  4. Seiberg--Witten invariants for $1$-parameter families
  5. Special chambers
  6. Connected sum and blowup formulas for $1$-parameter families
  7. Seiberg--Witten invariants of Kähler surfaces with $b_1 = 0$ and $b_+ = 1$
  8. Exotic diffeomorphisms
  9. Non-finitely generated mapping class groups

Key Result

Theorem 1.1

For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 15 more