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Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

Dave Auckly, Daniel Ruberman

TL;DR

This work demonstrates that smooth 4-manifolds harbor exotic, higher-parameter families of diffeomorphisms whose associated invariants detect infinitely many independent elements in the higher homotopy and homology of Diff^0, yet vanish after stabilization or in the homeomorphism category. The authors develop a parameterized Seiberg-Witten framework, introducing invariants SW^{\pi_k} and SW^{H_k} and a parameterized irreducible-reducible gluing theorem to compute them along a recursive stabilization process. They show these invariants yield Z^∞-summands in kernels of Diff^0(X) → Homeo^0(X) and propagate to classifying spaces and Torelli-type subgroups, with extensions to embedding spaces and spaces of PSC metrics. The results reveal profound contrasts with higher-dimensional stabilization and have wide-ranging consequences for the algebraic topology of Diff, embeddings, and PSC geometry in dimension four, providing new tools to probe smooth structures via gauge theory.

Abstract

We construct infinite rank summands isomorphic to $\mathbb{Z}^\infty$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with $S^2 \times S^2$. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give $\mathbb{Z}^\infty$ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and $3$-manifolds in various $4$-manifolds, and for the space of positive scalar curvature metrics on standard PSC $4$-manifolds.

Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

TL;DR

This work demonstrates that smooth 4-manifolds harbor exotic, higher-parameter families of diffeomorphisms whose associated invariants detect infinitely many independent elements in the higher homotopy and homology of Diff^0, yet vanish after stabilization or in the homeomorphism category. The authors develop a parameterized Seiberg-Witten framework, introducing invariants SW^{\pi_k} and SW^{H_k} and a parameterized irreducible-reducible gluing theorem to compute them along a recursive stabilization process. They show these invariants yield Z^∞-summands in kernels of Diff^0(X) → Homeo^0(X) and propagate to classifying spaces and Torelli-type subgroups, with extensions to embedding spaces and spaces of PSC metrics. The results reveal profound contrasts with higher-dimensional stabilization and have wide-ranging consequences for the algebraic topology of Diff, embeddings, and PSC geometry in dimension four, providing new tools to probe smooth structures via gauge theory.

Abstract

We construct infinite rank summands isomorphic to in the higher homotopy and homology groups of the diffeomorphism groups of certain -manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with . The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and -manifolds in various -manifolds, and for the space of positive scalar curvature metrics on standard PSC -manifolds.
Paper Structure (34 sections, 38 theorems, 136 equations, 1 figure)

This paper contains 34 sections, 38 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Invariants of families
  3. Family moduli spaces
  4. Subgroups of Diff(X)
  5. Regularity of family moduli spaces
  6. Invariants of homotopy groups and families
  7. Definition of the homotopy group invariant SWk
  8. Relation to Konno's characteristic classes SW
  9. Homology invariants of Diff(X)
  10. Moduli spaces parameterized by chains
  11. Definition of the invariants
  12. Basic properties of the invariants
  13. Non-isotopic diffeomorphisms
  14. Families of diffeomorphisms and collections of manifolds
  15. Commutators and spheres in Diff(Z)
  16. ...and 19 more sections

Key Result

Theorem 1.1

For any $p > 0$ there are $4$-manifolds ${\bf Z}^p$ such that for all $0< j \leq p$ with $j \equiv p \pmod{2}$ the following groups contain $\mathbb Z^\infty$ summands: For $j = p=0$ the groups $\mathop{\mathrm{Diff}^{\space0\space}}\nolimits({\bf Z}^p)$ in items (1) and (2) should be replaced by $\mathop{\mathrm{Diff}}\nolimits({\bf Z}^p,\mathfrak{s},\mathcal{O})$ for a particular choice of $\op

Figures (1)

  • Figure 1: Independence from metric and perturbation

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 94 more