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Infinite families of standard Cappell-Shaneson spheres

Kazunori Iwaki

TL;DR

This work advances the study of Cappell-Shaneson spheres by establishing numerous new infinite families that are standard, i.e., diffeomorphic to the 4-sphere $S^4$. Using the Latimer-MacDuffee-Taussky correspondence to translate CS matrices into ideal classes in C(ℤ[θ_n]), together with Gompf equivalence and symmetry between traces, the authors derive arithmetic criteria (via congruences) that guarantee standardness; they provide concrete triples (c,p,n0) yielding X_{c,p,p^2k+n0} not similar to any CS matrix from the A_n family. The results extend the prior Kim–Yamada framework, enumerate many new cases (including 146 triples and 145 new infinite families with p>7), and illustrate a robust method to generate standard CS spheres by number-theoretic means. Overall, the paper broadens the landscape of known standard CS spheres and informs the ongoing program related to the smooth 4-dimensional Poincaré conjecture.

Abstract

Cappell-Shaneson homotopy 4-spheres (CS spheres) are potential counterexamples of the smooth 4-dimensional Poincaré conjecture. Akbulut proved that infinite CS spheres are diffeomorphic to the standard 4-sphere by Kirby calculus. Kim and Yamada found another family of CS spheres which is composed of standard CS spheres. In this paper, we prove more CS spheres are standard. We give 145 new infinite families of CS spheres which are diffeomorphic to the standard 4-sphere.

Infinite families of standard Cappell-Shaneson spheres

TL;DR

This work advances the study of Cappell-Shaneson spheres by establishing numerous new infinite families that are standard, i.e., diffeomorphic to the 4-sphere . Using the Latimer-MacDuffee-Taussky correspondence to translate CS matrices into ideal classes in C(ℤ[θ_n]), together with Gompf equivalence and symmetry between traces, the authors derive arithmetic criteria (via congruences) that guarantee standardness; they provide concrete triples (c,p,n0) yielding X_{c,p,p^2k+n0} not similar to any CS matrix from the A_n family. The results extend the prior Kim–Yamada framework, enumerate many new cases (including 146 triples and 145 new infinite families with p>7), and illustrate a robust method to generate standard CS spheres by number-theoretic means. Overall, the paper broadens the landscape of known standard CS spheres and informs the ongoing program related to the smooth 4-dimensional Poincaré conjecture.

Abstract

Cappell-Shaneson homotopy 4-spheres (CS spheres) are potential counterexamples of the smooth 4-dimensional Poincaré conjecture. Akbulut proved that infinite CS spheres are diffeomorphic to the standard 4-sphere by Kirby calculus. Kim and Yamada found another family of CS spheres which is composed of standard CS spheres. In this paper, we prove more CS spheres are standard. We give 145 new infinite families of CS spheres which are diffeomorphic to the standard 4-sphere.
Paper Structure (18 sections, 23 theorems, 15 equations, 1 table)

This paper contains 18 sections, 23 theorems, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. History of Cappell-Shaneson spheres
  3. Main results
  4. Preliminaries
  5. Construction
  6. Gompf equivalence
  7. Latimer-MacDuffee-Taussky correspondence
  8. Symmetry between Cappell-Shaneson matrices
  9. Ideal class monoid C(Z[theta])
  10. Dedekind-Kummer theorem
  11. When C(Z[theta]) is not a group
  12. Finding representatives of ideal classes
  13. Representatives of ideal classes for given trace n
  14. Other representatives for given ideal classes
  15. Table of representatives of C(Z[theta])
  16. ...and 3 more sections

Key Result

Theorem 1.4

CS spheres corresponding to $X_{2,7,49k+27}$ is standard for any $k$. $X_{2,7,49k+27}$ is not similar to $A_n$ for any $k$ and for any $n$. In other words, they gave an infinite family of CS spheres which are diffeomorphic to $S^4$ and the family is different from $\Sigma_{A_n}^\varepsilon$.

Theorems & Definitions (53)

  • Conjecture 1.1: The smooth 4-dimensional Poincaré conjecture
  • Conjecture 1.2
  • Conjecture 1.3: Gompf:2010-1
  • Theorem 1.4: Kim-Yamada:2017-1
  • Theorem 1.5: Theorem \ref{['theorem:infiniteseries']}
  • Corollary 1.6
  • Proposition 2.1: Cappell-Shaneson:1976-1, Issa:2017-1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 43 more