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Connected sums of Brieskorn contact $5$-spheres

Florian Buck, Kai Zehmisch

TL;DR

The work investigates the extent to which Brieskorn contact structures on $S^5$ are closed under the contact connected sum. It leverages Brieskorn’s topological criterion and the $S^1$-equivariant mean Euler characteristic from symplectic homology to construct infinitely many pairs of Brieskorn spheres whose contact connected sum is not Brieskorn, and to analyze invertibility within the resulting monoid. A key contribution is the monoidomorphic behavior of $\chi_{\mathrm m}$, enabling explicit non-Brieskorn sums such as $\xi_{\mathbf a}\#\xi_{\mathbf a}$ to have contrasting signs with their factors. The results illuminate how pairwise coprimeness of exponents and invariant Brieskorn submanifolds constrain positivity invariants, thereby showing limits of Brieskorn-constructible contact structures under sums and informing the landscape of contact structures on $S^5$.

Abstract

In dimension $5$, the contact connected sum of Brieskorn contact spheres is, in general, not a Brieskorn contact sphere.

Connected sums of Brieskorn contact $5$-spheres

TL;DR

The work investigates the extent to which Brieskorn contact structures on are closed under the contact connected sum. It leverages Brieskorn’s topological criterion and the -equivariant mean Euler characteristic from symplectic homology to construct infinitely many pairs of Brieskorn spheres whose contact connected sum is not Brieskorn, and to analyze invertibility within the resulting monoid. A key contribution is the monoidomorphic behavior of , enabling explicit non-Brieskorn sums such as to have contrasting signs with their factors. The results illuminate how pairwise coprimeness of exponents and invariant Brieskorn submanifolds constrain positivity invariants, thereby showing limits of Brieskorn-constructible contact structures under sums and informing the landscape of contact structures on .

Abstract

In dimension , the contact connected sum of Brieskorn contact spheres is, in general, not a Brieskorn contact sphere.
Paper Structure (10 sections, 6 theorems, 36 equations)

This paper contains 10 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Topological invariants
  3. Brieskorn's criterion
  4. Positivity of equivariant Euler characteristic
  5. Invariant Brieskorn submanifolds
  6. Contact homological invariant
  7. Positivity of the mean Euler characteristic
  8. Pairwise coprime
  9. Monoidomorphic contact connected sum
  10. Inverse elements

Key Result

Theorem 1

There are infinitely many pairs of $4$-tuples $\mathbf a$ and $\mathbf b$ of natural numbers such that the contact structure $\xi_{\mathbf a}\#\xi_{\mathbf b}$ on $S^5$, defined by contact connected sum of the Brieskorn contact spheres $(\Sigma(\mathbf a),\xi_{\mathbf a})$ and $(\Sigma(\mathbf b),\x

Theorems & Definitions (19)

  • Theorem 1
  • Example 2.1.1
  • Example 2.1.2
  • Remark 2.2.1
  • Proposition 2.3.1
  • proof
  • Remark 2.3.2
  • Lemma 3.1.1
  • proof
  • Corollary 3.1.2
  • ...and 9 more