The $E_8$-boundings of homology spheres and negative sphere classes in $E(1)$

TL;DR

This paper introduces new invariants related to definite spin boundings of homology spheres, computes them for specific examples, and constructs $E_8$-boundings for certain Brieskorn spheres, revealing negative sphere classes in $E(1)$.

Contribution

It defines novel invariants for homology spheres, computes these invariants for specific cases, and constructs explicit $E_8$-boundings, including negative sphere classes in $E(1)$.

Findings

Computed invariants for some homology spheres.

Constructed $E_8$-boundings for Brieskorn spheres.

Identified negative classes represented by spheres in $E(1)$.

Abstract

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifolds. We compute these invariants for some homology spheres. We construct $E_8$-boundings for some of Brieskorn 3-spheres $\Sigma(2,3,12n+5)$ by handle decomposition. As a by-product of the construction, some negative classes which consist of addition of several fiber classes plus one sectional class in $E(1)$ are represented by spheres.