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We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are themaximal and minimal second Betti number divided by $8$ among definite spinboundings 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 somehomology 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 theconstruction, some negative classes which consist of addition of several fiberclasses plus one sectional class in $E(1)$ are represented by spheres.

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