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.
Categories: math.GT
