The E8-boundings of homology spheres and negative sphere classes in E(1)

We define topological invariants of homology 3-sphere, dsds and ds_, which are the maximal and minimal second Betti number divided by 8 among definite spin boundings of the homology sphere. We also define similar invariants g8g8 and g8_ by the maximal (or minimal) product sum of the quadratic form E8E8 of bounding 4-manifolds. The aim of these invariants is to measure the size of bounding definite spin 4-manifold. We give several ways to construct definite spin boundings. In particular, we construct uncommon E8E8-boundings for Σ(2,3,12n+5)Σ(2,3,12n+5) by using handle decomposition. As a by-product of this construction, we show that some negative 2nd homology classes k[f]−[s]k[f]−[s] in E(1)E(1) are represented by a sphere, where f and s   are a fiber and sectional class of E(1)E(1).