The \(E_{8}\)-boundings of homology spheres and negative sphere classes in \(E(1)\) (Q260553)

For a closed 3-manifold \(Y\) that is a homology sphere a spin bounding \(X\) of \(Y\) is definite if the intersection matrix of \(X\) has either no positive or no negative eigenvalues. If there is a bounding whose intersection form is \(nE_{8}\) that is called an \(E_{8}\) bounding. The aim of this paper is to find negative-definite spin boundings or \(E_{8}\)-boundings for some types of Brieskorn homology spheres. The author defines two pairs of topological invariants of homology spheres, one pair consists of the maximal and minimal second Betti number divided by \(8\) of the spin boundings and the other is the maximal and minimal product sum of the quadratic form \(E_{8}\) of the bounding \(4\)-manifolds. The author proves a number of properties of these invariants and gives a table of the negative definite \(E_{8}\)-boundings for a number of Brieskorn homology spheres.