Let \(\Gamma\) be a nonsingular symmetric bilinear form over \(\mathbb{Z}\). A homology 3-sphere \(\Sigma\) is said to bound the form \(\Gamma\) if and only if \(\Sigma\) bounds a compact, oriented, homologically 1-connected smooth 4-manifold \(W\) whose intersection form defined on \(H_2(W)\) is isomorphic to \(\Gamma\). Let \(H\) be the hyperbolic form, i.e., the intersection form of \(S^2\times S^2\). For the paper under review the basic definition is following. The bounding genus \(|\Sigma|\) of \(\Sigma\) is a minimum of \(\{n\mid\Sigma\) bounds \(nH\}\) if the Rohlin invariant \(\mu(\Sigma)\) of \(\Sigma\) is zero. In case \(\mu(\Sigma)=1, |\Sigma|=\infty\). It was introduced by Y. Matsumoto in [J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 29, 287–318 (1982; Zbl 0506.57006)]. The second important object of the paper under review is the \(\omega\)-invariant. Let \((\Sigma,X,c)\) be a triple composed of a homology 3-sphere \(\Sigma\), a compact smooth spin 4-\(V\)-manifold \(X\) [see I. Satake, J. Math. Soc. Japan 9, 464–492 (1957; Zbl 0080.37403)] with boundary \(\partial X=\Sigma\), and \(V\)-spin\(^c\) structure \(c\) on \(X\). We define \(\omega(\Sigma,X,c):=\text{in}d_VD(X \cup_\Sigma W)+\frac{\text{Sign}(W)} {8}\), where \(W\) is a smooth spin 4-manifold with boundary \(\partial W=\Sigma\) and \(\text{ind}_VD(\;)\) is the \(V\)-index of the positive chiral Dirac operator. The main results of the paper try to compare the value of \(|\Sigma|\) with \(\omega(\Sigma,X, c)\). The last four pages illustrate the results on some Brieskorn 3-spheres.