Homology cobordism and classical knot invariants (Q699683)

In recent years, gauge theoretical methods have been applied to study the integral homology cobordism group of integral homology 3-spheres (establishing for example that various Brieskorn homology spheres have infinite order in this group). In the present paper, the \(\mathbb Z_2\)-homology cobordism group of the (more frequent) \(\mathbb Z_2\)-homology 3-spheres is studied (i.e. of 3-manifolds with the homology, with coefficients in \(\mathbb Z_2\), of the 3-sphere, obtained for example as 2-fold branched coverings of knots or by surgery on a knot with odd framing). Based on a result of Furuta on the intersection forms of smooth 4-dimensional spin manifolds, two invariants of \(\mathbb Z_2\)-homology 3-spheres are defined and studied which turn out to be cobordism invariants, and which are closely related to classical knot invariants like signature and slice genus. As applications it is shown that many lens spaces have infinite order in the \(\mathbb Z_2\)-homology cobordism group, and a lower bound for the slice genus of a knot is given on which integral surgery yields a given \(\mathbb Z_2\)-homology sphere. Also, some new examples of 3-manifolds are obtained not arising by integral surgery on a knot.