Vanishing theorem for 2-torsion instanton invariants (Q5939522)

Let \(X\) be a closed, oriented, simply connected spin \(4\)-manifold with \(b^{+}>1\) and even. In [\textit{R. Fintushel} and \textit{R. J. Stern}, J. Am. Math. Soc. 6, No. 2, 299-339 (1993; Zbl 0790.57010)] a polynomial invariant with values in \({\mathbb Z}_{2}\) is defined. This is an analogue of the Donaldson invariants introduced in [\textit{S. K. Donaldson}, Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)], which is obtained from the moduli space of anti-self-dual \(SU(2)\)-connections by evaluating a non-trivial torsion class in \(H^{1}\) of the moduli space of irreducible connections modulo gauge (this class only exists for spin manifolds) on a suitable intersection of representatives of \(\mu\) classes. In [Fintushel-Stern, loc. cit.] it is proved that for a manifold of the form \(X'=X \# (S^{2} \times S^{2})\), these invariants do not always vanish. NEWLINENEWLINENEWLINEIn this paper, the author proves that these invariants do vanish for any connected sum \(X_{1} \# X_{2}\) when both manifolds have \(b^{+}>1\) and one takes a collection of homology classes not all of them in one of the two summands. The proof is a prototypical dimension counting for moduli spaces parametrizing glued instantons in both pieces when the neck is shrunk to a point.