On spin\(\mathbb{Z}/2^p\)-actions on spin 4-manifolds (Q1588336)

Suppose that \(X\) is a smooth, closed, spin 4-manifold with orientation chosen so that the intersection form is \(-kE_8 \oplus mH\) and \(k > 0,m = b_2^+(X) > 0, b_1(X) = 0. \) In this situation the 11/8 conjecture asserts that \(m \geq 3k.\) Furuta, in an unpublished but widely circulated preprint from 1995, proved a weaker form of this conjecture which says \(m \geq 2k + 1.\) In the presence of a spin action of \(\mathbb Z/2^p\) which is of odd type, \textit{J. Bryan} [Seiberg-Witten theory and \(\mathbb{Z}/2^p\) actions on spin 4-manifolds, Math. Res. Lett. 5, No. 1-2, 165-183 (1998)] improved Furuta's inequality to \(m \geq 2k + p + 1\) under some nondegeneracy assumptions -- he gets the same result for even type actions when \(p=1.\) The present paper uses techniques modelled on Bryan's to prove the same inequality as Bryan for even type actions under analogous nondegeneracy conditions. This is applied to classify spin, even type \(\mathbb Z/4\) actions on homotopy \(K3\) and related complex surfaces.