$\mathrm{Pin}(2)$–equivariant KO–theory and intersection forms of spin $4$–manifolds

Abstract

Using the Seiberg–Witten Floer spectrum and $Pin\left(2\right)$–equivariant $KO$–theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$–spheres. We then give explicit constrains on the intersection forms of spin $4$–manifolds bounded by Brieskorn spheres $\pm \Sigma \left(2,3,6k\pm 1\right)$. Along the way, we also give an alternative proof of Furuta’s improvement of $\frac{10}{8}$–theorem for closed spin $4$–manifolds.