We review constructions of Manolescu’s Floer homotopy type, which gives a homotopical refinement of monopole Floer homology. Based on it, we will introduce some homology cobordism/ knot concordance invariant. Using these invariants, we provide relative versions of 10/8 inequalities for 4-manifolds with boundary or surfaces in 4-manifolds. In particular, I’ll explain Manolescu’s relative 10/8 inequality, real 10/8 inequality, and Montague’s 10/8 inequality.

We first review fundamental concepts about Seiberg-Witten theory for closed 4-manifolds. Subsequently, we will introduce a refinement of Seiberg-Witten invariant, called Bauer—Furuta invariant. Using Bauer—Furuta invariant, I will explain how to prove Furuta’s 10/8 inequality and its variant for group actions proven by Bryan and Kato.