Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants, and Normalized Ricci Flow

Abstract

In this article, we produce infinite families of 4-manifolds with positive first Betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg–Witten invariants of their connected sums. Elementary building blocks used in Ishida and Sasahira ( arXiv:0804.3452 , 2008) are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov’s simplicial volume is nontrivial, Perelman’s $\bar{\lambda}$ invariant is negative, and the relevant Gromov–Hitchin–Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. Fang et al. (Math. Ann. 340:647–674, 2008) conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov’s simplicial volume and negative Perelman’s $\bar{\lambda}$ invariant implies the Gromov–Hitchin–Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds.