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

R. İnanç Baykur, Masashi Ishida

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

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 λ 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 λ 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.

Original languageEnglish
Pages (from-to)1716-1736
Number of pages21
JournalJournal of Geometric Analysis
Volume24
Issue number4
DOIs
Publication statusPublished - 2014 Oct
Externally publishedYes

Keywords

  • Normalized Ricci flow
  • Simplicial volume
  • Smooth four-manifolds
  • Stable cohomotopy Seiberg-Witten invariants

ASJC Scopus subject areas

  • Geometry and Topology

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