Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number

Abstract

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a partial, but strong affirmative answer to a conjecture concerning non-vanishing of the invariant. Various new applications of the non-vanishing theorem are also given. For example, we shall introduce variants ${\bar{\lambda }}_k$ of Perelman's $\bar{\lambda }$ invariants for real numbers $k$ and compute the values for a large class of 4-manifolds including connected sums of certain K{ä}hler surfaces. The non-vanishing theorem is also used to construct the first examples of 4-manifolds with non-zero simplicial volume and satisfying the strict Gromov-Hitchin-Thorpe inequality, but admitting infinitely many distinct smooth structures for which no compatible Einstein metric exists. Moreover, we are able to prove a new result on the existence of exotic smooth structures.

78 pages

References in corpus (5)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants
• Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants
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Cited by in corpus (3)

• Families of 4-manifolds with nontrivial stable cohomotopy Seiberg-Witten invariants, and normalized Ricci flow
• Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces
• The simple type conjecture for mod 2 Seiberg-Witten invariants