The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics

Abstract

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact $4$-manifolds with the same homology as $S^1\times S^3$. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of $H_3(X;Z)$ with the $4$-dimensional Casson invariant ${\lambda }_{SW}(X)$ defined by Mrowka-Ruberman-Saveliev. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.

v3: Added Section 4.2 for a detailed proof of exponential decay

References in corpus (4)

• A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$
• Dirac operators on manifolds with periodic ends
• A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture
• Seiberg-Witten theory on 4-manifolds with periodic ends

Cited by in corpus (2)

• Naturality of the Contact Invariant in Monopole Floer Homology Under Strong Symplectic Cobordisms
• Positive scalar curvature and $10/8$-type inequalities on $4$-manifolds with periodic ends