Dirac operators on manifolds with periodic ends

Abstract

This paper studies Dirac operators on end-periodic spin manifolds of dimension at least 4. We provide a sufficient condition for such an operator to be Fredholm for a generic end-periodic metric; this condition is shown to be necessary in dimension 4. We make use of end-periodic Dirac operators to give an analytical interpretation of an invariant of non-orientable smooth 4-manifolds due to Cappell and Shaneson. From this interpretation we show that some exotic non-orientable 4-manifolds do not admit a metric of positive scalar curvature.

22 pages. Additional references and acknowledgements added

References in corpus (4)

• Surgery and Harmonic Spinors
• Surgery and the Spectrum of the Dirac Operator
• Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture
• Real secondary index theory

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• On the Frøyshov invariant and monopole Lefschetz number
• The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics
• Seiberg-Witten theory on 4-manifolds with periodic ends
• Positive scalar curvature and $10/8$-type inequalities on $4$-manifolds with periodic ends
• Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant
• Positive scalar curvature and homology cobordism invariants