An index theorem for end-periodic operators

Casson-Type Invariants from the Seiberg–Witten Equations

“…In this section we express the invariant λ SW (X) solely in terms of the manifold X, without referring to auxiliary end-periodic manifolds. This formula will follow from the general index theorem for end-periodic operators proved in [17].…”

“…We further developed this theory, which allowed us to prove the well-definedness of our invariant in [16]. It also led us in [17] to a general index theorem for end-periodic Dirac operators in the spirit of the Atiyah-Patodi-Singer theorem [3], complete with a new η-invariant. A special case of this theorem is described in Section 5.…”

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This is a survey of our recent work [16,17,23] with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gauge theory.

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Index theory of the de Rham complex on manifolds with periodic ends

“…The first author was partially supported by NSF Grant 0805841, the second author was partially supported by NSF Grant 1105234, and the third author was partially supported by NSF Grant 1065905. [14] and can be viewed as a continuation of our research in [10] and [11] on the index theory of elliptic operators on such manifolds.…”

“…More generally, Atiyah, Patodi and Singer [1] computed the index for certain elliptic operators (that is, elliptic complexes of length two) on manifolds with cylindrical ends. In our paper [11] we extended their result to general manifolds with periodic ends; our index formula involves a new end-periodic η-invariant, generalizing the η-invariant of Atiyah, Patodi and Singer from the cylindrical setting. It would be interesting to compare the index formula of Theorem 1.2 with that of [11] for the operator d + d * obtained by wrapping up the de Rham complex.…”