On the spectral sets of Inoue surfaces

Abstract

The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin${}^c$ structure, and the corresponding chiral Dirac operators twisted by a flat ${\mathbb{C}}^{\ast }$--connection. The twisting connection is determined by $z\in {\mathbb{C}}^{\ast }$, and the points for which the twisted Dirac operators ${\mathcal{D}}_z^{\pm }$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus ${\alpha }^{-1/4}<|z|<{\alpha }^{1/4}$, where $\alpha >1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$.

15 pages. Final version, to appear in Open Book Series volume entitled 'Gauge theory and low-dimensional topology: progress and interaction.'

References in corpus (2)

• Harmonic Forms on the Kodaira-Thurston Manifold
• On the monopole Lefschetz number of finite order diffeomorphisms

Cited by in corpus (1)

• Leafwise flat forms on Inoue-Bombieri surfaces