Semiflat Orbifold Projections

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We compute the semiflat positive cone K₀+SF(Aθσ) of the K₀-group of the irrational rotation orbifold Aθσ under the noncommutative Fourier transform σ and show that it is determined by classes of positive trace and the vanishing of two topological invariants. The semiflat orbifold projections are 3-dimensional and come in three basic topological genera: (2,0,0), (1,1,2), (0,0,2). (A projection is called semiflat when it has the form h + σ(h) where h is a flip-invariant projection such that hσ(h)=0.) Among other things, we also show that every number in (0,1) ∩ (2Z + 2Zθ) is the trace of a semiflat projection in Aθ. The noncommutative Fourier transform is the order 4 automorphism σ: V → U → V⁻¹ (and the flip is σ²: U → U⁻¹,V → V⁻¹), where U,V are the canonical unitary generators of the rotation algebra Aθ satisfying VU = e²πⁱθ UV.