In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of Cr⁎(Gad∘) (where Gad∘ is the adiabatic deformation G ...restricted to the interval 0,1)). Our construction directly generalises the cases treated in 29,30, in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps. This extends the construction of the lower shriek map in 6. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for secondary invariants.
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real ...structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are 64 such real index pairings of real K-theory with real K-homology. For 16 of them, the index of the Fredholm operator representing the pairing vanishes, but there is a secondary Z2-valued invariant. The first set of results provides index formulas expressing each of these 16 Z2-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from an unbounded Fredholm module and shows that the Z2-invariant can be computed as the sign of the Pfaffian of the skew localizer and in 8 of the cases as the sign of the determinant of the off-diagonal entry of the skew localize. This is of relevance for the numerical computation of invariants of topological insulators.
In this paper, we construct a new index theory, Sn index theory, which is an improvement of the S1 index theory given by Benci in 1. Also S1 index theory is a powerful tool in the study of the ...multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional Φ is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.
This paper is concerned with the stationary solutions of the Dirac equation −i∑k=13αk∂ku+aβu+ωu+V(x)u=Gu(x,u),where G is asymptotically quadratic and is not assumed to be C2. We present a new ...approach to construct an index theory for the associated linear Dirac equation and define the relative Morse index to measure the difference between the nonlinearity at the origin and at infinity. By building upon the idea of combining the index theory and a generalized linking theorem, we obtain existence and multiplicity of stationary solutions.
A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is a Fredholm operator. Its symbol determines an element in the K-theory of the noncommutative algebra of Heisenberg ...symbols. We construct a periodic cyclic cocycle which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. Our index formula is local, i.e. given as a local expression in terms of the principal symbol of the operator and a connection on TM and its curvature. We prove our index formula by reduction to Boutet de Monvel's index theorem for Toeplitz operators.
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