131 research outputs found

### Orbifold index and equivariant K-homology

We consider a invariant Dirac operator D on a manifold with a proper and
cocompact action of a discrete group G. It gives rise to an equivariant
K-homology class [D]. We show how the index of the induced orbifold Dirac
operator can be calculated from [D] via the assembly map. We further derive a
formula for this index in terms of the contributions of finite cyclic subgroups
of G. According to results of W. Lueck, the equivariant K-homology can
rationally be decomposed as a direct sum of contributions of finite cyclic
subgroups of G. Our index formula thus leads to an explicit decomposition of
the class [D].Comment: minor correction in Sec.3.

### Foliated manifolds, algebraic K-theory, and a secondary invariant

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated
manifold with a stable framing and with a partially flat vector bundle. This
invariant can be expressed in terms of integration in differential $K$-theory,
or alternatively, in terms of $\eta$-invariants of Dirac operators and local
correction terms. Initially, the construction of the element in
$\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of
the foliation is sufficiently small, then this element is independent of these
choices and therefore an invariant of the data listed above. We show that the
invariant comprises various classical invariants like Adams' $e$-invariant, the
$\rho$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant
from foliation theory. Using methods from differential cohomology theory we
construct a regulator map from the algebraic $K$-theory of smooth functions on
a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$
coefficients. Our main result is a formula for the invariant in terms of this
regulator and integration in algebraic and topological $K$-theory.Comment: 58 pages (typos corrected, references added, small improvements of
presentation

### A theta function for hyperbolic surfaces with cusps

For a Riemann surface with cusps we define a theta function using the
eigenvalues of the Laplacian and the singularities of the scattering
determinant. We provide its meromorphic continuation and discuss its
singularities.Comment: 16 pages, Latex, reportSFB288-11

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