1,669 research outputs found
Real secondary index theory
In this paper, we study the family index of a family of spin manifolds. In
particular, we discuss to which extend the real index (of the Dirac operator of
the real spinor bundle if the fiber dimension is divisible by 8) which can be
defined in this case contains extra information over the complex index (the
index of its complexification). We study this question under the additional
assumption that the complex index vanishes on the k-skeleton of B. In this
case, using local index theory we define new analytical invariants \hat c_k\in
H^{k-1}(B;\reals/\integers). We then continue and describe these invariants in
terms of known topological characteristic classes. Moreover, we show that it is
an interesting new non-trivial invariant in many examples.Comment: LaTeX2e, 56 pages; v2: final version to appear in ATG, typos fixed,
statement of 4.5.5 improve
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 -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 -theory,
or alternatively, in terms of -invariants of Dirac operators and local
correction terms. Initially, the construction of the element in
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' -invariant, the
-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 -theory of smooth functions on
a manifold to its connective -theory with
coefficients. Our main result is a formula for the invariant in terms of this
regulator and integration in algebraic and topological -theory.Comment: 58 pages (typos corrected, references added, small improvements of
presentation
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