4,214 research outputs found
Groupoids and an index theorem for conical pseudo-manifolds
We define an analytical index map and a topological index map for conical
pseudomanifolds. These constructions generalize the analogous constructions
used by Atiyah and Singer in the proof of their topological index theorem for a
smooth, compact manifold . A main ingredient is a non-commutative algebra
that plays in our setting the role of . We prove a Thom isomorphism
between non-commutative algebras which gives a new example of wrong way
functoriality in -theory. We then give a new proof of the Atiyah-Singer
index theorem using deformation groupoids and show how it generalizes to
conical pseudomanifolds. We thus prove a topological index theorem for conical
pseudomanifolds
Remarks on flat and differential K-theory
In this note we prove some results in flat and differential -theory. The
first one is a proof of the compatibility of the differential topological index
and the flat topological index by a direct computation. The second one is the
explicit isomorphisms between Bunke-Schick differential -theory and
Freed-Lott differential -theory.Comment: 9 pages. Comments are welcome. Final version. To appear in Annales
Mathematiques Blaise Pasca
Equivariant embedding theorems and topological index maps
The construction of topological index maps for equivariant families of Dirac
operators requires factoring a general smooth map through maps of a very simple
type: zero sections of vector bundles, open embeddings, and vector bundle
projections. Roughly speaking, a normally non-singular map is a map together
with such a factorisation. These factorisations are models for the topological
index map. Under some assumptions concerning the existence of equivariant
vector bundles, any smooth map admits a normal factorisation, and two such
factorisations are unique up to a certain notion of equivalence. To prove this,
we generalise the Mostow Embedding Theorem to spaces equipped with proper
groupoid actions. We also discuss orientations of normally non-singular maps
with respect to a cohomology theory and show that oriented normally
non-singular maps induce wrong-way maps on the chosen cohomology theory. For
K-oriented normally non-singular maps, we also get a functor to Kasparov's
equivariant KK-theory. We interpret this functor as a topological index map
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