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Coassembly is a homotopy limit map
We prove a claim by Williams that the coassembly map is a homotopy limit map.
As an application, we show that the homotopy limit map for the coarse version
of equivariant -theory agrees with the coassembly map for bivariant
-theory that appears in the statement of the topological Riemann-Roch
theorem.Comment: Accepted version. Several improvements from the referee, including a
more elegant proof of Lemma 3.
Noncommutative gauge theory for Poisson manifolds
A noncommutative gauge theory is associated to every Abelian gauge theory on
a Poisson manifold. The semi-classical and full quantum version of the map from
the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten
map) is given explicitly to all orders for any Poisson manifold in the Abelian
case. In the quantum case the construction is based on Kontsevich's formality
theorem.Comment: 12 page
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
The Wronski map and shifted tableau theory
The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a
number of interesting consequences. Among them is a well-behaved correspondence
between certain points on a Grassmannian - those sent by the Wronski map to
polynomials with only real roots - and (dual equivalence classes of) Young
tableaux.
In this paper, we restrict this correspondence to the orthogonal Grassmannian
OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and
only if the corresponding tableau has a certain type of symmetry. From this we
recover much of the theory of shifted tableaux for Schubert calculus on
OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule
for OG(n,2n+1).Comment: 11 pages, color figures, identical to v1 but metadata correcte
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