2,794 research outputs found
Rough index theory on spaces of polynomial growth and contractibility
We will show that for a polynomially contractible manifold of bounded
geometry and of polynomial volume growth every coarse and rough cohomology
class pairs continuously with the K-theory of the uniform Roe algebra. As an
application we will discuss non-vanishing of rough index classes of Dirac
operators over such manifolds, and we will furthermore get higher-codimensional
index obstructions to metrics of positive scalar curvature on closed manifolds
with virtually nilpotent fundamental groups. We will give a computation of the
homology of (a dense, smooth subalgebra of) the uniform Roe algebra of
manifolds of polynomial volume growth.Comment: v4: final version, to appear in J. Noncommut. Geom. v3: added a
computation of the homology of (a smooth subalgebra of) the uniform Roe
algebra. v2: added as corollaries to the main theorem the multi-partitioned
manifold index theorem and the higher-codimensional index obstructions
against psc-metrics, added a proof of the strong Novikov conjecture for
virtually nilpotent groups, changed the titl
Homotopy theory with bornological coarse spaces
We propose an axiomatic characterization of coarse homology theories defined
on the category of bornological coarse spaces. We construct a category of
motivic coarse spectra. Our focus is the classification of coarse homology
theories and the construction of examples. We show that if a transformation
between coarse homology theories induces an equivalence on all discrete
bornological coarse spaces, then it is an equivalence on bornological coarse
spaces of finite asymptotic dimension. The example of coarse K-homology will be
discussed in detail.Comment: 220 pages (complete revision
Isospectral Alexandrov Spaces
We construct the first non-trivial examples of compact non-isometric
Alexandrov spaces which are isospectral with respect to the Laplacian and not
isometric to Riemannian orbifolds. This construction generalizes independent
earlier results by the authors based on Schueth's version of the torus method.Comment: 15 pages, no figures; minor clarification
Coarse cohomology theories
We propose the notion of a coarse cohomology theory and study the examples of
coarse ordinary cohomology, coarse stable cohomotopy and coarse cohomology
theories obtained by dualizing coarse homology theories. Our investigations of
coarse stable cohomotopy lead to a solution of J. R. Klein's conjecture that
the dualizing spectrum of a group is a coarse invariant. We further investigate
coarse cohomological -theory functors and explain why (an adaption of) the
functor of Emerson--Meyer does not seem to fit into our setting.Comment: 55 page
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