558 research outputs found
Geometric versus homotopy theoretic equivariant bordism
By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric
G-equivariant bordism to homotopy theoretic equivariant bordism is injective
for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated
fixed point square is a pull back square, thus confirming a recent conjecture
of D. Sinha. This is used in order to determine the image of the
Pontrjagin-Thom map for toral G.Comment: 19 pages, minor inaccuracies removed, to appear in Math. An
Enlargeability and index theory
Let M be a closed enlargeable spin manifold. We show non-triviality of the
universal index obstruction in the K-theory of the maximal -algebra of the
fundamental group of M. Our proof is independent from the injectivity of the
Baum-Connes assembly map for the fundamental group of M and relies on the
construction of a certain infinite dimensional flat vector bundle out of a
sequence of finite dimensional vector bundles on M whose curvatures tend to
zero.
Besides the well known fact that M does not carry a metric with positive
scalar curvature, our results imply that the classifying map
sends the fundamental class of M to a nontrivial homology class in H_n(B
\pi_1(M) ; \Q). This answers a question of Burghelea (1983).Comment: 32 pages, final version accepted for publication, added relation to
Gromov's 1-systole, typos corrected; to appear in Journal of Differential
Geometr
The space of metrics of positive scalar curvature
We study the topology of the space of positive scalar curvature metrics on
high dimensional spheres and other spin manifolds. Our main result provides
elements of infinite order in higher homotopy and homology groups of these
spaces, which, in contrast to previous approaches, are of infinite order and
survive in the (observer) moduli space of such metrics.
Along the way we construct smooth fiber bundles over spheres whose total
spaces have non-vanishing A-hat-genera, thus establishing the
non-multiplicativity of the A-hat-genus in fibre bundles with simply connected
base.Comment: 24 pages, v2: minor additions and corrections, based in particular on
comments of referees, v3: minor corrections, final version, to appear in
Publ.Math. IHE
Positive scalar curvature on manifolds with odd order abelian fundamental groups
We introduce Riemannian metrics of positive scalar curvature on manifolds
with Baas-Sullivan singularities, prove a corresponding homology invariance
principle and discuss admissible products. Using this theory we construct
positive scalar curvature metrics on closed smooth manifolds of dimension at
least five which have odd order abelian fundamental groups, are nonspin and
atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of
manifolds with finite fundamental groups.Comment: 31 pages; 2 figures; minor edits; published versio
K-cowaist of manifolds with boundary
We extend the K-cowaist inequality to generalized Dirac operators in the
sense of Gromov and Lawson and study applications to manifolds with boundary.Comment: 6 page
Large and small group homology
For several instances of metric largeness like enlargeability or having
hyperspherical universal covers, we construct non-large vector subspaces in the
rational homology of finitely generated groups. The functorial properties of
this construction imply that the corresponding largeness properties of closed
manifolds depend only on the image of their fundamental classes under the
classifying map.
This is applied to construct examples of essential manifolds whose universal
covers are not hyperspherical, thus answering a question of Gromov (1986), and,
more generally, essential manifolds which are not enlargeable.Comment: 24 pages, small corrections and improvement
Boundary conditions for scalar curvature
Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction
to positive scalar curvature metrics with mean convex boundaries on spin
manifolds of infinite K-area. We also characterize the extremal case. Next we
show a general deformation principle for boundary conditions of metrics with
lower scalar curvature bounds. This implies that the relaxation of boundary
conditions often induces weak homotopy equivalences of spaces of such metrics.
This can be used to refine the smoothing of codimension-one singularites a la
Miao and the deformation of boundary conditions a la Brendle-Marques-Neves,
among others. Finally, we construct compact manifolds for which the spaces of
positive scalar curvature metrics with mean convex boundaries have nontrivial
higher homotopy groups.Comment: minor change
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