480 research outputs found
Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
In this paper we continue the study of positive scalar curvature (psc)
metrics on a depth-1 Thom-Mather stratified space with singular
stratum (a closed manifold of positive codimension) and associated
link equal to , a smooth compact manifold. We briefly call such spaces
manifolds with -fibered singularities. Under suitable spin assumptions we
give necessary index-theoretic conditions for the existence of wedge metrics of
positive scalar curvature. Assuming in addition that is a simply connected
homogeneous space of positive scalar curvature, , with the semisimple
compact Lie group acting transitively on by isometries, we investigate
when these necessary conditions are also sufficient. Our main result is that
our conditions are indeed sufficient for large classes of examples, even when
and are not simply connected. We also investigate the
space of such psc metrics and show that it often splits into many cobordism
classes
K-duality for stratified pseudomanifolds
This paper is devoted to the study of Poincar\'e duality in K-theory for
general stratified pseudomanifolds. We review the axiomatic definition of a
smooth stratification \fS of a topological space and we define a groupoid
T^{\fS}X, called the \fS-tangent space. This groupoid is made of different
pieces encoding the tangent spaces of the strata, and these pieces are glued
into the smooth noncommutative groupoid T^{\fS}X using the familiar procedure
introduced by A. Connes for the tangent groupoid of a manifold. The main result
is that C^{*}(T^{\fS}X) is Poincar\'e dual to , in other words, the
\fS-tangent space plays the role in -theory of a tangent space for
Triangulations of 3-dimensional pseudomanifolds with an application to state-sum invariants
We demonstrate the triangulability of compact 3-dimensional topological
pseudomanifolds and study the properties of such triangulations, including the
Hauptvermutung and relations by Alexander star moves and Pachner bistellar
moves. We also provide an application to state-sum invariants of 3-dimensional
topological pseudomanifoldsComment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-24.abs.htm
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
Intersection homology with field coefficients: -Witt spaces and -Witt bordism
We construct geometric examples of pseudomanifolds that satisfy the Witt
condition for intersection homology Poincare duality with respect to certain
fields but not others. We also compute the bordism theory of -Witt spaces
for an arbitrary field , extending results of Siegel for .Comment: 26 pages; also available at http://faculty.tcu.edu/gfriedman/ -
Corrected version (proof of Lemma 4.11 corrected
Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds
This paper examines the issues involved with concretely implementing a sum
over conifolds in the formulation of Euclidean sums over histories for gravity.
The first step in precisely formulating any sum over topological spaces is that
one must have an algorithmically implementable method of generating a list of
all spaces in the set to be summed over. This requirement causes well known
problems in the formulation of sums over manifolds in four or more dimensions;
there is no algorithmic method of determining whether or not a topological
space is an n-manifold in five or more dimensions and the issue of whether or
not such an algorithm exists is open in four. However, as this paper shows,
conifolds are algorithmically decidable in four dimensions. Thus the set of
4-conifolds provides a starting point for a concrete implementation of
Euclidean sums over histories in four dimensions. Explicit algorithms for
summing over various sets of 4-conifolds are presented in the context of Regge
calculus. Postscript figures available via anonymous ftp at
black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-
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