57 research outputs found
Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities
In this paper we study manifolds, , with fibred singularities,
more specifically, a relevant space of
Riemannian metrics with positive scalar curvature. Our main goal is to prove
that the space is homotopy invariant under
certain surgeries on .Comment: 27 pages, 4 figure
Positive scalar curvature on manifolds with fibered singularities
A (compact) manifold with fibered -singularities is a (possibly) singular
pseudomanifold with two strata: an open nonsingular stratum
(a smooth open manifold) and a closed stratum (a closed
manifold of positive codimension), such that a tubular neighborhood of is a fiber bundle with fibers each looking like the cone on a fixed closed
manifold . We discuss what it means for such an with fibered
-singularities to admit an appropriate Riemannian metric of positive scalar
curvature, and we give necessary and sufficient conditions (the necessary
conditions based on suitable versions of index theory, the sufficient
conditions based on surgery methods and homotopy theory) for this to happen
when the singularity type is either or , and and the
boundary of the tubular neighborhood of the singular stratum are simply
connected and carry spin structures. Along the way, we prove some results of
perhaps independent interest, concerning metrics on spin manifolds with
positive "twisted scalar curvature," where the twisting comes from the
curvature of the spin line bundle.Comment: 30 pages, 1 figure. An error was corrected in the statement and proof
of the second main theorem, which is now Theorem 3.1
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