438 research outputs found
The two definitions of the index difference
Given two metrics of positive scalar curvature metrics on a closed spin
manifold, there is a secondary index invariant in real -theory. There exist
two definitions of this invariant, one of homotopical flavour, the other one
defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and
detailed proof of the folklore result that both constructions yield the same
answer. Moreover, we generalize this to the case of two families of positive
scalar curvature metrics, parametrized by a compact space. In essence, we prove
a generalization of the classical "spectral-flow-index theorem" to the case of
families of real operators.Comment: Revised versio
On the homotopy type of the Deligne-Mumford compactification
An old theorem of Charney and Lee says that the classifying space of the
category of stable nodal topological surfaces and isotopy classes of
degenerations has the same rational homology as the Deligne-Mumford
compactification. We give an integral refinement: the classifying space of the
Charney-Lee category actually has the same homotopy type as the moduli stack of
stable curves, and the etale homotopy type of the moduli stack is equivalent to
the profinite completion of the classifying space of the Charney-Lee category.Comment: 14 pages, published versio
Generalised Miller-Morita-Mumford classes for block bundles and topological bundles
The most basic characteristic classes of smooth fibre bundles are the
generalised Miller-Morita-Mumford classes, obtained by fibre integrating
characteristic classes of the vertical tangent bundle. In this note we show
that they may be defined for more general families of manifolds than smooth
fibre bundles: smooth block bundles and topological fibre bundles.Comment: 18 page
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