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INTERSECTION SPACES, HOMOLOGY TRUNCATION AND STRING THEORY.
We pursue here the following research program:
To a stratified singular space X, associate new spaces IpX, its
perversity p-intersection spaces, such that when X is a closed, oriented pseudomanifold,
the ordinary rational cohomology of IpX is Poincaré dual to the ordinary
rational homology of IqX if
p and
q are complementary perversities. The homology
of IpX is not isomorphic to intersection homology so that a new duality theory
for pseudomanifolds is obtained, which addresses certain needs in string theory
related to the existence of massless D-branes in the course of conifold
transitions and their faithful representation as cohomology classes. While
intersection homology accounts correctly for all massless D-branes in type IIA
string theory, the homology of intersection spaces accounts correctly for all
massless D-branes in type IIB string theory.
In fact, for singular Calabi-Yau conifolds, the two theories are mirrors
of each other in the sense of mirror symmetry. The new theory also allows for
certain types of cap products that are known not to exist for intersection
homology. Using these products, we show that capping with the symmetric
L-homology fundamental class induces an isomorphism between the rational
symmetric L-cohomology of ImX and the rational L-homology of InX. Perversity
p-intersection vector bundles on X may be defined as actual vector bundles on
IpX. In our Springer
LNM volume 1997 (2010), the construction of IpX is carried out for isolated
singularities and, more generally, for two-strata spaces with
trivial link bundle. It is based on an in-depth and autonomous homotopy
theoretic analysis of spatial homology truncation, where an emphasis was placed
on investigating functoriality.
In the meantime, our intersection space
approach has had numerous applications in algebraic geometry, differential
geometry, analysis and equivariant topology. The applications in algebraic
geometry concern improved stability under nearby smooth deformations of singular
spaces (compared to intersection homology). The applications in differential
geometry concern de Rham methods for flat link bundles whose structure group is
the isometries of the link, yielding e.g. spectral sequence collapse results.
The latter are applied to computing equivariant cohomology for certain actions.
In global analysis, our approach has led to a purely topological method for
calculating spaces of weighted L2-harmonic forms on a noncompact manifold
equipped with a scattering metric.
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