Stratified singular spaces and generalized geometric Poincaré complexes
Final Report Abstract
The ordinary cohomology of spaces with singularities does not enjoy Poincaré duality. Classical expedients are the intersection cohomology IH ∗ of Goresky-MacPherson and the L2-cohomology of Cheeger. A new solution has been found, by lifting local truncation processes to the space level, in previous work of the PI. This method associates to certain types of singular spaces their intersection space, in such a way that the ordinary rational cohomology of the intersection space satisfies Poincaré duality across complementary perversities. The resulting cohomology theory HI ∗ is very different from classical intersection cohomology IH ∗. In the present project, we investigated the internal structure of the new theory (for which spaces is it defined? What are alternative descriptions of the theory?), as well as interdisciplinary applications to algebraic geometry, global analysis and theoretical physics. Surprisingly, we found that the methodology of intersection spaces leads to new results even in classical intersection homology and manifold theory. The grant allowed the PI to establish/continue contact to research groups in algebraic geometry and global analysis. It also enabled members of his research group to attend workshops and conferences. The PI has taken first steps in constructing intersection spaces for higher stratification depths. The PI’s PhD student M. Spiegel showed under a local torsion-freeness assumption that the complex K-theory of intersection spaces satisfies Poincaré duality. Our PhD student F. Gaisendrees extended the class of stratified spaces for which intersection spaces can be defined. In collaboration with B. Chriestenson, we give the up to now most widely applicable construction of intersection spaces via equivariant Moore approximations. Through this method, we obtain new vanishing results for the classical signature of neighborhoods of singular strata. The PI has given a de Rham description of HI ∗ by global differential forms on the regular part of the space, under the assumption that the link bundle is isometrically flat. As an application, this description allows us to show that the spectral sequence (with real coefficients) of such a link bundle collapses. Note that this is a stand-alone result about smooth manifolds which prima facie has no relation to intersection spaces and singularities. A special case is a theorem of Milnor on the Euler class of flat sphere bundles. The PI has also obtained calculations of equivariant cohomology groups through these methods. The PI’s PhD student T. Essig is extending this de Rham picture of HI ∗ to higher stratification depth. Applications in complex algebraic geometry have been pursued together with L. Maxim. For complex projective algebraic hypersurfaces with an isolated singularity, we show that HI ∗ is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. (This is very far from being true for intersection cohomology). The monodromy operator is trivial for instance in physicists’ conifold transition on Calabi-Yau 3-folds, which plays an important role in type II string theory. With N. Budur and L. Maxim, we have shown for complex projective hypersurfaces with an isolated singularity that HI ∗ is the hypercohomology of a perverse sheaf on the hypersurface. Moreover, this perverse sheaf underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. In collaboration with P. Albin, E. Leichtnam, R. Mazzeo and P. Piazza we show that J. Cheeger’s self-dual analytic L2 complex, based on choices of ideal boundary conditions, on a pseudomanifold which need not satisfy the Witt condition coincides in the derived category with the PI’s topological Verdier self-dual perverse sheaf complex, based on choices of Lagrangian subsheaves. With G. Laures and J. McClure, we give a detailed construction of the L• -homology orientation class of a pseudomanifold, which may be viewed as an integral refinement of the Goresky-MacPherson L-class. We apply our construction in deriving the stratified Novikov conjecture, provided the fundamental group satisfies the Novikov conjecture. With E. Hunsicker, we describe HI ∗ via extended L2 -harmonic forms with respect to the scattering metric.
Publications
- Stratified Spaces: Joining Analysis, Topology and Geometry, Report No. 56/2011, Oberwolfach Reports, vol. 8, no. 4, 3217 – 3286, European Math. Soc.
Markus Banagl, U. Bunke and Sh. Weinberger
- Deformation of Singularities and the Homology of Intersection Spaces, J. Topol. Anal. 4 (2012), no. 4
Markus Banagl, L. Maxim
(See online at https://doi.org/10.1142/S1793525312500185) - First Cases of Intersection Spaces in Stratification Depth 2, J. Singularities 5 (2012), 57 – 84
Markus Banagl
- Isometric Group Actions and the Cohomology of Flat Fiber Bundles, Groups Geom. Dyn. 7 (2013), no. 2, 293 – 321
Markus Banagl
(See online at https://doi.org/10.4171/GGD/183) - Intersection Spaces, Perverse Sheaves and Type IIB String Theory, Adv. Theor. Math. Phys. Volume 18, Number 2 (2014), 363-399
Markus Banagl, N. Budur and L. Maxim
(See online at https://doi.org/10.4310/ATMP.2014.v18.n2.a3)
