Konstruktion, Berechnung und Anwendung von topologischen Invarianten für singuläre stratifizierte Räume mit Hilfe von selbstdualen perversen Garbenkomplexen
Final Report Abstract
Poincaré duality is a cornerstone of classical manifold theory. The ordinary cohomology of spaces with singularities does not enjoy this duality. One solution is intersection cohomology, which can be constructed using so-called perverse sheaves with duality properties. A singular space satisfies the Witt condition if and only if its middleperversity intersection sheaf is self-dual. For such spaces we have defined a symmetric signature based on work of Eppelmann carried out In Heidelberg. The symmetric signature is an important invariant for classifying manifolds in a surgery theoretic setting. On spaces that do not satisfy the Witt condition, one must use Lagrangian subsheaves to construct self-dual sheaves. The grant allowed the PI to establish contact to research groups in global analysis, with the aim of describing the cohomology of such self-dual sheaves analytically. Elementary descriptions of such sheaves in the framework of MacPherson-Vilonen have been found (Bormann). Contact with the Japanese school of singularities has been established, particularly with S. Yokura (Kagoshima); a joint project also involving Jörg Schürmann (Münster) on fiberwise bordism and bivariant theory has been pursued and is nearing completion. On the computational front, the PI in joint work with S. Cappell (New York) and J. Shaneson (Univ. Penn.) has refined formulae that compute the Goresky-MacPherson L-class of a PL pseudomanifold piecewise-linearly embedded in a PL manifold (in a possibly nonlocally flat way). These refinements concern situations where twisted local coefficient systems do not extend into the singularities of the embedding and build on earlier work of the Pl on hwisted signatures. The most innovative breakthrough, however, was achieved in the construction of a new cohomology theory for singular spaces. Intersection cohomology has a series of drawbacks: much of the internal algebraic structure available in the ordinary cohomology of a space, such as an internal cup product, is lacking. It is highly unstable under deformation of singularities. Its chain-theoretic definition makes it hard to define generalized intersection cohomology theories. Furthermore, it is the wrong theory to describe massless fields in type IIB string theory. We have succeeded in associating 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. In particular, one obtains a new cohomology theory for stratified spaces, which overcomes the above-mentioned drawbacks. The disadvantage of the new theory is that it is expected to be not as generally definable as intersection homology, which is the price to pay for its richer internal algebraic structure. The PI's doctoral student Florian Gaisendrees, whose position was financed by the grant, made substantial contributions to the construction of intersection spaces for spaces with twisted link bundles. The PI's PhD student Matthias Spiegel (not financed through the grant) has obtained results on the K-theory of intersection spaces. DFG support also enabled the PI to get Laurentiu Maxim (Univ. Wisconsin at Madison) on board to study jointly the deformation stability of the new cohomology theory. This new theory immediately raises many questions that we plan to study through a separate new research proposal.
Publications
- Singular Spaces and Generalized Poincaré Complexes, Electron. Res. Announc. Math. Sci. 16 (2009), 63-73
Markus Banagl
- Intersection Spaces, Spatial Homology Truncation, and String Theory, Series: Springer Lecture Notes in Mathematics 1997, 2010, XVI, 217 pages, ISBN: 978-3-642-12588-1
Markus Banagl
- Rational Generalized Intersection Homology Theories, Homology, Homotopy Appl. 12 (2010), no. 1, 157-185
Markus Banagl
- Knots, Singular Embeddings, and Monodromy, in: The Mathematics of Knots: Theory and Application (Banagl, Vogel, eds.), Contributions in Mathematical and Computational Sciences 1, Springer Verlag, 2011
M. Banagl, S. E. Cappell and J. L. Shaneson
- The Signature of Singular Spaces and its Refinements to Generalized Homology Theories, in: Topology of Stratified Spaces (Friedman, Hunsicker, Libgober, Maxim, eds.), Mathematical Sciences Research Institute Publications 58, Cambridge University Press, New York, 2011
Markus Banagl
