Markus Banagl

Extending intersection homology type invariants to non-Witt spaces

Introduction The algebraic framework Ordered resolutions The cobordism group

Intersection spaces, spatial homology truncation, and string theory

\Omega_\ast^{SD}

DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF INTERSECTION SPACES

Lagrangian structures and ordered resolutions Appendix A. On signs Bibliography.

COMPUTING TWISTED SIGNATURES AND L -CLASSES OF NON-WITT SPACES

To a stratified singular space X, we associate new spaces I p̄X, its perversity p̄-intersection spaces, such that when X is a closed, oriented pseudomanifold, the ordinary rational cohomology of I p̄X is Poincaré dual to the ordinary rational homology of I q̄X if p̄ and q̄ are complementary perversities. The homology of I p̄X 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 Im̄X and the rational L-homology of In̄X. Perversity p̄-intersection vector bundles on X may be defined as actual vector bundles on I p̄X. In the present monograph, the construction of I p̄X 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.

THE SIGNATURE OF PARTIALLY DEFINED LOCAL COEFFICIENT SYSTEMS

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first authors cohomology of intersection spaces 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. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Delignes mixed Hodge structure on the ordinary cohomology of the singular hypersurface.

Intersection spaces, perverse sheaves and type IIB string theory

In previous joint work with Cappell and Shaneson, we have established an Atiyah?Lusztig?Meyer-type multiplicative characteristic class formula for the twisted signature and, more generally, the twisted

Refined intersection homology on non-Witt spaces

L

Triangulations of 3–dimensional pseudomanifolds with an application to state-sum invariants

-class, of a stratified Witt space. The present paper shows that these formulae hold even when the stratified space does not satisfy the Witt condition. It constitutes one of the first applications of signature homology.

Isometric group actions and the cohomology of flat fiber bundles

Intersection homology enables the definition of a twisted signature of a stratified pseudomanifold with coefficients in a local system that is typically only given on the top stratum. If the local system extends to the entire space, then the twisted signature can be computed by a version of a characteristic class formula first observed by Atiyah for smooth fiber bundles. In the first part of this paper, we construct examples of singular spaces, equipped with local systems that do not extend, such that the above characteristic class formula fails. In the second part, we consider smooth codimension two embeddings of manifolds. We view the target as a stratified space with bottom stratum the image of the embedding and top stratum the complement. When the embedded manifold is a sphere, we establish various formulae that compute the twisted signature even when the local system does not extend from the top stratum to the entire space.

Knots, Singular Embeddings, and Monodromy

The method of intersection spaces associates rational Poincare complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA the- ory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the inter- section space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincare duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.