Is topology useful in computer science?

Many spaces arising in mathematics and its applications are, however, not uniform but have singularities. Similarly, smooth maps between smooth manifolds generally have singularities. It was realized early on by Whitney, Thom and Mather that it is advantageous to organize points in the space into equisingular strata. Now, Poincaré duality breaks down in the presence of singularities, but can be restored by changing to different cohomology theories: There are presently three theories that exhibit a generalized form of Poincaré duality: Intersection cohomology, discovered by Goresky and MacPherson, -cohomology with respect to conical metrics, discovered by Cheeger, and the cohomology of intersection spaces, discovered by Banagl. Each of these theories depends on a parameter, and for a self-dual choice of parameter, the two first named theories are isomorphic, while the third generally has quite different Betti numbers. The construction of intersection spaces can be viewed as a homotopy theoretic desingularization of a space.

The above theories are not invariant under arbitrary homotopies and it is natural to consider and further develop stratified homotopy theory. For example, work carried out by Banagl, Mäder, Sadlo and Waas in topological data science within the Cluster of Excellence STRUCTURES lead to the discovery of concrete elementary simplicial collapses and expansions that imply the existence of a stratified simple homotopy theory. Often, duality issues on singular spaces are best understood by considering Verdier self-dual complexes of sheaves with appropriate stalk and costalk vanishing conditions. If there are strata of odd codimension, the existence of such complexes may be obstructed, but if existent, they lead to cohomology theories with duality closely related to intersection cohomology. In the case of the reductive Borel-Serre compactification of Hilbert modular surfaces, for instance, the obstructions have been found to vanish. Once duality is restored, and the self-duality obstructions vanish, many of the invariants of manifolds alluded to above can be generalized to singular spaces, but their computation presents many interesting challenges, since the functorial bundle theoretic calculus used in the manifold case is not available in the singular case.

Nevertheless, it turns out that important L-theoretic objects can still be constructed: One particular recent breakthrough was the construction of a Dold-type orientation class in symmetric L-homology for singular spaces by Banagl, Laures (Bochum), and McClure (Purdue). This implies the existence of invariants such as higher and symmetric signatures for singular spaces. Closely related classes in the K-homology of singular spaces can be constructed topologically or using analytic methods. The Banagl-Hunsicker Hodge-Theorem allows for a topological interpretation of -cohomology spaces associated to scattering metrics in terms of the cohomology of intersection spaces. For complex algebraic varieties, there are strong ties to Saito's theory of mixed Hodge modules, designed for singular varieties. We are only beginning to understand the relation of the above topological invariants to those coming from MHMs such as the intersection Hodge module.