For manifolds, topological properties such as Poincaré duality and invariants such as the signature and characteristic classes, results and techniques from complex algebraic geometry such as the Hirzebruch-Riemann-Roch theorem, and results from global analysis such as the Atiyah-Singer index theorem, worked hand in hand in the past to weave a tight web of knowledge. Individually, many of the above results are in the meantime available for singular stratified spaces as well. The 2011 Oberwolfach workshop “Stratified Spaces: Joining Analysis, Topology and Geometry” discussed these with the specific aim of cross-fertilization in the three contributing fields.
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Markus Banagl, Ulrich Bunke, Shmuel Weinberger, Stratified Spaces: Joining Analysis, Topology and Geometry. Oberwolfach Rep. 8 (2011), no. 4, pp. 3217–3286DOI 10.4171/OWR/2011/56