Obstructions, vanishing topology and homological indices

Resumen:

Singular spaces arise as fibers of smooth or holomorphic maps which are not manifolds (i.e. the fibers are not locally Euclidean). Nevertheless Sard's theorem and its generalizations in complex analytic geometry tell us that any singular fiber has smooth neighboring fibers in arbitrarily close vicinity. For so-called "isolated singularities" the topology of these nearby fibers can be considered an invariant of the singularity itself.

We will review old and new results on this "vanishing topology" and discuss how these invariants can actually be computed in concrete examples. This involves relating topological obstruction theory for complex vector bundles to Euler characteristics of coherent sheaves via Riemann-Roch type theorems and goes as far as computations of spectral sequences for higher direct images in algebraic geometry.

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