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Characteristic Classes

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Handbook of Geometry and Topology of Singularities III

Abstract

The theory of characteristic classes for singular varieties experiences a huge development, in quantity and in quality, since the work of Marie-Hélène Schwartz, Wu Wen-Tsün and Robert MacPherson. An impressive number of researchers are extending the field of applications of characteristic classes and their ingredients, including in applied mathematics and physics.

In 1537, in Messina, Francesco Maurolico observed, for the five Platonic polyhedra, the formula that has been called later Euler formula. The Poincaré-Hopf Theorem says that the Euler-Poincaré characteristic is the obstruction to the construction of continuous non-vanishing vector fields tangent to a compact manifold. That opened the door for the construction of characteristic classes by obstruction theory: for manifolds, by Eduard Stiefel and Hassler Whitney in the real case and by Shiing-shen Chern in the complex case, then, in the singular framework, by Marie-Hélène Schwartz. The functorial definition by Robert MacPherson is the starting point of a huge development of the theory and applications of Chern-Schwartz-MacPherson classes and their ingredients: local Euler obstruction, Wu-Mather classes, Milnor classes, Segre classes, bivariant theory, motivic characteristic classes, etc.

This survey intentionally includes a brief history of the creation of characteristic classes from their very beginning as well as the detailed definition, by obstruction theory, of Schwartz classes giving rise to Chern-Schwartz-MacPherson classes.

In memory of Roberto Callejas-Bedregal

who died from covid19 on April 6, 2021.

We will never forget your joy, your laughter

and your enthusiasm for mathematics.

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Notes

  1. 1.

    There are several ways to write Wu Wen-Tsün name in Latin characters. We use the one he used to sign his articles during his French period, that is the one of main papers cited here.

  2. 2.

    Given a triangulation of X for which x is a vertex, the link of x is the union of simplexes τ which are faces of simplexes σ whose x is a vertex but such that x is not a vertex of τ.

  3. 3.

    The map is labelled by (6) in [156]. Here Cons(X) is the set of constructible functions, that we denote by \(\mathcal {F}(X)\).

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    Jean-Paul Brasselet

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    José Seade

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Brasselet, JP. (2022). Characteristic Classes. In: Cisneros-Molina, J.L., Dũng Tráng, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities III. Springer, Cham. https://doi.org/10.1007/978-3-030-95760-5_5

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