分野別セミナー

The famous Poincar\'e-Hopf Theorem says that Euler-Poincar\'e
characteristic measures the obstruction to construct a continuous
vector field tangent to a (compact) manifold. The Whitney classes
in the real case and the Chern classes in the complex case
can be seen as generalizations of the Euler-Poincar\'e
characteristic in the sense that they are measure to the construction
of frames on manifolds.

In the lecture, we will recall these constructions. We will give
explicit
examples showing that, in case of a singular variety, the Poincar\'e-
Hopf
Theorem fails to be true if one considers any vector field.
One has to consider special vector fields called ``radial vector fields".
After providing examples, we will show how one can use the obstruction
theory to define characteristic classes in the singular case.