
05192618
j
2007b.00369
Persson, Ulf
The Euler number. (Eulertalet.)
Normat. Nord. Mat. Tidskr. 54, No. 4, 173183 (2006).
2006
Nationellt Centrum f\"or Matematikutbildning, G\"oteborg
SV
H70
I60
G90
I90
differential geometry
topology
GaussBonnet theorem
vector fields
Summary: The Eulernumber is often defined as the alternating sum of the Bettinumbers, or equivalently as the alternate sum of the number of simplices of a given dimension. The problem is that this requires an explicit triangulation. For purposes of actual computation it is much more efficient (and illuminating) to take a more ``functorial'' approach, namely the Eulernumber which behaves like a cardinal number, but with the difference that we can freely mix dimensions. In the article this is illustrated by a variety of examples, compact real surfaces, projective spaces and Grassmannians, curves and hypersurfaces. We also show how to relate the Eulernumber to the types of singularities of a vector field and to prove the GaussBonnets formula, relating the integral of the Gaussian curvature to the Eulernumber of the surface.