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\input zb-matheduc
\iteman{ZMATH 2007b.00369}
\itemau{Persson, Ulf}
\itemti{The Euler number. (Eulertalet.)}
\itemso{Normat. Nord. Mat. Tidskr. 54, No. 4, 173-183 (2006).}
\itemab
Summary: The Euler-number is often defined as the alternating sum of the Betti-numbers, 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 Euler-number 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 Euler-number to the types of singularities of a vector field and to prove the Gauss-Bonnets formula, relating the integral of the Gaussian curvature to the Euler-number of the surface.
\itemrv{~}
\itemcc{H70 I60 G90 I90}
\itemut{differential geometry; topology; Gauss-Bonnet theorem; vector fields}
\itemli{}
\end