Euler Characteristic

Let a closed surface have genus . Then the polyhedral formula generalizes to the Poincaré formula

(1)

where

(2)

is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case .

The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus (Dodson and Parker 1997, p. 125). The following table gives the Euler characteristics for some common surfaces (Henle 1994, pp. 167 and 295; Alexandroff 1998, p. 99).

surface
cylinder	0
double torus
Klein bottle	0
Möbius strip	0
projective plane	1
sphere	2
torus	0

In terms of the integral curvature of the surface ,

(3)

The Euler characteristic is sometimes also called the Euler number. It can also be expressed as

(4)

where  is the th Betti number of the space.

REFERENCES:

Alexandroff, P. S. Combinatorial Topology. New York: Dover, 1998.

Armstrong, M. A. "Euler Characteristics." §7.3 in Basic Topology, rev. ed. New York: Springer-Verlag, pp. 158-161, 1997 Coxeter, H. S. M. "Poincaré's Proof of Euler's Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165-172, 1973.

Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997.

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997.

Henle, M. A Combinatorial Introduction to Topology. New York: Dover, p. 167, 1994.