University of Sheffield
Abstract Far beyond the realm where we can count 'vertices minus edges', there are spaces that, nevertheless, appear to have a well-defined Euler characteristic. For example, the Julia set of any rational function f seems to have an Euler characteristic, a number giving basic information about the dynamical behaviour of f. But to define the Euler characteristic of such spaces, we first need to define the Euler characteristic of a category. This involves generalizing the Möbius inversion formula of classical number theory. We'll see, for instance, that the Euler characteristic of the category of finite sets and bijections is e = 2.718... . Throughout, our motto is: 'Euler characteristic is generalized cardinality'.
Slides In this pdf file (603K). (The images of Julia sets are reproduced from Milnor, Dynamics in One Complex Variable.)