John Baez

Lecture at the Fields Institute Program on Applied Homotopy Theory

September 20, 2003

Euler Characteristic versus Homotopy Cardinality

Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups. The Euler characteristic is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive fractional values. The two quantities have many of the same properties, but it's hard to tell if they're the same, since like Jekyll and Hyde, they're almost never seen together: there are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series - and the two then agree! We give examples of this phenomenon and beg the audience to find some unifying concept which has both Euler characteristic and homotopy cardinality as special cases.

Click here to see the slides as a PDF file:

Authors' names underlined with a wiggly line refer to papers that are discussed further here:

For more, try these:

© 2003 John Baez