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
welldefined. However, in
many cases where one is welldefined, 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:

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© 2003 John Baez
baez@math.removethis.ucr.andthis.edu