July 14, 2005
The Mysteries of Counting:
We all know what it means for a set to have 6 elements, but what sort of
thing has -1 elements, or 5/2? Believe it or not, these questions have
nice answers. The
Euler characteristic of a space is a generalization of cardinality that
admits negative integer values, while the homotopy cardinality is a
generalization that admits positive real values. These concepts shed
new light on basic mathematics. For example, the space of finite sets
turns out to have homotopy cardinality e, and this explains the key
properties of the exponential function. Euler characteristic and
homotopy cardinality share many properties, but it's hard to tell if
they are the same, because there are very few spaces for which both
are 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! The challenge of unifying them remains open.
Euler Characteristic versus Homotopy Cardinality
The Mysteries of Counting -
transparencies in PDF format.
For more information, try these papers:
Also try these
expository treatments in This
Week's Finds in Mathematical Physics:
"Negative sets" and Euler characteristic:
André Joyal, Regle des signes en algebre combinatoire, Comptes
Rendus Mathematiques de l'Academie des Sciences, La Societe Royale
du Canada VII (1985), 285-290.
Steve Schanuel, Negative sets have Euler characteristic and dimension,
Lecture Notes in Mathematics 1488,
Springer Verlag, Berlin, 1991, pp. 379-385.
Daniel Loeb, Sets with a negative number of elements, Adv. Math.
91 (1992), 64-74.
Resumming divergent Euler characteristics:
William J. Floyd and Steven P. Plotnick, Growth functions on
Fuchsian groups and the Euler characteristic, Invent. Math.
88 (1987), 1-29.
- R. I. Grigorchuk, Growth functions, rewriting systems and Euler
characteristic, Mat. Zametki 58 (1995), 653-668, 798.
James Propp, Exponentiation and Euler measure, Algebra Universalis
49 (2003), 459-471. Also available as
James Propp, Euler measure as generalized cardinality.
Available as math.CO/0203289.
Euler characteristics and supersymmetry:
Matthias Blau and George Thompson, N = 2 topological gauge
theory, the Euler characteristic of moduli spaces, and the Casson
invariant, Comm. Math. Phys. 152 (1993), 41-71.
Euler characteristics of tame spaces:
Lou van den Dries, Tame Topology and O-Minimal Structures
Cambridge U. Press, Cambridge, 1998. Chapter 4.2: Euler Characteristic.
Euler characteristics of groups:
G. Harder, A Gauss-Bonnet formula for discrete arithmetically
defined groups, Ann. Sci. Ecole Norm. Sup. 4
Jean-Pierre Serre, Cohomologie des groups discretes,
Ann. Math. Studies 70 (1971), 77-169.
Kenneth S. Brown, Euler characteristics, in
Cohomology of Groups, Graduate Texts in Mathematics
182, Springer, 1982, pp. 230-272.
Euler characteristics of categories and n-categories:
Tom Leinster, The Euler characteristic of a category.
Available as arXiv:math/0610260.
Tom Leinster, The Euler characteristic of a category as a
sum of a divergent series. Available as
Euler characteristics of chain complexes of graded vector spaces:
Bijective proofs in combinatorics:
Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg.
1-21. Also available at:
Robbie Gates, On the generic solution to P(X) = X in distributive
categories, Jour. Pure Appl. Alg. 125 (1998), 191-212.
Marcelo Fiore and Tom Leinster,
An objective representation of the Gaussian integers,
Jour. Symb. Comp. 37 (2004), 707-716.
Also available as
Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers,
Adv. Math. 190 (2005), 264-277. Also
Marcelo Fiore, Isomorphisms of generic recursive polynomial
types, to appear in 31st Symposium on Principles of Programming
Languages (POPL04). Also available at
Applications to quantum theory:
John Baez and James Dolan,
From finite sets to Feynman diagrams,
in Mathematics Unlimited - 2001
and Beyond, vol. 1, eds. Björn Engquist and Wilfried
Schmid, Springer, Berlin, 2001, pp. 29-50.
Also available as
Categorified algebra and quantum mechanics,
TAC 16 (2006), 785-854.
For much more, try these course notes:
- Week 147 -
Categorification, Euler characteristic versus homotopy
cardinality, and the cardinality of the fundamental group
of a Riemann surface.
Week 184 -
q-deformed cardinalities: how cardinalities of Grassmannians
over finite fields relate to Euler characteristics of Grassmannians
over R and C.
Week 185 -
Vector spaces over the field with q elements
as a q-deformation of finite sets; more on q-deformed cardinalities.
Week 186 -
The q-polynomial of a simple algebraic group, and using it
to compute the cardinalities of flag manifolds over finite
fields, and their Euler characteristics over R and C.
Week 187 -
The q-polynomial of a simple algebraic group - the classical
groups treated in detail.
Week 202 -
Joyal's species and reasoning with complex cardinalities.
Week 203 -
An object whose cardinality is the golden ratio.
Week 244 -
Leinster's work on the Euler characteristic of a category.
Finally, here's an earlier talk on this subject:
John Baez and Derek Wise,
Quantization and Categorification.
The imaginary expression √-a and the negative expression
-b resemble each other in that each one, when they seem the solution
of a problem, they indicate that there is some inconsistency or
nonsense. - Augustus De Morgan
Mathematics is not the rigid and rigidity-producing schema that the
layman thinks it is; rather, in it we find ourselves at that meeting
point of constraint and freedom that is the very essence of human
nature. - Hermann Weyl
© 2005 John Baez