From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: This week in the mathematics arXiv (28 Aug - 1 Sep)
Date: 5 Sep 2000 21:00:09 -0500
Newsgroups: sci.math.research
Summary: [missing]

[deletia --djr]

This week is a big week for algebraic topology (math.AT) in the
mathematics arXiv.  One article that caught my attention is "The
cohomology of the Sylow 2-subgroup of the Higman-Sims group", by
Adem, Carlson, Karagueuzian, and Milgram [math.AT/0008230].  If you
are wondering whatever happened to the study of finite simple groups,
one answer is that the algebraic topologists picked them up in order to
compute their cohomology. In topologists' terms, they want the cohomology
of a classifying space BG = K(G,1) of a finite group G, meaning a space
with fundamental group G whose universal cover is contractible.  I don't
know why they want this information, but it's certainly cool. In fact I
only know lesson #1 in this theory, which is already interesting enough.
Lesson #1 is that in studying the cohomology with coefficients in Z/p^n,
you really want the cohomology of the Sylow p-subgroup G_p of G and not
G itself.  The reason is that BG_p is a partial cover of G whose index is
coprime to p, which implies that there is a very interesting push-down map

H^*(BG_p;Z/p^n) -> H^*(BG;Z/p^n)

given by averaging over sheets of the covering.   This push-down map is
a one-sided inverse to the pull-back map

H^*(BG;Z/p^n) -> H^*(BG_p;Z/p^n),

which implies that the pull-back map is an embedding.  Thus
H^*(BG_p;Z/p^n) has all of the information that H^*(BG;Z/p^n) does,
and then some.  This is why math.AT/0008230 is devoted to the Sylow
2-subgroup of Higman-Sims and not to Higman-Sims itself. For reasons
that I understand less well, the prime 2 is more interesting than other
primes in this game.  In fact there is a joke among algebraic topologists
that the only primes worth studying are 2 and the infinite prime.

Speaking of Z/p^n and its close cousin, the p-adic integers, Henry Cohn
interprets domino tilings of a square in terms of the 2-adics integers
Z_2 [math.CO/0008222].  Counting domino tilings of a square is one of my
favorite examples of the merit of recreational mathematics; the question
is almost invariably to mathematicians and non-mathematicians alike.
It so happens that the number of domino tilings of a 2n x 2n square is
always 2^n times an odd perfect square, which Cohn writes as f(n)^2.
This fact is tied up with the closed-form expression for the number that
was found by Kasteleyn and Kasteleyn's method for finding that expression.
Cohn establishes that f(n) extends to a continuous function from Z_2 to
Z_2, and that it satisfies the functional equation

f(-1-n) = (-1)^(n choose 2) f(n).

"This week in the mathematics arXiv" may be freely redistributed
with attribution and without modification.

[deletia --djr]

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