On a Theorem of Ossa
David Copeland Johnson
University of Kentucky
Lexington, Kentucky 40506
johnson@ms.uky.edu
and
W. Stephen Wilson
Johns Hopkins University
Baltimore, Maryland 21218
wsw@math.jhu.edu
21 December 1995
Abstract
If V is an elementary abelian 2-group, Ossa proved that the con-
nective K-theory of BV splits into copies of Z=2 and of the connective
K-theory of the infinite real projective space. We give a brief proof of
Ossa's theorem.
Introduction. We have been asked whether our work, [1] and [2] on
the Brown-Peterson homology of BV , V an elementary p-group, gives a nice
structure of the connective K-theory of BV . The answer is that the approach
of [1] leads to the elegant structure theorem of Ossa [3]. Although the ap-
proach is motivated by our [1] and [2], the proof is independent of that work.
In this reproof of an established theorem we shall limit our exposition to the
p = 2 case. For us, the notation makes this the easiest case, but for Ossa,
it was the more difficult one. With obvious modifications, the odd-primary
version of our argument follows the same outline.
1
Notation. Let bu be the connective K-theory spectrum and let P denote
BZ=2 (also known as infinite real projective space). Let HZ=2 be the Z=2
Eilenberg-MacLane spectrum.
Theorem 1 (Ossa) With the above notation, there is a homotopy equiva-
lence of spectra
_
bu ^ P ^ P ' [ 2i+2j-2HZ=2] _ [2bu ^ P ]:
0* 0} give a basis for an E-module M* isomorphic
to H*(S2 ^ P ; Z=2). Modulo M*, H*(P ^ P ; Z=2) is isomorphic to a free
E-module with basis {s2i-1^ t2j-1 : i; j > 0}. Clearly in dimension 2,
(bu ^ ss)* O ae* : H2(bu ^ S2; Z=2) -! H2(bu ^ P ; Z=2)
is an isomorphism. Thus g*1takes H2(bu ^ S2 ^ P ; Z=2) isomorphically onto
A==EW M*. By the construction of g0, we see that g*0takes
H*( 0** 0}. Thus g induces an isomorphism in mod 2
cohomology and thus is an equivalence. 2
References
[1]D. C. Johnson and W. S. Wilson, The Brown-Peterson homology of
elementary p-groups, Amer. J. Math. 102 (1982), 427-454.
[2]D. C. Johnson, W. S. Wilson, and D. Y. Yan, Brown-Peterson homology
of elementary p-groups II, Topology and its Applications 59 (1994) 117-
136.
[3]E. Ossa, Connective K-theory of elementary abelian groups, Transfor-
mation Groups, Osaka 1987, K. Kawakubo (ed.), Springer Lecture Notes
in Mathematics 1375 (1989) 269-275.
3
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