Contemporary Mathematics
K(n + 1) equivalence implies K(n) equivalence
W. Stephen Wilson
Abstract.We give an entirely different proof of a recent result of Bousf*
*ield's
which states that if there is a map of spaces inducing an isomorphism on*
* the
(n + 1)stMorava K-theory then it also induces an isomorphism on the nth
Morava K-theory. The result relies heavily on the fundamentals introduced
to prove the results of [RWY ] which in turn relies on the Boardman-Wil*
*son,
[BW ], generalization of Quillen's theorem that MU*(X) is generated by *
*non-
negative degree elements when X is a finite complex.
1. Introduction
Bousfield has shown that a map which induces an isomorphism on the Morava
K-theory, K(n + 1)*(-), also induces an isomorphism on K(n)*(-). This is easily
seen, by taking the cofibre, to be equivalent to showing that if K(n + 1)*(X) is
acyclic then so is K(n)*(X). We state a cohomology version which is what we will
prove. It is equivalent to the homology version by duality, i.e. X is K(n) homo*
*logy
acyclic if and only if it is K(n) cohomology acyclic. Our version is weaker th*
*an
Bousfield's because we assume X is of finite type and he has no restrictions on*
* X.
Theorem 1.1 (Bousfield, [Bou ]).Let X by a space of finite type with K(n +
1)*(X) trivial, then K(n)*(X) is trivial for n > 0.
Our proof has little, if anything, in common with Bousfield's and so we hope
it will be of independent interest. Ravenel, in [Rav84 , Theorem 2.11], proves *
*this
result for X finite. Our proof makes heavy use of work on phantom maps and the
Atiyah-Hirzebruch spectral sequence in [RWY ] which was preparatory to the ma*
*in
theorems of that paper. This work in turn was made possible by the generalized
Quillen theorem in [BW ] which says that P (n)*(X), for X finite, is generated*
* by
non-negative degree elements.
We use reduced theories throughout. Special thanks go out to Pete Bousfield,
Dan Christensen, Douglas Ravenel, Hal Sadofsky, and Neil Strickland, all of whom
have help me to clarify my thoughts on these matters.
2. Proof
We recall the generalized cohomology theory E(n; n+1), n > 0, with coefficie*
*nt
ring E(n; n + 1)* ' Fp[vn; vn+1; v-1n+1].
cO0000 (copyright holder)
1
2 W. STEPHEN WILSON
Lemma 2.1. Let X be a space of finite type. If K(n + 1)*(X) is trivial then
E(n; n + 1)*(X) is trivial.
Proof. There is a stable cofibration E(n; n + 1) vn!E(n; n + 1) ! K(n). This
gives rise to a long exact sequence in cohomology theories and since K(n)*(X) is
trivial, multiplication by vn is an isomorphism on E(n; n + 1)*(X). However, by
[RWY , Corollary 4.11] there are no infinitely vn divisible elements in E(n;_*
*n +
1)*(X) so this must be trivial. |__|
Proof of Theorem 1.1. By [RWY , Corollary 4.8] we have no phantom
maps for either K(n)*(X) or E(n; n + 1)*(X) and so the Atiyah-Hirzebruch spec-
tral sequence for these theories converges. By the same result, we can obtain
the Atiyah-Hirzebruch spectral sequence for these theories by tensoring K(n)* or
E(n; n + 1)* with the Atiyah-Hirzebruch spectral sequence for P (n)*(X) where
P (n) is the theory with coefficient ring BP *=In where BP *' Z(p)[v1; v2; : :]:
and In = (p; v1; : :;:vn-1). Combining this with [RWY , Lemma 4.4] we have,
for an arbitrary fixed s, some m for which Es;*m' Es;*1for all of the theories
P (n), K(n) and E(n; n + 1). Our acyclic assumption and Lemma 2.1 tell us
that Es;*m(P (n)*(X)) P(n)*E(n; n + 1)* is zero. Es;*m(P (n)*(X)) is a finitely*
* pre-
sented P (n)*(P (n)) module and so has a Landweber filtration (from [Yag76 ] and
[Yos76 ]), see [RWY , Theorem 3.10]. Again from Yagita and Yosimura, tensoring
with K(n)* or E(n; n + 1)* is exact, see [RWY , Theorem 3.9]. The quotients of
the Landweber filtration are P (n)*=Iq;nwhere Iq;n= (vn; : :;:vq-1). By exactne*
*ss
we know that E(n; n + 1)* tensored with each quotient must be zero. That means
that each q must be greater than n + 1. Tensoring those quotients with K(n)* al*
*so
gives zero. Thus we have shown that Es;*m(P (n)*(X)) P(n)*K(n)* is zero. This
tells us the whole Atiyah-Hirzebruch spectral sequence converges to zero and_our
result is complete. |__|
References
[Bou] A. K. Bousfield. On Morava K-equivalences of spaces. In preparation.
[BW] J. M. Boardman and W. S. Wilson. k(n)-torsion-free H-spaces and P(n)-coho*
*mology. In
preparation.
[Rav84]D. C. Ravenel. Localization with respect to certain periodic homology th*
*eories. American
Journal of Mathematics, 106:351-414, 1984.
[RWY] D. C. Ravenel, W. S. Wilson, and N. Yagita. Brown-Peterson cohomology fro*
*m Morava
K-theory. K-Theory. To appear.
[Yag76]N. Yagita. The exact functor theorem for BP*=In-theory. Proceedings of t*
*he Japan Acad-
emy, 52:1-3, 1976.
[Yos76]Z. Yosimura. Projective dimension of Brown-Peterson homology with*
* modulo
(p; v1; : :;:vn-1) coefficients. Osaka Journal of Mathematics, 13:289-309*
*, 1976.
Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: wsw@math.jhu.edu