Date: Mon, 15 Jul 2002 10:00:43 -0400 (EDT)
From: "Douglas C. Ravenel"
Subject: Zhou's paper on the existence of V(n)
Dear Topologists,
A few months ago Zhou Xueguang posted a paper on the Hopf archive
entitled "Smith-Toda spectrum V(\infty) exists for all p \geq 5."
It is 50 pages long. Several of you have asked me what I think
of it. I am very skeptical about it, and I sent a message
similar to this one to Zhou 5 months ago. He has not replied.
As the title indicates, the paper claims to prove that the
Smith-Toda spectrum $V(n)$ exists for all $n$ for primes larger
than 3. This is surprising since several authors including
myself have proved nonexistence results for these complexes.
Zhou is asserting that we are all mistaken. In particular he
denies Toda's result about the vanishing of $\alpha_1 \beta_1^p$
in the stable homotopy of the sphere at an odd prime. The
paper includes an appendix explaining why both of Toda's proofs
of this theorem are incorrect. Adding it was a good idea because
most people would be very reluctant to read Zhou's paper unless
they believed that Toda was wrong.
I have not read the body of Zhou's paper, so I cannot point to an
error in his method of proof. However I have read the appendix,
and I do not believe it. I will give my reasons in detail here.
In what follows, citation and reference numbers are taken from
Zhou's preprint.
Zhou discusses two proofs of Toda's relation given in his papers
[14] and [15]. The first involves a space $B_m(p)$ which is a
$S^{2m+1}$-bundle over $S^{2m+2p+1}$. Zhou disputes Toda's claim
that $H_*(\Omega B_m(p))$ is a polynomial alagebra on two
variables, saying that he fails to prove that the Serre SS for
the fibration
$$
\Omega S^{2m+1} \to \Omega B_m(p) \to \Omega S^{2m+2p+1}.
$$
collapses. However both fiber and base have cells only in even
dimensions, so it must collapse.
Toda's argument in [15] relies on the fact that the Steenrod
operation $P^1$ acts nontrivially in the cohomology of the finite
complex $ep^{p-2}M(Z/p, l)$. Proposition 19.1 says it acts
trivially there. In Zhou's proof of it I am assuming that he
means
$$
T_1 = M(3l) \vee M(3l+1) \vee M(3l+1)
$$
and $T = T_1 \vee T_2$. Since $W$ is equivalent to $T$ there are
maps $\overline{q}:T_1 \to W$, $\overline{_4q}:T_2 \to W$,
$\overline{r}:W \to T_1$ and $r_4: W \to T_2$ such that
$\overline{q} \vee q_4$ and $\overline{r} \vee r_4$ are homotopy
equivalences.
In the argument that $P((\alpha \odot \beta)\odot \gamma)$ is
homotopic to $id((\alpha \odot \beta)\odot \gamma)$, one cannot
use the symbols $\alpha$, $\beta$ and $\gamma$ interchangeably
even though they denote the identity maps on isomorphic spectra.
All that Zhou's argument shows is that the two maps behave the
same in homology and that the composite $r_4 (P-id) q_4$ is null.
This does NOT imply $(P-id) q_4$ is null or that $\Sigma T_2$ is
a wedge summand of $C(P-id)$. For that conclusion one would need
to show that the composite $\overline{r} (P-id) q_4$ is also
null.
Toda proves that $\overline{r} (P-id) q_4$ is not null because it
is homotopic to the composite $ i \alpha_1 j$, where $i:S^{3l}
\to T_1$ is the inclusion of the bottom cell, and $j:T_2 \to
S^{3l+3}$ is projection onto the top cell.
Thus I do not believe that Zhou has shown that either of Toda's
proofs that $\alpha_1 \beta_1^p = 0$ is incorrect.
Doug
Douglas C. Ravenel, Chair |918 Hylan Building
Department of Mathematics |drav@math.rochester.edu
University of Rochester |(585) 275-4413
Rochester, New York 14627 |FAX (585) 273-4655
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