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 Personal home page: http://www.math.rochester.edu/u/drav/ Department of Mathematics home page: http://www.math.rochester.edu/ Math 443 home page: http://www.math.rochester.edu/courses/2002-spring/MTH443/ Faculty Senate home page: http://www.cc.rochester.edu/Faculty/senate/