Localized J homomorphism

Question

Let $X$ be a simply connected finite CW complex, $\xi$ and $\eta$ vector bundles over $X$ of the same dimensions and their dimension is big enough, so they are stable bundles. Let $p$ be a prime. Are the following two conditions equivalent?

1. $J(\xi)$ and $J(\eta)$ as elements in $J(X)$ have equal $p$-primary components. (That is there is a $q$ not divisible by $p$ such that $q(J(\xi) - J(\eta) ) = 0.$)

2. The sphere bundles $S(\xi)$ and $S(\eta)$ are fiberwise $p$-equivalent. (That is there is a fiberwise map $S(\xi) \to S(\eta)$ that induces isomorphism in homologies with $Z/pZ$ coefficients.)

Answer 1

That 2. implies 1. is a consequence of Adams' mod k Dold Theorem (Theorem 1.1 of On the groups J(X)-I. Topology 2 (1963) pp. 181-195).

The following is an argument for 1. implies 2. (using ideas similar to those in Adams' proof of the above as well as Sullivan's theory of p-local spherical fibrations contained in his MIT Geometric Topology notes (Chapter 4)).

If $E \to X$ is a spherical fibration ($p$-local or not) over $X$, I will write $\Sigma E$ for its fiberwise suspension. It is sufficient to produce a map $\Sigma^m S(\xi) \to \Sigma^m S(\eta)$ with fiberwise degree prime to $p$ for some large $m$, as obstruction theory and the stability assumption on $\xi$ and $\eta$ allow us to compress such a map down to a map $S(\xi) \to S(\eta)$.

[Edit:] In more detail, let $n$ be the dimension of the fibers of $S(\xi)$ and $S(\eta)$. The stability assumption is that $\dim X<n$. A fiberwise map from $S(\xi)$ to $S(\eta)$ is a section of a bundle over $X$ with fiber $Map(S^n,S^n)$. Since the inclusion $Map(S^n,S^n) \hookrightarrow Map(S^{n+m},S^{n+m})$ determined by suspension is $(n-1)$-connected there is no obstruction to desuspending a fiberwise map $\Sigma^m S(\xi) \to \Sigma^m S(\eta)$ to a map $S(\xi) \to S(\eta)$.

Condition 1. implies that the fiberwise localizations $S(\xi)_{(p)}$ and $S(\eta)_{(p)}$ are equivalent.

[Edit:] In more detail, let $BG$ denote the classifying space of stable spherical fibrations, let $BG_{p-loc}$ denote the classifying space for $p$-local spherical fibrations and let $BSG$ and $BSG_{p-loc}$ be their universal covers. Since $X$ is simply connected we have $[X,BSG]=[X,BG]$. $J(X)$ is the image of $[X,BSO]=[X,BO]$ in $[X,BSG]$ under the canonical map. Since the map $BSG \to BSG_{p-loc} = BSG_{(p)}$ is localization at $p$ (see Sullivan's notes Theorem 4.2 (iii) p. 93), Condition 1. implies that the maps $X \to BSG$ classifying $S(\xi)$ and $S(\eta)$ become homotopic when composed with the localization map $BSG \to BSG_{(p)}$.

Thus there exists a fiberwise map $f \colon S(\xi)_{(p)} \to S(\eta)_{(p)}$ with fiberwise degree $k/l \in \mathbb{Z}_{(p)}^\times$ (with $k,l$ integers prime to $p$). Using the suspension coordinate we can define a map $lf \colon \Sigma S(\xi)_{(p)} \to \Sigma S(\eta)_{(p)}$ with fiberwise degree $k \in \mathbb{Z}$. Precomposing with the localization map we have a map $\Sigma S(\xi) \to \Sigma S(\eta)_{(p)}$ of fiberwise degree prime to $p$ and all that remains is to compress this map down along the inclusion $\Sigma S(\eta) \hookrightarrow \Sigma S(\eta)_{(p)}$.

[Edit:] Clearly this is possible over the $0$-skeleton of $X$. This compression will be extended inductively over the cells of $X$ at the expense of suspending and "multiplying" the maps fiberwise by integers prime to $p$.

In the inductive step we need to extend a compression over a $k$-cell of $X$ with $k>1$. As the space of fiberwise maps $\Sigma^m S(\xi) \to \Sigma^m S(\eta)_{(p)}$ is the space of sections of a fibration with fiber $Map(S^{n+m},S^{n+m}_{(p)})$ the problem reduces to finding a long diagonal lift in the commutative diagram $$ \begin{array}{ccc} S^{k-1} & \longrightarrow & Map(S^{n+m},S^{n+m}) & \stackrel{j}{\longrightarrow} & Map(\Sigma S^{n+m}, \Sigma S^{n+m}) \\ \downarrow & & \downarrow & & \downarrow\\ D^k & \longrightarrow & Map(S^{n+m},S^{n+m}_{(p)}) & \stackrel{j}{\longrightarrow} & Map(\Sigma S^{n+m},\Sigma S^{n+m}_{(p)}) \end{array} $$ where the horizontal maps $j$ denote fiber multiplication by a natural number $j$ (using the suspension coordinate).

The middle vertical map is $p$-localization on higher homotopy groups, and $j$ induces multiplication by $j$ on $\pi_{k-1}$ so we are done.