The Sullivan conjecture on maps from classifying spaces originated in Sullivan’s work on localizations and completions in topology, which together with étale homotopy theory led him to a proof of the classical Adams conjecture. The purpose of this post is to briefly explain the conjecture; in the next post I’ll discuss Sullivan’s proof.

1. Vector bundles and spherical fibrations

Let ${X}$ be a finite CW complex. Given a real ${n}$-dimensional vector bundle ${V \rightarrow X}$, one can form the associated spherical fibration ${S(V) \rightarrow X}$ with fiber ${S^{n-1}}$ by endowing ${V}$ with a euclidean metric and taking the vectors of length one in each fiber. The Adams conjecture is a criterion on when the sphere bundles associated to vector bundles are fiber homotopy trivial. It will be stated in terms of the following definition:

Definition 1 Let ${J(X)}$ be the quotient of the Grothendieck group ${KO(X)}$ of vector bundles on ${X}$ by the relation that ${V \sim W}$ if ${V , W}$ have fiber homotopy equivalent sphere bundles (or rather, the group generated by it).

Observe that if ${V, W}$ and ${V', W'}$ have fiber homotopy equivalent sphere bundles, then so do ${V \oplus V', W \oplus W'}$; for example, this is because the sphere bundle of ${V \oplus V'}$ is the fiberwise join of that of ${V}$ and ${V'}$. It is sometimes more convenient to work with pointed spherical fibrations instead: that is, to take the fiberwise one-point compactification ${S^V}$ of a vector bundle ${V \rightarrow X}$ rather than the sphere bundle. In this case, the fiberwise join is replaced with the fiberwise smash product; we have

$\displaystyle S^{V \oplus W} \simeq S^V \wedge_X S^W,$

where ${\wedge_X}$ denotes a fiberwise smash product.

One reason is that this is of interest is that the group ${KO(X)}$ of vector bundles on a space ${X}$ is often very computable, thanks to Bott periodicity which identifies the ${KO}$-groups of a point. The set of spherical fibrations is much harder to describe: to describe the spherical fibrations over ${S^n}$ essentially amounts to computing a bunch of homotopy groups of spheres.

2. An abstract reformulation

It is possible to phrase the above more abstractly. Given a vector bundle ${V \rightarrow X}$, we can think of it as a “family of vector spaces” ${V_x}$ parametrized by the points ${x \in X}$. One-point compactifying at each level, we get a family of pointed spheres ${S^{V_x}, x \in X}$. Taking suspension spectra, we get a family of spectra ${\Sigma^\infty S^{V_x}, x \in X}$. We can think of this as the “stable spherical fibration” associated to the vector bundle ${V}$: a family of sphere spectra parametrized by ${X}$. A way of making this precise is the Grothendieck construction, which identifies spaces fibered over ${X}$ with functors from ${X}$ (considered as an ${(\infty, 1)}$-category) into the ${(\infty, 1)}$-category of spaces.

More explicitly, we have a map of group-like topological monoids

$\displaystyle O(n) \rightarrow \mathrm{Equiv}_*(S^n, S^n)$

from ${O(n)}$ to the monoid of self-equivalences of ${S^n}$ (preserving the basepoint). The classifying space ${BO(n)}$ classifies ${O(n)}$-bundles (i.e., vector bundles). The classifying space ${B \mathrm{Equiv}_*(S^n, S^n)}$ classifies pointed ${S^n}$-fibrations. And the unstable ${J}$-homomorphism is the map

$\displaystyle BO(n) \rightarrow B \mathrm{Equiv}_*(S^n, S^n).$

Taking the colimit, we get a map

$\displaystyle BO \rightarrow B \mathrm{gl}_1(S) \stackrel{\mathrm{def}}{=}B \mathrm{Equiv}^{\mathrm{Sp}}_*( S^0, S^0)$

from ${BO}$ to the classifying space of the monoid of self-equivalences of the sphere spectrum ${S^0}$. The space ${B \mathrm{gl}_1(S)}$ classifies “families of sphere spectra.” This is the stable ${J}$-homomorphism, and the group ${J(X)}$ is the image of ${[X, BO]}$ in ${[X, B \mathrm{gl}_1(S)]}$.

In fact, it is an infinite loop map, as it arises as the group-completion of the map of symmetric monoidal topological categories

$\displaystyle \mathrm{Vect}^{\simeq } \rightarrow \mathrm{Spheres}, \quad V \mapsto S^V$

which sends a vector space to its one-point compactification.

The spectrum ${\mathrm{gl}_1(S) = \Omega B \mathrm{gl}_1(S)}$ can be thought of the “spectrum of units” of the sphere spectrum (see e.g. this paper of Ando-Blumberg-Gepner-Hopkins-Rezk for a modern exposition); as an infinite loop space, it is the union of the components ${\pm 1}$ of the infinite loop space ${(\Omega^\infty( S^0))}$ of ${S^0}$. The (infinite loop) group law comes from multiplication in the sphere spectrum, though. In particular, we have

$\displaystyle \pi_i B \mathrm{gl}_1(S) = \begin{cases} 0 & \text{if } i = 0 \\ \mathbb{Z}/2 & i = 1 \\ \pi_{i-1}^s(S^0) & i > 1. \end{cases}$

We get a map

$\displaystyle \pi_i BO \rightarrow \pi_{i-1}^s(S^0),$

which is the classical formulation of the ${J}$-homomorphism.

3. Dold’s theorem

In order to motivate the Adams conjecture, we’ll need to give some examples of vector bundles which are not equivalent but have equivalent spherical fibrations. A criterion of Dold will be useful here.

Example 1 (Dold) Let ${f_1, f_2: X \rightarrow B \mathrm{gl}_1(S)}$ classify (pointed) stable spherical fibrations ${T \rightarrow X, T' \rightarrow X}$ over ${X}$ (which can be thought of as parametrized spectra or spherical fibrations of large dimension). Suppose that there is a fiberwise map

which is of degree ${k}$ on each fiber. In other words, the parametrized spectra become equivalent after fiberwise inversion of ${k}$. There is a space (in fact, an infinite loop space) ${B \mathrm{gl}_1( S[1/k])}$ which classifies parametrized ${S[1/k]}$‘s in a similar manner.

In particular, the maps

$\displaystyle X \stackrel{f_1, f_2}{\rightrightarrows} B \mathrm{gl}_1(S) \rightarrow B \mathrm{gl}_1(S[1/k])$

are homotopic. But the map

$\displaystyle B \mathrm{gl}_1(S)[1/k] \rightarrow B \mathrm{gl}_1(S[1/k])[1/k]$

can be identified with a covering map. It follows that the maps ${X \stackrel{f_1, f_2}{\rightrightarrows} B \mathrm{gl}_1(S) \rightarrow B \mathrm{gl}_1(S)[1/k]}$ are homotopic, or that (since ${X}$ is a finite complex!)

$\displaystyle k^N (f_1 - f_2) = 0 \in J(X), \quad N \gg 0.$

That is, after “multiplying by ${k^N}$” (i.e., by taking fiberwise smash products ${k^N}$ times) the spherical fibrations associated to ${f_1, f_2}$ become fiber homotopy equivalent.

Example 2 (Adams) In general, it is possible for non-equivalent vector bundles to have equivalent spherical fibrations. Let ${\mathcal{L}}$ be a complex line bundle (equivalently, an oriented real ${2}$-dimensional bundle). Consider the spherical fibrations (${S^1}$-bundles) associated to ${\mathcal{L}}$ and ${\mathcal{L}^{k}}$. There is a map

$\displaystyle S^{\mathcal{L})} \rightarrow S^{ \mathcal{L}^k}, \quad z \mapsto z^k,$

which is of degree ${k}$ on each fiber. As a result, the element

$\displaystyle k^N ( \mathcal{L}^k - \mathcal{L}) \in K(X) , \quad N \gg 0$

maps to zero in ${J(X)}$.

One notes now that ${\mathcal{L}^k}$ can also be described as ${\psi^k(\mathcal{L})}$, where ${\psi^k}$ is the Adams conjecture. Adams conjectured that this natural generalization worked for any vector bundle:

Adams conjecture: Let ${V}$ be a real vector bundle on ${X}$. Then, for ${N \gg 0}$, ${k^N ( \psi^k(V) - V)}$ maps to zero in ${J(X)}$.

In other words, the stable spherical fibration associated to ${k^N ( \psi^k(V) - V)}$ is fiber homotopy trivial. In more abstract language, we can state:

$\displaystyle BO \stackrel{\psi^k - 1}{\rightarrow} BO \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S) \rightarrow B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic.

For simplicitly, let’s see what this implies when ${BO}$ is replaced by the complex version ${BU}$. We have

$\displaystyle \pi_{2n} BU \simeq \mathbb{Z} ,$

and if ${\iota_n \in \pi_{2n} BU}$ is a generator, then ${\psi^k \iota_n = k^n \iota_n}$. In particular, it follows that

$\displaystyle J( k^N(k^n - 1) \iota_n) = 0, \quad N \gg 0,$

for each ${k}$. Taking the least common multiple over all ${k}$, Adams observed that this conjecture enabled one to thus bound above the image of the ${J}$-homomorphism.