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)}.

4. The Adams conjecture

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:

Adams conjecture (restated): The map

\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.