The purpose of this post is to show the existence of a phantom map . To say that is a phantom map means that is nonzero, but is nontrivial for each : in particular, induces the zero map in any homology theory. (This is essentially following a paper of Brayton Gray, “Spaces of the same -type for all .”)

In order to do this, we note that is the homotopy colimit of the spectra , and consequently we have a Milnor exact sequence

For convenience, we have abbreviated to simply .

Our goal will be to show that the term is nonzero. It will then follow that we have a “phantom” map , as desired.

**1. Failure of the Mittag-Leffler condition**

To do so, we consider the inverse system of groups where denotes homotopy classes of stable maps. The claim is that this does not satisfy the Mittag-Leffler condition. In fact, let’s look at the images of the maps

Rational stable cohomotopy is the same as rational cohomology, so the map is a -isomorphism: in particular, the image has finite index. However, the claim is that given any class in , it is not in the image of for sufficiently large.

In fact, choose a class ; suppose induces multiplication by on . Suppose , and we had a stable map in restricting to on . Then we would have a map of suspension spectra

inducing multiplication by on . But, for any odd prime , the Steenrod th power acts on by raising the generating class to the power . If , then acts nontrivially on the for the generator of . On the contrary, it acts trivially on . This is a contradiction, and we find that there is no stable map restricting to a degree map , if is sufficiently large (bigger than the first prime after ).

So we’ve seen that the inverse system does *not* satisfy the ML condition. In the next section, we will show that this implies the term is actually nonzero, which will prove the existence of a phantom map.

**2. A lemmas on inverse limits**

An inverse system of abelian groups satisfying the Mittag-Leffler condition has vanishing ; the purpose of the next lemma is to prove a weak converse.

Lemma 1Let be an inverse system of countable abelian groups, and suppose that it does not satisfy the Mittag-Leffler condition. Then .

Let’s assume for simplicity that the ML condition is not satisfied at the zeroth stage: that is, that the decreasing sequence of subgroups

does not stabilize. Then we have a map of inverse systems

which is a surjection, and consequently induces a surjection on (because there is no for a -indexed tower of abelian groups). Consequently, it suffices to prove the lemma for an inverse system of *injections.*

So let’s do that. Suppose we have an inverse system of countable abelian groups

We have a short exact sequence of inverse systems

and if , then we have a short exact sequence

But is a complete topological group under the inverse limit topology. Also, there is a neighborhood basis at consisting of the images of for each (these are nonzero because the sequence does not stabilize), and in particular the inverse limit is not discrete, and has the property that the complement of any point is dense. By the Baire category theorem, it follows that is actually uncountable. This contradicts the existence of a surjection .

## Leave a Reply