Let {X} be a connective spectrum with finitely generated homotopy groups. Then the lowest homotopy group in {\pi_*(X \wedge X)} is the tensor square of the lowest homotopy group in {\pi_*(X)}: in particular, X \wedge X is never zero (i.e., contractible). The purpose of this post is to describe an example of a nontrivial spectrum {I} with {I \wedge I \simeq 0}. I learned this example from Hovey and Strickland’s “Morava {K}-theories and localization.”

1. A non-example

To start with, here’s a spectrum which does not work: {H \mathbb{Q}/\mathbb{Z}}. This is a natural choice because

\displaystyle \mathbb{Q}/\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z} = 0.

On the other hand, from the cofiber sequence

\displaystyle H \mathbb{Z} \rightarrow H \mathbb{Q} \rightarrow H \mathbb{Q}/\mathbb{Z} \rightarrow \Sigma H \mathbb{Z},

we obtain a cofiber sequence

\displaystyle H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z} \rightarrow 0 \rightarrow (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \rightarrow \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z}

which shows in particular that

\displaystyle (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \simeq \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z};

in particular, its {\pi_1} is isomorphic to {\mathbb{Q}/\mathbb{Z}}, not zero. (more…)