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…)