p-adic K(n) Euler Characteristic

This is a definition of Yanovski.

Defn Let $X$ be a $p$-small space ${\chi }_n(X):={\text{dim}}_{K(n{)}^{\bullet }(pt)}K(n{)}^{\text{even}}(X)-{\text{dim}}_{K(n{)}^{\bullet }(pt)}K(n{)}^{\text{odd}}(X)$

Example $n=0$, $K(0)\simeq H\mathbb{Q}$, we recover the rationalization of the usual Euler characteristic. To get the integral version, we might set $n=-1$ and use the heuristic $K(-1):=H\mathbb{Z}$.

Setting

The relevant class of objects is

Defn (p-small spaces) ${\mathcal{S}}^{p-sm}\subset \mathcal{S}$ subcategory generated by $\pi$-finite $p$-spaces under finite colimits.

That is, it’s the homotopical version of finite $p$-groups.

Example $B^{n}G$, for $G$ a finite $p-$group, and any finite colimits built out of these.

Example B^n\mathbb Z_\hat p are $p$-small. This is a key example, as they are the tori in this setting: A ${\mathbb{T}}^n$-action on a $p$ space is equivalent to an action by B^n\mathbb Z_\hat p