The connective -theory of the Eilenberg–MacLane space

We compute $ku^*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ and $ku_*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ , the connective $KU$ -cohomology and connective $KU$ -homology groups of the mod- $p$ Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,2\right)$ , using the Adams spectral sequence. We obtain a striking interaction between $h_0$ -extensions and exotic extensions. The mod- $p$ connective $KU$ -cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.