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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.

We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

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The connective $K$-theory of the Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,\textrm{2}\right)$

MORAVA K-THEORY AND FILTRATIONS BY POWERS

THE GROMOV-LAWSON-ROSENBERG CONJECTURE FOR THE SEMI-DIHEDRAL GROUP OF ORDER 16

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The connective $K$-theory of the Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,\textrm{2}\right)$

MORAVA K-THEORY AND FILTRATIONS BY POWERS

THE GROMOV-LAWSON-ROSENBERG CONJECTURE FOR THE SEMI-DIHEDRAL GROUP OF ORDER 16