For a prime $p \geq 5$ , we compute the algebraic $K$ -theory modulo $p$ and $v_{1}$ of the mod $p$ Adams summand, using topological cyclic homology. On the way, we evaluate its modulo $p$ and $v_{1}$ topological Hochschild homology. Using a localization sequence, we also compute the $K$ -theory modulo $p$ and $v_{1}$ of the first Morava $K$ -theory.

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The smooth Whitehead spectrum of a point at odd regular primes.

Hopf algebra structure on topological Hochschild homology.

Differentials in the homological homotopy fixed point spectral sequence.

A chain rule in the calculus of homotopy functors.

The Hatcher-Waldhausen map is a spectrum map.

Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere.

Algebraic K -theory of the first Morava K -theory

Algebraic $K$ -theory of the first Morava $K$ -theory