Relative Loday constructions and applications to higher THH-calculations

We define a relative version of the Loday construction for a sequence of commutative S-algebras $A \rightarrow B \rightarrow C$ and a pointed simplicial subset $Y \subset X$. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.