  We introduce the notion of "covering homology" of a commutative ring spectrum
with respect to certain families of coverings of topological spaces. The
construction of covering homology is extracted from Bokstedt, Hsiang and
Madsen's topological cyclic homology. In fact covering homology with respect to
the family of orientation preserving isogenies of the circle is equal to
topological cyclic homology.
  Our basic tool for the analysis of covering homology is a cofibration
sequence involving homotopy orbits and a restriction map similar to the
restriction map used in Bokstedt, Hsiang and Madsen's construction of
topological cyclic homology.
  Covering homology with respect to families of isogenies of a torus is
constructed from iterated topological Hochschild homology. It receives a trace
map from iterated algebraic K-theory and the hope is that the rich structure,
and the calculability of covering homology will make covering homology useful
in the exploration of J. Rognes' ``red shift conjecture''.
