  We compute the equivariant complex K-theory ring of a cohomogeneity-one
action of a compact Lie group at the level of generators and relations and
derive a characterization of K-theoretic equivariant formality for these
actions. Less explicit expressions survive for a range of equivariant
cohomology theories including Bredon cohomology and Borel complex cobordism.
The proof accordingly involves elements of equivariant homotopy theory,
representation theory, and Lie theory.
  Aside from analysis of maps of representation rings and heavy use of the
structure theory of compact Lie groups, a more curious feature is the essential
need for a basic structural fact about the Mayer--Vietoris sequence for any
multiplicative cohomology theory which seems to be otherwise unremarked in the
literature, and a similarly unrecognized basic lemma governing the equivariant
cohomology of the orbit space of a finite group action.
