  We consider genera of polyhedra (finite cell complexes) in the stable
homotopy category. Namely, the genus of a polyhedron X is the class of
polyhedra Y such that all localizations of Y are stably isomorphic to the
corresponding localizations of X. We prove that Y is in the genus of X if and
only if the wedge XvB is stably isomorphic to YvB, where B is the wedge of all
spheres S^n such that the n-th stable homotopy group of X is not torsion. We
also prove that if XvX and XvY are stably isomorphic, so are also X and Y.
Several examples of calculations of genera are considered.
