in the late 1960s and early 1970s, several definitions of higher "k"-theory were proposed. swan and gersten both produced definitions of "k" for all "n", and gersten proved that his and swan's theories were equivalent, but the two theories were not known to satisfy all the expected properties. nobile and villamayor also proposed a definition of higher "k"-groups. karoubi and villamayor defined well-behaved "k"-groups for all "n", but their equivalent of "k" was sometimes a proper quotient of the bass–schanuel "k". their "k"-groups are now called "kv" and are related to homotopy-invariant modifications of "k"-theory.