the dennis trace map to topological hochschild homology facts through topological cyclic homology, providing an even more detailed tool for calculations. in 1996, dundas, goodwillie, and mccarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic "k"-theory, so that if a calculation in "k"-theory or topological cyclic homology is possible, then many other "nearby" calculations follow.