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In this paper we are concerned with the four functors .9*, .9: iff , ~ , and £:-00 , which map from the category of topological spaces X to the category d'ff of n-spectra. The functor .9* ( ) (or .9* 1 ( )) maps the space X to the nspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces .9 i (X) , .9 i diff (X) , i ;::: 0, are Hatcher's deloopings of .9 0 (X) , .9~iff(X) (cf. )). The functor ~() maps a path-connected space X to the algebraic Ktheoretic n-spectrum fordoi:10.1090/s0894-0347-1993-1179537-0 fatcat:34nxr4h6kvfivodnazel7c7rvq