Topology Vol.11.~~. 349-330. P-qmnon F'rcsr, 1972.Printcd inGreatBritain DELOOPING CLASSIFYING SPACES IN ALGEBRAIC K-THEORY J. B. WAGONER* (Receizxxi 17 September 1971) FOR ANY associative ring with identity A let BGL(A)+ denote the “classifying space” for algebraic K-theory given in [I51 where a definition for the higher K-theory functors Ki, i >, 1, is proposed by Quillen as K,(A) = n,(BGL(A)+). This K, and KZ agree with the K, of Bass [2] and the K2 of Milnor [ 141 and the theory has other very pleasant properties [ 151. Now Anderson [l] and Segal [17] have shown how to associate a generalized cohomology theory K’(X; q) to any category %? with a “commutative” and “associative” internal operation ?? x %’ -+ e. If one takes the category 9’ of finitely generated projective modules over A with morphisms the isomorphisms and internal operation the direct sum, then K,(A) = K-‘(pt; 9’) for i 2 0. See [18]. Our purpose here is to show that K,(A) x BGL(rl)+ E n(BGL(@)+) where PA is the ring of .* bounded operators modulo compact operators”. This givesa more specific construction of the R-spectrum for algebraic K-theory. In $4 we show how to define relative K-groups K,Cf) for a ring homomorphism f‘: R ---fS so that there is a long exact sequence . . . + K,(R) + K,(S) -+ Ki(f) + K,_,(R) -+ IT,_.~(S) -+ This amounts to identifying, at least theoretically, the fiber ofthe map BGL(R)+ -+ BGL(S)+. The first six sections of this paper are concerned with algebraic K-theory; the last, which relies only on $1, $2, and (3.1), briefly speculates on Fredholm map germs in homotopy theory. We work with the Bf construction of [15] which in our applications to algebraic K-theory is just the integral completion functor of [3]. Recall the definition of ,uA from [7]: Let /A denote the ring of locallyfinite matrices over A; that is, those infinite matrices (mij) with entries in A such that each row and each column has at most finitely many non-zero entries (1 < i, j < co). Let mA c LA be the ideal ofjinite matrices; that is, those matrices with at most finitely many non-zero entries. Define ,uLA = /A/nzrl. From an algebraic viewpoint the “cone” CA and “suspension” .SA of [l l] could be used in place of /A and ptA in this paper. Recall that CA c /A is the subring gener- ated by the “permuting” matrices; that is, by those matrices of the form P. D where P is an infinite permutation matrix and D is an infinite diagonal matrix with entries coming from a finite subset of A. The suspension is defined as SA = CAhA. * Research supported in part by NSF grant GP 22723. 349