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The loop space of the Q-construction. (English) Zbl 0628.55011
Ill. J. Math. 31, 574-597 (1987); erratum ibid. 47, No. 3, 745-748 (2003).
Quillen’s Q-construction associates to an exact category M a new category QM, and the $$(i+1)st$$ homotopy group of $$| QM|$$ is the ith algebraic K-group, $$K_ iM$$. In this paper the authors provide us with a simplicial set GM which is homotopy-equivalent to the loop space of $$| QM|$$; thus $$K_ iM=\pi_ i| GM|$$. The paper contains three applications of this construction: (1) A simplified proof that $$S^{-1}S$$ is homotopy equivalent to the loops on $$| QM|$$; (2) an explicit algebraic representation for elements of $$K_ 1M$$; and (3) a new definition of products in K-theory.
Erratum: We correct an error in our paper, and provide a new shorter proof of Theorem B’.
Reviewer: M.Stein

##### MSC:
 55U35 Abstract and axiomatic homotopy theory in algebraic topology 55U10 Simplicial sets and complexes in algebraic topology 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 55P35 Loop spaces