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Kan extension and stable homology of Eilenberg-MacLane spaces. (English) Zbl 0858.55006
Forty years ago S. MacLane defined the homology theory of rings [Centre Belge Rech. math., Colloque de Topologie algébrique, Louvain les 11, 12 et 13 juin 1956, 55-80 (1957; Zbl 0084.26703)]. Recently it was proved that this homology theory is isomorphic to the Bökstedt’s topological Hochschild homology [the author and F. Waldhausen, J. Pure Appl. Algebra 82, 81-98 (1992; Zbl 0767.55010)] and to the stable \(K\)-theory of Waldhausen [B. I. Dundas and R. McCarthy, Ann. Math., II. Ser. 140, 685-701 (1994; Zbl 0833.55007)]. The original definition of MacLane was based on the cubical construction, which assigns a chain complex \(Q_* (A)\) to each abelian group \(A\) (see [S. Eilenberg and S. MacLane, Trans. Am. Math. Soc. 71, 294-330 (1951; Zbl 0043.25403); M. Dzhibladze and the author, J. Algebra 137, 253-296 (1991; Zbl 0724.18005); S. Mac Lane, loc. cit.]). This complex has the following property:
Theorem: The homology of \(Q_* (A)\) is isomorphic to the stable homology of the Eilenberg MacLane spaces: \[ H_n \bigl( Q_* (A)\bigr) \cong H_{n+k} \bigl(K(A,k) \bigr), \quad n\leq k-1. \] The original proof of this theorem requires two papers of S. Eilenberg and S. MacLane, namely the one cited above and [Can. J. Math. 7, 43-53 (1955; Zbl 0064.02701)], and based on the theory of the generic cycles. It was mentioned in the introduction of the ‘Collected Works’ of S. Eilenberg and S. MacLane (1986; Zbl 0613.01018) that this theory is somewhat mysterious. Here we give a new, simple, proof of this fact.

MSC:
55P20 Eilenberg-Mac Lane spaces
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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