Algebraic cycles and infinite loop spaces
Let \(\mathbb{Z} \times BU\) be the space representing complex \(K\)-theory, and let \(K(\mathbb{Z},\text{even}) = \prod^ \infty_{j = 0} K(\mathbb{Z},2j)\) be the product of Eilenberg-MacLane spaces representing the functor \(\prod^ \infty_{j = 0} H^{2j}(-)\). The authors construct an infinite loop space structure on \(K(\mathbb{Z},\text{even})\) such that the total Chern class \(c: \mathbb{Z} \times BU \to K(\mathbb{Z},\text{even})\) is an infinite loop map, and they prove a similar result for the total Stiefel-Whitney class. The constructions use algebraic cycles [see \textit{H. B. Lawson jun.}, Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006)]. The \(H\)-space structure induced on \(K(\mathbb{Z},\text{even})\) by the infinite loop space structure represents the cup product. There is another infinite loop space structure on \(K(\mathbb{Z},\text{even})\) representing the cup product due to \textit{G. Segal} [Quart. J. Math., Oxford II. Ser. 26, 289-293 (1975; Zbl 0321.55017)]. But the infinite loop space structure used here is different from Segal's; indeed, with Segal's structure, \(c\) is not an infinite loop map.
scientific article; zbMATH DE number 549659
scientific article