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Umbral calculus on multi-valued formal groups and Adams projectors in K- theory. (Russian) Zbl 0673.55017

In this paper the author studies the umbral analysis on the m-valued cyclic formal groups [V. M. Bukhshtaber and S. P. Novikov, Mat. Sb., Nov. Ser. 84(126), 81-118 (1971; Zbl 0222.55008); V. M. Bukhshtaber and A. N. Kholodov, Izv. Akad. Nauk SSSR, Ser. Mat. 46, 3-27 (1982; Zbl 0501.55003); S. Roman: The umbral calculus (1984; Zbl 0536.33001)]. As an application the description of the non- stable additive cohomology operations (or H-maps of classifying spaces) in K-theory, via Adams projectors [J. F. Adams, Lect. Notes Math. 99, 1-138 (1969; Zbl 0193.517)] is obtained. Namely, the author considers the projector \[ E_{m,\alpha}=\sum_{n\geq 0}\log (1+u)^{mn+\alpha}/(mn+\alpha)! \] in complex K-theory, and if for this classifying spaces functor is \(BUE_{m,\alpha}\), \(0\leq \alpha \leq m\), then every map f: \(BUE_{m,\alpha}\to BUE_{m,\alpha}\) induces some homomorphisms of homotopy groups \[ f_ i: \pi_ i(BUE_{m,\alpha})\to \pi_ i(BUE_{m,\alpha}),\quad i=0,1,.... \] The author establishes some algebraic conditions on the homomorphisms \(f_ i\) for which there exists the H-map f, with the H-structure given by Whitney sum. In the cases \(m=\alpha =1\) and \(m=\alpha =2\) some results of F. Clarke [Math. Proc. Camb. Philos. Soc. 89, 491-500 (1981; Zbl 0491.55010)] and G. R. Mamedov [Usp. Mat. Nauk 41, No.3(249), 191-192 (1986; Zbl 0627.55012)] are reproved. Also, using the methods of the umbral analysis, the author obtains some combinatorial identities between Steenrod operations and the central factorial numbers.
Reviewer: Ioan Pop (Iaşi)

MSC:

55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55S05 Primary cohomology operations in algebraic topology
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