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Stirling and Bernoulli numbers for complex oriented homology theory. (English) Zbl 0698.55002
Algebraic topology, Proc. Int. Conf., Arcata/Calif. 1986, Lect. Notes Math. 1370, 362-373 (1989).
The classical Stirling and Bernoulli numbers are defined as coefficients of certain power series with rational coefficients. More generally, they can be defined in terms of one-dimensional formal group laws, the classical case being that of the multiplicative formal group. In this setting the Stirling and Bernoulli numbers become elements in the ring over which the formal group law is defined. The universal example is known to topologists as the complex cobordism ring \(MU_*\). The author gives explicit formulae for these elements there and for their images in the coefficient rings for Morava \(K\)-theory. The integrality of the classical Stirling numbers generalizes to the fact that these universal Stirling numbers lie in the subring \(\mathbb{CP}_*\subset MU_*\) generated by the classes of the complex projective spaces. He also gives a formula for the reduction of the universal Bernoulli numbers modulo \(\mathbb{CP}_*\) which generalizes the classical congruence of von Staudt.
[For the entire collection see Zbl 0661.00012.]
Reviewer: D. C. Ravenel

MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N15 Topological \(K\)-theory
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers