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Characters and elliptic cohomology. (English) Zbl 0729.55003

Advances in homotopy theory, Proc. Conf. in Honour of I.M. James, Cortona/Italy 1988, Lond. Math. Soc. Lect. Note Ser. 139, 87-104 (1989).
[For the entire collection see Zbl 0682.00015.]
A complex oriented cohomology theory E gives rise to a ring homomorphism, a genus, MU(pt)\(\to E(pt)\), and, inversely, by the so-called exact functor theorem, suitable ring homomorphisms define cohomology theories. Complex cobordism, MU, when localized at a prime p, splits as a sum of shifted copies of Brown-Peterson cohomology, BP, and \(BP(pt)={\mathbb{Z}}_{(p)}[\nu_ 1,\nu_ 2,...].\) Killing \(\nu_{n+1},\nu_{n+2},..\). and localizing at \(\nu_ n\), this gives a ring homomorphism \[ MU(pt)\to {\mathbb{Z}}_{(p)}[\nu_ 1,...,\nu_{n- 1}][\nu_ n,\nu_ n^{-1}]. \] The corresponding cohomology theories are denoted by \(E_ n\), and by results of the authors and others these theories are crucial in detecting the behaviour of stable homotopy. Th purpose of this note is to explain and discuss the main result of a preprint by the author, N. J. Kuhn and D. C. Ravenel [Generalized group characters and complex oriented cohomology theories]. This main result concerns the classifying spaces cohomologies \(E^*_ n(BG)\) and should be seen in terms of higher genus analogues of Atiyah’s celebrated result linking the K-theory of BG to the representation ring RG of a group G.

MSC:

55N22 Bordism and cobordism theories and formal group laws in algebraic topology

Citations:

Zbl 0682.00015