Symbolic calculus: a 19th century approach to MU and BP.

*(English)*Zbl 0651.55004
Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 195-238 (1987).

[For the entire collection see Zbl 0628.00011.]

The aim of this partly expository paper is to show that umbral calculus over graded rings, augmented by operator calculus, provides an apt framework for calculations in complex cobordism theory and other complex- oriented cohomology theories. The foundations for this study have been laid in the author’s article in Adv. Math. 61, 49-100 (1986; Zbl 0631.05002). The author groups under the heading of symbolic calculus both the umbral calculus (founded by John Blissard, and given a modern presentation by S. Roman and G.-C. Rota) and the operator calculus of Oliver Heaviside; the introduction includes a bit of history. The relevance of the symbolic calculus to bordism theory is its close connection with complex-oriented Thom isomorphisms. A wide selection of combinatorial and number-theoretic aspects of this approach are explored.

The aim of this partly expository paper is to show that umbral calculus over graded rings, augmented by operator calculus, provides an apt framework for calculations in complex cobordism theory and other complex- oriented cohomology theories. The foundations for this study have been laid in the author’s article in Adv. Math. 61, 49-100 (1986; Zbl 0631.05002). The author groups under the heading of symbolic calculus both the umbral calculus (founded by John Blissard, and given a modern presentation by S. Roman and G.-C. Rota) and the operator calculus of Oliver Heaviside; the introduction includes a bit of history. The relevance of the symbolic calculus to bordism theory is its close connection with complex-oriented Thom isomorphisms. A wide selection of combinatorial and number-theoretic aspects of this approach are explored.

Reviewer: P.Landweber

##### MSC:

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

57R77 | Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism) |