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The topological \(q\)-expansion principle. (English) Zbl 0924.55004
In the theory of modular forms, the \(q\)-expansion principle expresses the relation between a modular form and its \(q\)-expansion. The paper develops a topological version of the \(q\)-expansion principle as a natural transformation from elliptic homology to \(K\)-theory \(E_*\to K_*((q))\) given by taking the \(q\)-expansion. In cohomology, this says elements in elliptic cohomology are described as power series in virtual bundles over \(X\) which rationally behave in a modular fashion.
Using these ideas, the author determines the structure of the cooperations in elliptic cohomology and computes the first two lines in the Adams-Novikov spectral sequence. The author also uses the \(q\)-expansion principle to give orientations to elliptic cohomology which satisfy Riemann-Roch theorems.

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
19L10 Riemann-Roch theorems, Chern characters
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55N15 Topological \(K\)-theory
55T15 Adams spectral sequences
19L41 Connective \(K\)-theory, cobordism
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