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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.

##### MSC:
 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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