Laures, Gerd The topological \(q\)-expansion principle. (English) Zbl 0924.55004 Topology 38, No. 2, 387-425 (1999). 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. Reviewer: R.E.Stong (Charlottesville) Cited in 10 ReviewsCited in 14 Documents 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 Keywords:elliptic homology; \(K\)-theory; cooperations in elliptic cohomology; Adams-Novikov spectral sequence; Riemann-Roch theorems PDFBibTeX XMLCite \textit{G. Laures}, Topology 38, No. 2, 387--425 (1999; Zbl 0924.55004) Full Text: DOI