The topological \(q\)-expansion principle.

*(English)*Zbl 0924.55004In 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.

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)

##### 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 |