Topological modular forms

In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space ${\displaystyle \operatorname {tmf} ^{n}}$, and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set ${\displaystyle \operatorname {tmf} ^{n}(X)}$ of homotopy classes of continuous maps from X to ${\displaystyle \operatorname {tmf} ^{n}}$. One feature that distinguishes tmf is the fact that its coefficient ring, ${\displaystyle \operatorname {tmf} ^{0}}$(point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.

The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop spaces of manifolds. tmf was first constructed by Michael Hopkins and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Hopkins, Mark Mahowald, Miller, Charles Rezk, and Tilman Bauer.