Topological modular forms

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, topological modular forms (tmf) is the name of a <a href="/facts/Spectrum_(homotopy_theory)/mWho3roy">spectrum</a> that describes a generalized <a href="/facts/Cohomology_theory/J7FJzTNh">cohomology theory</a>. In concrete terms, for any integer n there is a topological space 
 
 
 
 
 tmf
 
 n
 
 
 
 
 {\displaystyle \operatorname {tmf} ^{n}}
 
, and these spaces are equipped with certain maps between them, so that for any <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X, one obtains an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> structure on the set 
 
 
 
 
 tmf
 
 n
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {tmf} ^{n}(X)}
 
 of homotopy classes of continuous maps from X to 
 
 
 
 
 tmf
 
 n
 
 
 
 
 {\displaystyle \operatorname {tmf} ^{n}}
 
. One feature that distinguishes tmf is the fact that its <a href="/facts/Eilenberg%E2%80%93Steenrod_axioms/2c42f4He">coefficient ring</a>, 
 
 
 
 
 tmf
 
 0
 
 
 
 
 {\displaystyle \operatorname {tmf} ^{0}}
 
(point), is almost the same as the <a href="/facts/Graded_ring/AIUjkSaP">graded ring</a> of holomorphic <a href="/facts/Modular_forms/PcnbYR1f">modular forms</a> with integral <a href="/facts/Cusp_form/m0zHUrmc">cusp</a> expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of <a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion</a> information in the coefficient ring.
The spectrum of topological modular forms is constructed as the global sections of a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> of <a href="/facts/Highly_structured_ring_spectrum/GC2bU7L7">E-infinity</a> <a href="/facts/Ring_spectrum/fB4IaL5T">ring spectra</a> on the <a href="/facts/Moduli_stack/9sqrdpwR">moduli stack</a> of (generalized) <a href="/facts/Elliptic_curves/60jvaHSA">elliptic curves</a>. This theory has relations to the theory of <a href="/facts/Modular_forms/PcnbYR1f">modular forms</a> in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the <a href="/facts/Homotopy_groups_of_spheres/YMENlGgd">homotopy groups of spheres</a>, and conjectural <a href="/facts/Atiyah%E2%80%93Singer_index_theorem/qwcSxamI">index theories</a> on <a href="/facts/Loop_space/S5xkdCkx">loop spaces</a> of <a href="/facts/Manifold/rXPERJtu">manifolds</a>. tmf was first constructed by <a href="/facts/Michael_J._Hopkins/nM5Jqqnx">Michael Hopkins</a> and <a href="/facts/Haynes_Miller/51VqT8nh">Haynes Miller</a>; many of the computations can be found in preprints and articles by Paul Goerss, Hopkins, <a href="/facts/Mark_Mahowald/0aJBEaeB">Mark Mahowald</a>, Miller, <a href="/facts/Charles_Rezk/ghh0xzNp">Charles Rezk</a>, and Tilman Bauer.

Topological modular forms open-in-new

Topological modular forms