nLab
Boardman homomorphism in tmf

Contents

Context

Elliptic cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Write 𝕊\mathbb{S} for the sphere spectrum and tmf for the connective spectrum of topological modular forms.

Since tmf is an E-∞ring spectrum, there is an essentially unique homomorphism of E-∞ring spectra

𝕊e tmftmf. \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,.

Regarded as a morphism of generalized homology-theories, this is called the Hurewicz homomorphism, or rather the Boardman homomorphism for tmftmf

Properties

Proposition

(Boardman homomorphism in tmftmf is 6-connected)

The Boardman homomorphism in tmf

𝕊e tmftmf \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf

induces an isomorphism on stable homotopy groups (hence from the stable homotopy groups of spheres to the stable homotopy groups of tmf), up to degree 6:

π 6(𝕊)π 6(e tmf)π 6(tmf). \pi_{\bullet \leq 6}(\mathbb{S}) \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow} \pi_{\bullet\leq 6}(tmf) \,.

(Hopkins 02, Prop. 4.6, DFHH 14, Ch. 13)

References

Last revised on September 7, 2020 at 15:57:55. See the history of this page for a list of all contributions to it.