Contents

cohomology

# Contents

## Definition

Write $\mathbb{S}$ 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

$\mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,.$

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

## Properties

###### Proposition

(Boardman homomorphism in $tmf$ is 6-connected)

$\mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf$

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:

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

## 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.