nLab Boardman homomorphism in tmf

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)

The Boardman homomorphism in tmf

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

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:

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

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

Related concepts

• stable cohomotopy

References

• Michael Hopkins, section 4 of Algebraic topology and modular forms, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (arXiv:math/0212397)

• Christopher Douglas, John Francis, André Henriques, Michael Hill, Chapter 13 of: Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)