# Contents

## Idea

Inside the compactified moduli stack of elliptic curves, at the cusp point corresponding to the nodal cubic curve sits the moduli stack of one dimensional tori $\mathcal{M}_{\mathbb{G}_m}$ (Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2

$\mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2$

(Lawson-Naumann 12, prop. A.4). Here the $\mathbb{Z}_2$-action is the inversion involution on abelian groups.

Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf $\mathcal{O}^{top}$ (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a $\mathbb{Z}_2$-∞-action. This is KU with its involution induced by complex conjugation, hence essentially is $KR$.

Accordingly, the global sections of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-homotopy fixed points of this action, hence is $KO$. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).

As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.

## References

Discussion of $KU$ with its $\mathbb{Z}_2$-action as the E-∞ ring-valued structure sheaf of the moduli stack of tori is due to

which is reviewed and amplified further in

• Akhil Mathew, section 3 of The homology of $tmf$ (arXiv:1305.6100)

• Akhil Mathew, section 2 of The homotopy groups of $TMF$, talk notes (pdf)

Last revised on April 9, 2014 at 05:05:10. See the history of this page for a list of all contributions to it.