nLab inversion involution

Idea

For $A$ an abelian group there is a group automorphism

$$- \;\colon \; A \stackrel{\simeq}{\longrightarrow} A$$

given by sending any element $a$ to its inverse $-a$.

This is an involution.

Examples

• The inversion involutuion on the abelian groups underlying elliptic curves (over the complex numbers) extends to an inversion action on the universal elliptic curve $\mathcal{E} \to \mathcal{M}_{ell}(\mathbb{C})$ over the compactified moduli stack of elliptic curves over the complex numbers (Hain 08,lemma 5.6). It extends along the inclusion $\mathcal{M}_{ell}\to \mathcal{M}_{\overline{ell}}$ to the compactified moduli stack to give the inversion involution on the nodal cubic curve (Hain 08, prop. 5.7).

• The inversion involution on the multiplicative group induces, via the Goerss-Hopkins-Miller theorem the real oriented cohomology theory structure on complex K-theory KU (namely KR-theory), see at moduli stack of tori for details.

References

• Richard Hain, Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)