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I am interested in the $\mathbf{Z}_2$ equivariant K-theory of $S^1$, but I cannot find any good references or methods to calculate it with the action I have in mind. The action on $S^1$ is an inversion, so $(x,y) \to (-x,-y)$. This action does not have any fixed points, so we can rewrite the equivariant K-theory as ordinary K-theory \begin{align} K_{\mathbf{Z}_2}(S^1) = K(S^1/\mathbf{Z}_2) \end{align} Does any one know how I can calculate $K^i(S^1/\mathbf{Z}_2)$ or does any one know a good reference for these kinds of calculations? Any other references regarding these types of calculations are also welcome :)

Let me add that I have found several calculations which use the relation with K-homology, most notably K-homology of certain group C*-algebras, but they consider a mirror symmetry in the $y$-axis.

Physicist
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1 Answers1

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With this action, the quotient is just $S^1$ again (or, if you prefer, $\mathbb{RP}^1$), so you're just asking for the ordinary K-theory of $S^1$. You can calculate this very directly. Let me assume you mean complex K-theory. Then

$$K^0(S^1) \cong \pi_0 [S^1, \mathbb{Z} \times BU] \cong \mathbb{Z}$$

and

$$K^1(S^1) \cong \pi_0 [S^1, U] \cong \mathbb{Z}.$$

Qiaochu Yuan
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