nLab cyclotomic space

Definition

Let $\rho_r: S^1 \to S^1/C_r$ be the homeomorphism $z \mapsto \sqrt[r] z$, $C_r \coloneqq \mathbb{Z}/r\mathbb{Z}$.

Given an $S^1$-space $X$, denote by $\rho^{\ast}_r X^{C_r}$ the $S^1$–space obtained by pulling back the $S^1/C_r$ action on $X^{C_r}$ via $\rho_r$, where $X^{C_r}$ is the homotopy fixed point space of the induced cyclic group action. Then $\rho^{\ast}_r(\rho^{\ast}_s X^{C_s})^{C_r} =\rho^{\ast}_{r s} X^{C_{r s}}$.

Definition

A cyclotomic space is

1. a topological space $X$

2. equipped with a continuous circle group action

   (hence a topological G-space for $G = S^1$)

3. and equipped with a family of $S^1$-equivariant weak homotopy equivalences $R_r: \rho^{\ast}_r X^{C_r} \to X$, for $r \geq 1$, such that $R_1 = id$, and $R_{r s} = R_r \circ \rho^{\ast}_r R_s^{C_r}$.

(Schl 09, def. 4.3, AGHL 12, def. 3.6)

Properties

A free loop space, $\mathcal{L} X$, is a cyclotomic space, where $S^1$ acts by rotation of loops.

Let $R X = \underset{R_r}{holim} \rho^{\ast}_r X^{C_r}$. Then $R$ is a comonad on the category of cyclotomic spaces.

The $S^1$-equivariant suspension spectrum of a cyclotomic space is a cyclotomic spectrum.

Related concepts

• cyclotomic spectrum

• cyclic space

References

• Christian Schlichtkrull, The cyclotomic trace for symmetric ring spectra, Geometry & Topology Monographs 16 (2009), 545–592, (pdf), (arXiv:0903.3495)

• Vigleik Angeltveit, Teena Gerhardt, Michael Hill, Ayelet Lindenstrauss, On the algebraic K-theory of truncated polynomial algebras in several variables (arXiv:1206.0247)