nLab
cyclotomic space

Contents

Definition

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

Given an S 1S^1-space XX, denote by ρ r *X C r\rho^{\ast}_r X^{C_r} the S 1S^1–space obtained by pulling back the S 1/C rS^1/C_r action on X C rX^{C_r} via ρ r\rho_r, where X C rX^{C_r} is the homotopy fixed point space of the induced cyclic group action. Then ρ r *(ρ s *X C s) C r=ρ rs *X C rs\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 XX

  2. equipped with a continuous circle group action

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

  3. and equipped with a family of S 1S^1-equivariant weak homotopy equivalences R r:ρ r *X C rXR_r: \rho^{\ast}_r X^{C_r} \to X, for r1r \geq 1, such that R 1=idR_1 = id, and R rs=R rρ r *R s C rR_{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, X\mathcal{L} X, is a cyclotomic space, where S 1S^1 acts by rotation of loops.

Let RX=holimR rρ r *X C rR X = \underset{R_r}{holim} \rho^{\ast}_r X^{C_r}. Then RR is a comonad on the category of cyclotomic spaces.

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

References

Last revised on July 25, 2017 at 07:19:15. See the history of this page for a list of all contributions to it.