nLab
Ravenel's spectrum

Contents

under construction

Contents

Idea

Due to Bott periodicity, the coprojection SU(n)USU(n) \hookrightarrow \mathrm{U} of any special unitary group SU(n)SU(n) into the stable unitary group induces a map ΩSU(n)BUBO\Omega SU(n) \longrightarrow B \mathrm{U} \to B\mathrm{O} from the based loop space of SU(n)SU(n) to the classifying space of U\mathrm{U} and hence of O\mathrm{O}. This may be regarded as defining a notion of Ω 2SU(n)\Omega^2 SU(n)-tangential structure.

The corresponding Thom spectra M(Ω 2SU(n))M \big(\Omega^2 SU(n)\big) were denoted “X(n)X(n)” in Ravenel 84, Section 3, used there for analysis of the Adams spectral sequence (see also Ravenel 86, Section 6.5) and influential on Ravenel's conjectures (notably the nilpotence theorem); and thus have come to be known as Ravenel’s spectra.

These spectra turn out to be finite-rank analogs of MU in complex oriented cohomology theory as one passes from full complex orientation to complex orientation up to rank (“degree”) nn (Hopkins 84, Section 1.2). For instance, just as MU is p-locally a wedge sum of suspensions of BP, so Ravenel’s spectra are pp-locally wedge sums of suspensions of spectra that Ravenel denoted T(k)T(k).

Definition

For nn \in \mathbb{N}, n1n \geq 1, consider the composite morphism of homotopy types

BΩ 2SU(n)ΩSU(n)Ω 2BSU(n)Ω 2BiΩ 2BUβBUBO, B \Omega^2 SU(n) \simeq \Omega SU(n) \simeq \Omega^2 B SU(n) \overset{ \Omega^2 B i }{ \longrightarrow } \Omega^2 B U \underoverset {\simeq} {\beta} {\longrightarrow} B U \longrightarrow B O \,,

where

  1. On the left we have looping and delooping equivalences (this Prop.), using that the based loop space ΩSU(n)\Omega SU(n) is connected;

  2. SU(n)iSUSUk(k)SU(n) \overset{i}{\longrightarrow} SU \coloneqq \underset{\underset{k}{\longrightarrow}} SU(k) is the coprojection of SU(n) into the special stable unitary group:

  3. β\beta is the Bott periodicity equivalence.

Regarded as a universal tangential structure, this induces the corresponding Thom spectrum M(Ω 2SU(n))M\big(\Omega^2 SU(n)\big) (introduced as “X(n)X(n)” in Ravenel 84, Section 3).

(…)

Properties

(…)

This carries the finite-rank analog of the universal complex orientation of MU: Hopkins 84, Prop. 1.2.1

(…)

Examples

For n=1n=1 we have that SU(1)=1SU(1) = 1 is the trivial group, so that Ravenel’s spectrum at this stage is the sphere spectrum

MΩ 2SU(1)M1𝕊 M \Omega^2 SU(1) \simeq M 1 \simeq \mathbb{S}

On the other hand, the colimit of Ravenel’s spectra as nn \to \infty is MU, essentially by construction:

limnMΩ 2SU(n)MΩ 2SUMU. \underset{ \underset{n}{\longrightarrow} }{lim} M \Omega^2 SU(n) \simeq M \Omega^2 SU \simeq M U \,.

Hence the tower of Ravenel’s spectra interpolates between the sphere spectrum and MU

𝕊MΩ 2SU(2)MΩ 2SU(3)MU. \mathbb{S} \longrightarrow M \Omega^2 SU(2) \longrightarrow M \Omega^2 SU(3) \longrightarrow \cdots \longrightarrow M U \,.

Accordingly, the corresponding tower of Whitehead generalized cohomology theories interpolated between stable Cohomotopy and complex cobordism cohomology.

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theoryMB\,M B (B-bordism):

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

References

Finite-dimensional complex orientation and Ravenel’s spectra

Discussion of complex orientation (in Whitehead generalized cohomology) on (only) those complex vector bundles which are pulled back from base spaces of bounded cell-dimension (Hopkins 84, 1.2, Ravenel 86, 6.5.2) – or rather, for the most part, of Ravenel's Thom spectra X(n)X(n) and T(m)T(m) (Ravenel 84, Sec. 3) which co-represent these:

Last revised on January 28, 2021 at 16:59:18. See the history of this page for a list of all contributions to it.