nLab Ravenel's spectrum

under construction

Idea

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

The corresponding Thom spectra $M \big(\Omega^2 SU(n)\big)$ were denoted “$X(n)$” 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”) $n$ (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 $p$-locally wedge sums of suspensions of spectra that Ravenel denoted $T(k)$.

Definition

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

$$B \Omega^2 SU(n) \simeq \Omega SU(n) \simeq \Omega^2 B SU(n) \overset{ \Omega^2 B i }{ \longrightarrow } \Omega^2 B U \underoverset {\simeq} {\beta} {\longrightarrow} B U \longrightarrow B O \,,$$

where

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

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

3. $\beta$ is the Bott periodicity equivalence.

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

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Properties

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

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Examples

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

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

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

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

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

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

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

Related concepts

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

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

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 $X(n)$ and $T(m)$ (Ravenel 84, Sec. 3) which co-represent these:

• Douglas Ravenel, section 3 of: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)

• Michael Hopkins, Stable decompositions of certain loop spaces, Northwestern 1984 (pdf)

• Douglas Ravenel, section 6.5 of: Complex cobordism and stable homotopy groups of spheres, 1986

• Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Theorem 3 of: Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (jstor:1971440)

• Doug Ravenel, The first Adams-Novikov differential for the spectrum $T(m)$, 2000 (pdf, pdf)

• Ippei Ichigi, Katsumi Shimomura, The Modulo Two Homotopy Groups of the $L_2$-Localization of the Ravenel Spectrum, CUBO A Mathematical Journal, Vol. 10, No 03, (43–55). October 2008 (cubo:1498)

• Gabe Angelini-Knoll, J. D. Quigley: The Segal Conjecture for topological Hochschild homology of the Ravenel spectra, Journal of Topology 4 3 (2011) 591-622 [doi:10.1112/jtopol/jtr015, arXiv:1705.03343]

• Jonathan Beardsley, A Theorem on Multiplicative Cell Attachments with an Application to Ravenel’s $X(n)$ Spectra, Journal of Homotopy and Related Structures volume 14, pages 611–624 (2019) (arXiv:1708.03042, doi:10.1007/s40062-018-0222-6)

• Jonathan Beardsley, Topological Hochschild homology of $X(n)$ (arXiv:1708.09486)

• Xiangjun Wang, Zihong Yuan, The homotopy groups of $L_2 T(m)/\big(p^{[\tfrac{m}{2}]+2}, v_1 \big)$ for $m \gt 1$, New York J. Math.24 (2018) 1123–1146 (pdf)