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).

(…)

Properties

(…)

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

(…)

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, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• 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 1984 §1.2, Ravenel 1986 §6.5.2, SS23 §3.8)_ – or rather, for the most part, of Ravenel's Thom spectra $X(n)$ and $T(m)$ (Ravenel 1984, Sec. 3) which co-represent these:

• Douglas Ravenel, §6.5.2 in: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics 106 2 (1984) 351-414 [doi:10.2307/2374308, jstor:2374308]

• Michael Hopkins, §1.2 in: Stable decompositions of certain loop spaces, PhD thesis, Northwestern (1984) [proquest:303306354, [HopkinsStableDecompositions.pdf:file]]]

• Douglas Ravenel, §6.5 of: Complex cobordism and stable homotopy groups of spheres, Academic Press Orland (1986), AMS Chelsea Publishing 347 (2004) [ISBN:978-0-8218-2967-7]

• Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Theorem 3 of: Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series, 128 2 (1988) 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 10 03 43–55 (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 14 (2019) 611–624 [doi:10.1007/s40062-018-0222-6, arXiv:1708.03042]

• 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]

Review of the bounded-dimensional orientations of Hopkins 1984 §1.2 and further discussion:

• Hisham Sati, Urs Schreiber: Ravenel orientations, §3.8 in: M/F-Theory as Mf-Theory, Reviews in Mathematical Physics 35 10 (2023) [doi:10.1142/S0129055X23500289, arXiv:2103.01877]