under construction
cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
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 $T(k)$.
For $n \in \mathbb{N}$, $n \geq 1$, consider the composite morphism of homotopy types
where
On the left we have looping and delooping equivalences (this Prop.), using that the based loop space $\Omega SU(n)$ is connected;
$SU(n) \overset{i}{\longrightarrow} SU \coloneqq \underset{\underset{k}{\longrightarrow}} SU(k)$ is the coprojection of SU(n) into the special stable unitary group:
$\beta$ 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).
(…)
(…)
This carries the finite-rank analog of the universal complex orientation of MU: Hopkins 84, Prop. 1.2.1
(…)
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
On the other hand, the colimit of Ravenel’s spectra as $n \to \infty$ is MU, essentially by construction:
Hence the tower of Ravenel’s spectra interpolates between the sphere spectrum and MU
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 theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
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 (arXiv:1705.03343, doi:10.1112/jtopol/jtr015)
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)
Last revised on January 28, 2021 at 21:59:18. See the history of this page for a list of all contributions to it.