nLab finite-dimensional complex orientation and Ravenel's spectra -- 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]