Universality of mod-\(p\) cohomology
type: Item
instance of: scholarly article
title: Universality of mod-\(p\) cohomology
zbMATH Open document ID: 1209.55003
author: Markus Spitzweck
author: Paul Arne Østvær
published in: Topology and its Applications
publication date: 2011-01-12T00:00:00Z
review text: \textit{D. Quillen} [Bull. Am. Math. Soc. 75, 1293--1298 (1969; Zbl 0199.26705)] showed that the canonical formal group law \(F_{MO}(x_1, x_2)\) over the coefficient ring \(MO^*\) of unoriented cobordism theory \(MO^*(-)\) is a universal mod 2 formal group law in the sense that given any such formal group law \(F(x_1, x_2)\) over an \(\mathbb{F}_2\)-algebra \(R\) there is a unique homomorphism \(MO^*(pt) \to R\) carrying \(F_{MO}(x_1, x_2)\) to \(F(x_1, _2)\), and proved that there is a canonical ring isomorphism \(MO^*(-)\cong H^*(- ; \mathbb{F}_2)\otimes_{\mathbb{F}_2}MO^*\). In this paper the authors attempt to prove the odd primary analogies of these results. The key to the approach here seems to lie in observing the Quillen isomorphism above where \(MO^*\) is isomorphic to the mod 2 Lazard ring \(L_2\). In order to do this, however one needs the notion of mod \(p\) orientations where \(p\) is an odd prime. Let \(E\) be a ring spectrum whose zeroth coefficient \(E_0\) is an \(\mathbb{F}_p\)-algebra. Then by the mod \(p\) orientation on \(E\) one means the pair of an ordinary complex orientation class \(c\in \tilde{E}(\mathbb{C}P^\infty)^2\) and a class \(e\in \tilde{E}^1(K(\mathbb{F}_p, 1))\) whose restriction under \(\iota : S^1 \to K(\mathbb{F}_p, 1)\) is the class in \(\tilde{E}^1(S^1)\) corresponding to the unit in \(E^0\) under the suspension isomorphism, where \(\iota\) represents the canonical generator of \(H^1(S^1; \mathbb{F}_p)\). This definition of orientability is consistent with the notion of mod \(p\) formal group laws, which was introduced by \textit{S. Bullett} [Math. Z. 141, 9--24 (1975; Zbl 0292.57024)]. In fact one finds that such an orientation on \(E\) determines a mod \(p\) formal group law \(F(e_1, c_1, e_2, c_2)\) over \(E^*\). Put \({MO_p}_*(-)=H{\mathbb{F}_p}_*(-)\otimes_{\mathbb{F}_p}L_p\), where \(L_p\) is the mod \(p\) Lazard ring, and let \(MO_p\) denote a ring spectrum representing this homology theory. Then one is led to the desir…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 5835566
zbMATH Keywords: cobordism
zbMATH Keywords: formal group law
MaRDI profile type: Publication
full work available at URL: j.topol.2010.09.004
OpenAlex ID: W2028245540
cites work: Z/p bordism
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cites work: On Thom Spectra, Orientability, and Cobordism
DOI: 10.1016/J.TOPOL.2010.09.004
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description: scientific article; zbMATH DE number 5835566