Universality of mod-\(p\) cohomology
\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 desired conclusion (Theorem 4.7) that there exists a mod \(p\) orientation on \(MO_p\) inducing the universal mod \(p\) formal group law over the coefficient ring \(L_p\).
scientific article; zbMATH DE number 5835566
scientific article