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Definition 2.1. A complex orientable spectrum is a multiplicative spectrum \(E\) such that \(\tilde E^2(\mathbb {C}P^\infty )\to \tilde E^2(\mathbb {C}P^1)\cong E^0\) is surjective.
Since \(E\) is multiplicative, \(E^0\) has a generator \(1_E\), which correspond to a generator \(t\in E^2(\mathbb {C}P^1)\). A complex orientation is a lift of \(t\) in \(E^2(\mathbb {C}P^\infty )\). A complex oriented cohomology is a tuple \((E,t_E)\) where \(E\) is a complex orientable cohomology with a complex orientation
We have a map \(E^*[t]/(t^{n+1})\to E^*(\mathbb {C}P^n)\), from the fact that there is a cover of \(n+1\) contractible open set of \(\mathbb {C}P^n\). As it turns out
Lemma 2.2. [Hop99] The map is an isomorphism.
Proof. For \(E=H\mathbb {Z}\), this is a classic result. We generalize it using the Atiyah-Hirzebruch SS, which is multiplicative, with one generator \(t\) that correspond to the element \(t\in E^2(\mathbb {C}P^n)\). We have that □
We can take one step further, from Milnor’s sequence \begin{align*} 0\to {\lim }^1E^{*-1}(\mathbb {C}P^n)\to E^*(\mathbb {C}P^\infty )\to {\lim }E^*(\mathbb {C}P^n)\to 0 \end{align*}
and the fact \(E^{*}(\mathbb {C}P^{n+1})\to E^{*}(\mathbb {C}P^n)\) is surjective, we have \begin{align*} E^*(\mathbb {C}P^\infty )\cong E^*[[t]] \end{align*}
The name complex orientation has a geometric meaning. We know complex line bundle is classified by the space \(\mathbb {C}P^\infty \simeq BU(1)\). Given a complex oriented cohomology, \(t\) is then a characteristic class of universal complex line bundle.
The tensor product of line bundle \(\mathbb {C}P^∞\times \mathbb {C}P^∞\to \mathbb {C}P^∞\) then induces a map \begin{align*} \psi _E:E^*(\mathbb {C}P^∞)\cong E^*[[t]]\to E^*(\mathbb {C}P^∞\times \mathbb {C}P^∞)\cong E^*[[x,y]] \end{align*}
The generator \(t\) then correspond to a formal power series in \(x,y\), \(F_E(x,y):=\psi _E(t)\).
Definition 2.3. A formal group law over a ring \(R\) is a formal power series \(F(x,y)\) such that
For \(R\) graded, we require \(F(x,y)\) to be homogeneous of degree \(2\).
We will write \(x+_Fy:=F(x,y)\).
Lemma 2.4. [Rud98] Let \(E\) be a complex oriented cohomology, \(F_E(x,y)\) is a graded formal group law.
Proof. This follows from the properties of tensor product of line bundle being unital, associative and commutative. □
Example 2.5. Let \(E=HR\) for some ring \(R\), \(f_E(x,y)=x+y\), as it is the only possible graded formal
group law.
Let \(E=K\) be the complex \(K\)-theory, we choose \(t\in K^2(\mathbb {C}P^\infty )\) to be the one that correspond to \([\mathcal {O}(1)]-1\), where \(\mathcal {O}(1)\)
is the universal line bundle and \(1\) is the trivial rank \(1\) bundle over \(\mathbb {C}P^\infty \), under the bott
map \(\beta : K^*\to K^{*+2}\). We have that given line bundle \begin{align*} [\mathcal {O}(1)]\otimes [\mathcal {O}(1)]-1= ([\mathcal {O}(1)]-1)\otimes ([\mathcal {O}(1)]-1)+ ([\mathcal {O}(1)]-1)\otimes 1 + 1\otimes ([\mathcal {O}(1)]-1) \end{align*}
So \(F_K(x,y)=x+y+\beta ^{-1}xy \)