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Could anyone point me to a reference where the formal group law of (topological or motivic) K-theory is computed in as much detail as possible?

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2 Answers 2

Well, I don't think there can be much detail here: the first Chern class of a line bundle in topological K-theory is defined by $c_1(L)=\pm1\pm[L]$; hence $c_1(L\otimes L')=c_1(L)+c_1(L')\pm c_1(L)c_1(L')$ — so the FGL is just $F(u,v)=u+v\pm uv$ (sign convention varies).

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Let $Vect^1(X)$ be isomorphism classes of complex line bundles on $X$. I assume you're familiar with the definition of topological $K^0(X)$.

Let's examine the map $c_1: Vect^1(X) \to K^2(X)$, (recalling that, due to Bott periodicity, $K^0(X) \simeq K^2(X)$). The group operators are the tensor product of line bundles (in $Vect^1(X)$) and the tensor product of virtual line bundles (in $K^0(X)$).

Recall that the dimensions multiply when you do the tensor product, so $L_1 \otimes L_2$ is still a line bundle.

Let's say that $c_1(L) = 1-L$ (there are other choices of Thom class), then we'd expect $c_1(L_1 \otimes L_2) = 1 - L_1 \otimes L_2$.

So, how do we express $c_1(L_1 \otimes L_2)$ in terms of a formal group law $F(x,y)$ where $x = c_1(L_1)$, and $y = c_1(L_2))$?

\begin{align*} F(x, y) & = x + y - xy \\ F(c_1(L_1), c_1(L_2)) & = c_1(L_1) + c_1(L_2) - c_1(L_1)c_1(L_2) \\ & = (1-L_1) + (1-L_2) - (1-L_1)(1-L_2) \\ & = 2 - L_1 - L_2 - 1 + L_1 + L_2 - L_1 \otimes L_2 \\ & = 1 - L_1 \otimes L_2 \end{align*}

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