Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$

Question

Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$

I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact sequence

$\cdots \to K^{-1}(M) \to H^{odd}(M;\mathbb{C}) \to K^{-1}\mathbb{C}/\mathbb{Z}(M) \to K(M) \xrightarrow{ch \otimes \mathbb{C}} H^{even}(M;\mathbb{C}) \to \cdots$ .

It therefore helps to know the groups $K^{\pm 1}(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z};\mathbb{C})$.

However I am unable to compute these groups. Could anyone please suggest some references where these groups may have been explicitly computed ? Or may be some hints as to how I may proceed ? Thanks so much !

Answer 1

First, you just have $K^n(\mathbb{Z}\times BU)=\text{Map}(\mathbb{Z},K^n(BU))$, so you can work with $K^*(BU)$, which is technically more convenient. In particular, this is the inverse limit of the rings $K^*(BU(n))$. If $V$ is a complex vector bundle of dimension $n$ over a base space $X$, we can consider the polynomial $g_V(u)=\sum_{k=0}^n[\lambda^k(V)]u^{n-k}\in K^0(X)[u]$. We can then define $c^{KU}_k(V)\in K^0(X)$ to be the coefficient of $t^{n-k}$ in $g_V(t-1)$. By considering the universal case, we get classes $c^{KU}_0=1,c^{KU}_1,\dotsc,c^{KU}_n\in K^0(BU(n))$. It is a standard fact that $K^0(BU(n))=\mathbb{Z}[[c^{KU}_1,\dotsc,c^{KU}_n]]$, with $c^{KU}_n$ mapping to zero in $K^0(BU(n-1))$. One way to prove this is to use the general framework of complex orientable cohomology theories. Another is to use the Atiyah-Hirzebruch spectral sequence, which is in (even,even) bidegrees and so collapses. By passing to the limit we get $$ K^0(BU) = \mathbb{Z}[[c^{KU}_1,c^{KU}_2,c^{KU}_3,\dotsc]]. $$ We also have $$ H^*(BU) = \mathbb{Z}[[c^{H}_1,c^{H}_2,c^{H}_3,\dotsc]], $$ with $c_k^H\in H^{2k}(BU)$ being the Chern classes in ordinary cohomology.