Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$.
A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong U\times \mathbf{A}^n$, and that $\pi\colon U\times \mathbf{A}^n \to U$ is the projection onto the second factor, for every open set $U\subseteq X$ in some Zariski open covering of $X$.


It is proved in Fulton's Intersection Theory that any such $\pi$ induces surjections on the Chow groups via pull-back $\pi^\*\colon A^\*(X) \to A^\*(Y)$, and that this map is an isomorphism if $\pi$ admits a structure of vector bundle.


I would like to know whether $\pi^\*$ is an isomorphism in general or not.


I suspect it to be so: Recall that $M^c(X)$ is the motive with compact support of $X$ in the sense of Voevodsky. He proved that there are


1. Localisation sequences for any closed subscheme $Z\subset X$ and its complement $U$
$$M^c(Z) \to M^c(X) \to M^c(U) \to, $$
2. Pull-back morphisms for any flat morphism $f\colon Y\to X$ of equidimension $n$
$$M^c(X)(n)[2n]\to M^c(Y),$$
3. Isomorphisms $M^c(X)(n)[2n]\cong M^c(X\times \mathbf{A}^n)$, and
4. Comparisons $\mathrm{Hom}\_{\mathbf{DM}^-}(\mathbf{Z}(k)[2k], M^c(X))\cong A^{\dim X-k}(X)$.


It appears to me that one can apply 1., 2., and 3. to obtain an isomorphism $M^c(X)(n)[2n]\to M^c(Y)$ for the locally trivial $\mathbf{A}^n$-fibration $\pi$, and then use 4. to conclude the isomorphisms 
$$A^{k}(X)\cong\mathrm{Hom}\_{\mathbf{DM}^-}(\mathbf{Z}(\dim X-k)[2(\dim X-k)], M^c(X))\cong\mathrm{Hom}\_{\mathbf{DM}^-}(\mathbf{Z}(\dim Y-k)[2(\dim Y-k)], M^c(X)(n)[2n])\cong\mathrm{Hom}\_{\mathbf{DM}^-}(\mathbf{Z}(\dim Y-k)[2(\dim Y-k)], M^c(Y))\cong A^{k}(Y).$$
Is anything wrong with the above arguments?



Chow groups of locally trivial affine fibrations
Not sure if your argument is correct, but the statement is Lemma 2.2 in Totaro's Group cohomology and algebraic cycles.


