  Let k be a field of characteristic 0. Let G be a reductive group over the
ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that G
has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of
fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map
H^1_{et}(R,G)\to H^1_{\et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop
reductive, i.e. contains a maximal R-torus. In particular, we settle in
positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that
H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a reductive
group over R of isotropic rank >=2, then the natural map of non-stable
K_1-functors K_1^G(R)\to K_1^G( k((x_1))...((x_n)) ) is injective, and an
isomorphism if G is moreover semisimple.
