The cohomslogy group of the classifying space

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For a smooth complex algebraic variety X, the group CH^I(X) of codimension I algebraic cycles modulo rational equivalence assemble to the Chow ring CH^*(X) = Σ_iCH^I(X). Totaro constructed a map c^^~l : CH^*(X) → BP^*(X) 【cross product】_<BP>・ Z_<(p)> such that the compositioncl : CH^I(X)_<(p)> →^^<c^^~l> BP^*(X) 【cross product】_<BP>・ Z_<(p)> → H^*(X)_<(p)>coincides with the cycle map. One of the main results of Totaro's paper is that there is a space X = BG for which the kernel of cl contains p-torsion elements. Here the Chow ring of a classifying space BG is defined [Tol,To2] as the limit Lim_<m→∞>CH^*((c^m - s)/G) of a system of algebraic varieties where G acts on C^m - S freely and codim(S) → ∞ as m → ∞. The group Totaro uses is G = Z/2 × 2^<1+4>_+, where 2^<1+4>_+ is the extraspecial 2-group of order 32, which is isomorphic to the central product of two copies of the dihedral group D_8 of order 8. Similar facts hold for the extraspecial 2-groups G = 2^<1+2n>_+ of order 2<1+2n>.Totaro computed the Chow rings of abelian groups and symmetric groups in and he and Pandharipande determined the Chow rings of O(n), SO(2n+1) aud SO(4). For these cases the cycle maps c^~l are isomorphisms, namely, CH^*(BG)_<(2)> =^~ BP^*(BG) 【cross product】_BP・Z_<(2)>. Field also determined the Chow ring of BSO(2n), but its BP-theory is unknown for n > 3. Vezzosi has shown that d^~ is epimorphic for X = BPGL_3c, p = 3. Totaro also gives many interesting theorems to study CH*(BG) in