The cohomslogy group of the classifying space

For a smooth complex algebraic variety X，the group CH^I(X) of codimension I algebraic cycles modulo rational equivalence assemble to the Chowring 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译文:< (p) >→h ^ * (x) _ < (p) > coincides with thecycle map. One of the main results of Totaro's paper is that there is a space X = BG for which thekernel of cl contains p-torsion elements. Here the Chow ring of a classifying space BG是定义[Tol,To2] as the limit Lim_<m→∞>CH^*((c^m - s)/G) of a system of algebraic varieties where G acts onC^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，其中isomorphic to the central product of two copies of the dihedral groupd_8 of order 8.Similar facts hold for the extraspecial 2-groups G = 2^<1+2n>_+ of order 2<1+2n>.Totaro computed theabelian groups and symmetric groups in and he and Pandharipande determined the Chow rings of abelian groups and symmetric groups in and he and Pandharipande determined the Chowrings 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 ofbso (2 n),but its wp -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

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