2002 Fiscal Year Final Research Report Summary
The cohomslogy group of the classifying space
| Project/Area Number |
13640006
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | IBARAKI UNIVERSITY |
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Principal Investigator |
YAGITA N IBARAKI Univ., College of Education, professor, 教育学部, 教授 (20130768)
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| Co-Investigator(Kenkyū-buntansha) |
KANEDA M Osaka City Univs., Fawlty of Nat., Sceince professor, 理学部, 教授 (60204575)
KUDOU K IBARAKI Univ., College of Education, Lecturer, 教育学部, 講師 (00114017)
OKAYASU T IBARAKI Univ., College of Education, associate professor, 教育学部, 助教授 (00191958)
TEZUKA M Ryukyus Univer., Fawlty of Nat., Sceince professor, 理学部, 教授 (20197784)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Cohomology group / Classifying space / BP-theory / Chow ring |
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| Research Abstract |
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 composition
cl : 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
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