TAILIEUCHUNG - A brief introduction to Quillen conjectures

This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors. | Science Technology Development Journal 22 2 235- 238 Review A brief introduction to Quillen conjecture Bui Anh Tuan Nguyen Anh Thi ABSTRACT Introduction In 1971 Quillen stated a conjecture that on cohomology of arithmetic groups a certain module structure over the Chern classes of the containing general linear group is free. Over time many efforts has been dedicated into this conjecture. Some verified its correctness some disproved it. So the original Quillens conjecture is not correct. However this conjecture still has great impacts on the field cohomology of group especially on cohomology of arithmetic groups. This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors. Method In this work we investigate the key reasons that makes Quillen conjecture fails. We review two of the reasons 1 the injectivity of the restriction map 2 the non-free of the image of the Quillen homomorphism. Those two reasons play important roles in the study of the correctness of Quillen conjecture. Results In section 4 we present the cohomology of ring H SL2 Z V 2 F2 which is isomorphic to the free moduleF2 e4 x2 x3 y3 z3 S3 x4 s4 s5 S6 over F2 e4 . This confirms the Quillen conjecture. Conclusion The scope of the conjecture is not correct in Quillens original statement. It has been disproved in many examples and also been proved in many cases. Then determining the conjectures correct range of validity still in need. The result in section 4 is one of the confirmation of the validity of the conjecture. Keywords Quillen conjecture Cohomology of group Arithmetic groups S-arithmetic groups Cohomology ring Faculty of Mathematics and Computer Science University of Science VNU-HCMC Correspondence Bui Anh Tuan Faculty of Mathematics and Computer Science University of Science VNU-HCMC Email batuan@ History Received 2018-12-05 Accepted 2019-03-31 Published 2019-06-14 DOI https

• Toán học
• Quillen conjecture
• Cohomology of group
• Arithmetic groups
• S arithmetic groups
• Cohomology ring
• A brief introduction to Quillen conjecture
• The original Quillens conjecture
• The Quillen homomorphism
• The injectivity of the restriction map
• The validity of the conjecture
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học
• Ring cohomology of p groups
• Periodic groups
• Invariant Theory
• Lyndon Hochschild Serre spectral sequence
• Integral homology