A brief introduction to Quillen conjecture

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[ √ −2] [ 1 2 ]) ;F2 ) which is isomorphic to the free module F2 [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.


PRELIMINARIES
In this section, we will review some concepts related to Quillen conjecture and then give a full and details of the conjecture. The key objects in the conjecture are arithmetic and S-arithmetic groups whose concepts arose when studying modular form and some other classical topics in number theory. However, the definition of arithmetic groups is very technical. We can, roughly, think of arithmetic groups as the intersection of SL(n, Z) (or GL(n, Z)) with some semi-simple Lie subgroup G of SL(n, R) (or GL(n, Z)). Full definition can be seen in 1 as follows: Definition 1 Let G be a semi-simple subgroup of SL(n, R). Group Γ is said to be an arithmetic subgroup of G if and only if Γ is a lattice in G and there exist a closed, connected, semisimple subgroup G ′ of some SL(n, R), such that G ′ is defined over Q, • a closed, connected, semisimple subgroup G ′ of some SL(n, R), such that G ′ is defined over Q, • compact normal subgroups K and K ′ of G • and G ′ , respectively, and • an isomorphism ψ : Reader can find definition of defined over, commensurable and other related concepts in 2 . Some common arithmetic groups are SL n (Z), PSL n (Z), PGL n (Z) modular groups, Siegel groups, etc. In order to generalize the concept of arithmetic groups, one natural method is replacing the ring Z of integers by a bigger ring. There are two ways that lead to two different branches. One is considering the ring of integers of some number fields, this leads to the concepts of Hilbert modular and Bianchi groups. Another is replacing Z by the ring of integers with some primes inverted Z . The fusion of these two ideas leads us to the definitions of Sintegers and S-arithmetic groups. Recall that given an algebraic number field K, an absolute value on K is a map |. One can prove that two absolute values are equivalent if there exists c > 0 such that |α| 1 = |α| c 2 . An equivalent class of non-trivial absolute values is called a place of K. Definition 2 Let K be a number field. Given a finite set S of places of K that containing all archimedean places. We define the ring of S-integers to be the set Let us consider one example as follows: Example 1 In this example, we will calculate the possibilities of S-integers in case Kis the field Qof rational numbers. Let pbe a prime, then each rational number q ∈ Q * can be written uniquely as q = p r m And the p-adicabsolute value associated to the prime pis defined |q| p = p −v p (q) . Ostrowskis theorem 3 says that every non-trivial absolute value on Qis equivalent to | · | p for some prime p ≤ ∞. Note that we consider the usual absolute value on Qas |.| ∞ , the case that p = ∞. Forconvenience we identify the absolute value v p by its related prime p. Thus the set of places Sis of the form P ∪ {∞}where P is a set of prime numbers. Suppose that S = {∞, p 1 , p 2 , . . . , p n } then And thus we have a class of S-arithmetic groups where m is square-free positive integer.

QUILLENS CONJECTURE AND RELATED RESULTS
In this section, we will present the details of Quillens conjecture and also give a comprehensive list of the results related to this conjecture. Conjecture 1 2 Let ℓ be a prime number. Let K be a number field with ζ ℓ ∈ K, and S a finite set of places containing the infinite places and the places over ℓ.
a free module over the cohomology ring H * cts (GL n (C); F ℓ ). This conjecture was re-stated as Over the last forty years, many efforts have been put into this conjecture and many results have been published. Some of them show the positiveness of the conjecture, some disproved it. Here we give a brief list of the outstanding results. Positive cases, in which the conjecture has been established, are: Despite of the above positive cases, many mathematicians have been trying to find a counterexample. In 1998, Mitchell 4 show that the restriction map

QUILLEN CONJECTURE FOR FARRELL-TATE COHOMOLOGY
In effort of making a refinement of Quillens conjecture, Rahm and Wend, in their paper 11 , state that the conjecture is true for Farrel-Tate cohomology in the following cases. Theorem 1 11 Let ℓ be a prime number. Let K be a number field with ζ ℓ ∈ K, and S a finite set of places containing the infinite places and the places over ℓ 1.
The Quillen conjecture is true for Farrell-Tate cohomology of SL 2 ( in cohomological degrees above the virtual cohomological dimension. They also refine the conjecture to Conjecture 3 11 Let K be a number field. Fix a prime ℓ such that ζ ℓ ∈ K, and an integer n < ℓ. Assume that S is a set of places containing the infinite places and the places lying over ℓ. If each cohomology class of GL n ( O K,S ) is detected on some finite subgroup, then H • cts (GL n (C), F ℓ ) is a free module over the image of the restriction map H • cts (GL n (C), F ℓ ) → H • (GL n (C), F ℓ ). By the above theorem, Rahm and Wendt have made the following observations for SL 2 over rings of Sintegers at odd prime ℓ:

RECENT RESULTS
In this section, we will introduce some results related to Quillen conjecture that the first author and his collaborator had published. Theorem 2 6 The cohomology ring H * ( is isomorphic to the free module This gives one more positive example for Quillen conjecture. Reader can find the details of the computation in 6 .

CONCLUSION
Over the last forty years, many results have been published about Quillen conjecture. Some give positive results, some introduce counterexamples in which the conjecture fails, and some make refinements. In 11 , Rahm and Wendt have stated that for large primes ℓ, we can determine precisely the module structure above the virtual cohomological dimension . Therefore, the future work on this conjecture should focus on the cases of small primes. In order to refine the conjecture or find other counterexamples, there are two sources that we can look at: of diagonal matrices.
• (ii) In 12 , Wendt found that the image of the Quillen homomorphism is not free in the cases that he is observing.
In order to examine a wide range of S-arithmetic groups, we may need to subdivide the spaces on that those groups act to get a rigid action. This problem may become a serious problem since those cell complexes can be very big and complicated. The first author and his collaborator have developed an algorithm named Rigid Facet Subdivision 13 to overcome this problem.