K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Equivalently, G is a duality group if its cohomology with group ring coe#cients is torsion free and concentrated in dimension n. The dualizing module is this case is H (G, ZG) A group G is said to be a virtual duality group if it has a finite index subgroup that is a duality group. See [2] and [7] for further information on duality properties for groups. Example 2.2. The simplest examples of duality groups are the free and free abelian groups. Finitely generated free groups are 1 dimensional duality groups and the free abelian group Z is an n dimensional duality group since it admits a ....

....G with ZG coe#cients is concentrated in dimension n and is Z torsion free. We express H # (G, ZG) in terms of the equivariant cohomology for the action of G on P , H # G ( P ) The standard equivariant spectral sequence arises from filtering a space by skeleta (see for example VII.7 in [7]) However, our filtration of by G equivariant subcomplexes that are more naturally related to the underlying poset structure. We let 0 denote the subcomplex constructed using only corank 0 elements; this should be thought of as the collection of vertices for this complex. In general, ....

K.S. Brown, Cohomology of Groups, Springer-Verlag, 1982.

....the image of H (#, Q p Z p ) # (# # , Q p Z p ) # (dual to the corestriction map) is contained in e(#, # # )H (# # , Q p , Z p ) # . Proof: Dually, we prove that the corestriction map (# # , Q p Z p ) #, Q p Z p ) vanishes on H (# # , Q p Z p ) e(#, # # ) It is shown in [Br] that H (# # , Q p Z p ) is generated by cup products of characters # # 1 , # # 2 (# # , Q p Z p ) Furthermore, if e(#, # # ) kills # # 1 # # 2 , then it kills one of the characters, say # # 2 . The corestriction map on characters is composition with the transfer map # # # , which is ....

K. S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.

....notice that the Poincare construction of the presentation can be subdivided to yield a finite dimensional simplicial complex. Since it is known that a finite dimensional K(G, 1) space implies that the cohomological dimension of G is finite, which in turn implies that G is torsion free (see [3]) the proof is complete. # 124 Lemma 12.14 is reminiscent of Lyndon s Theorem which states that the Poincare construction of a 1 relator group is a K(G, 1) space i# the relator is simple. The above lemma verifies several cases of Lyndon s Theorem and extends the result to the realm of general ....

K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982.

....is the set of all divisors of #. Then the resolution of Z constructed in [14] is isomorphic to the resolution of Proposition 3.2. Technically, the connection between the cells considered in [14] and ours is analogous to what happens when one goes from a standard resolution to a bar resolution [10] so it is just a change of variables. Proof. By definition, the n cells considered in [14] are of the form (# 1 , # n ) with # 1 , # n in M such that the product # 1 . # n belongs to (which implies that each # j belongs to ) We map such a cell to C # n by #( # 1 , ....

K.S. Brown, Cohomology of groups, Springer (1982).

....2 est d ordre premier p on sait que p 6 N 1. Il en rsulte que l action de sur X N permet de d nir un complexe V = Vn ; dn ) tel que Vn soit isomorphe au module libre engendr par n , n 0, et que l homologie de V concide, modulo SN 1 , avec l homologie quivariante de la paire (X N ) [2] VII Prop. 8.1, 16] Prop. 2.2) Hn (V ) H n (X N ; Z) mod SN 1 ) Proposition 2.2. i) Si = GL 5 (Z) on a, modulo S 5 , Z si n = 9; 14; ii) Si = GL 6 (Z) on a, modulo S 7 , Z si n = 10; 11; 15; iii) Si = SL 6 (Z) on a, modulo S 7 , si n = 15; Z si n = 10; 11; 12; ....

....St 4 ) H 4 (GL 2 (Z) St 2 ) 0 : H 1 (GL 6 (Z) St 6 ) H 2 (GL 5 (Z) St 5 ) H 4 (GL 3 (Z) St 3 ) H 5 (GL 2 (Z) St 2 ) 0 : Pour dmontrer cette proposition on utilise les rsultats du calcul amenant la Proposition 2.1 et les rsultats de [14] 7] 15] et [16] D aprs (3. 1) et [2] ou [16] on peut calculer les groupes Hm(GLN (Z) St N ) l aide d une suite spectrale dont le terme E 1 est une somme de groupes d homologie des stabilisateurs des cellules des n . L analyse de ces groupes conduit la Proposition 4.2. 4.2. A l aide de (4.1) et de [7] on dduit de la Proposition ....

Brown, K.; Cohomology of Groups, Springer GTM 87, New York (1982).

....and discussion, as well as another proof of the theorem for n 3, see [6] 2 An E spectral sequence converging to H (G) Let G stand for an arbitrary group. We define H (G) the integral homology of G as a discrete group, to be H (BG; Z) where BG is the classifying space of G (see [1], 8] Let us recall the sequence of constructions leading to H (BG) The space BG is the geometric realization of the simplicial nerve of G, NG : NG (0) f1g; NG (n) G Theta Delta Delta Delta Theta G (n times) The face maps d i are defined by d 0 (g 1 ; g n ) g 2 ; ....

....which it computes the homology, is a K(G; 1) Consider now a short exact sequence of groups 0 A Gamma G Gamma Q 1; 2) where A is abelian and A is normal in G. The quotient Q acts on A by conjugation, h Delta a = g ag, where a 2 A and g is any element of G so that (g) h (see [1], pp. 86 87) The group Q then acts on the integral homology of A. Let us also use the notation h Delta ff for this action (h 2 Q; ff 2 H k (A) The context will always make the domain of the action clear. Theorem 2.1 (a) There is a spectral sequence with E H q (A) converging to H p q ....

K.S. Brown, Cohomology of Groups (Springer, New York, 1982).

.... to the corestriction map) is contained in e( Gamma; Gamma 0 )H 2 ( Gamma 0 ; Q p ; Z p ) Proof: Dually, we prove that the corestriction map H 2 ( Gamma 0 ; Q p =Z p ) H 2 ( Gamma; Q p =Z p ) vanishes on H 2 ( Gamma 0 ; Q p =Z p ) e( Gamma; Gamma 0 ) It is shown in [Br] that H 2 ( Gamma 0 ; Q p =Z p ) is generated by cup products of characters 0 1 ; 0 2 2 H 1 ( Gamma 0 ; Q p =Z p ) Furthermore, if e( Gamma; Gamma 0 ) kills 0 1 [ 0 2 , then it kills one of the characters, say 0 2 . The corestriction map on characters is composition ....

K. S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.

....sequence of homology groups corresponding to the above short exact sequence of G modules. If we use Shapiro s Lemma for computing the homology groups of the middle term, we obtain an exact sequence # H 2 (G v0 , Z p ) # H 2 (G, Z p ) # H 1 (G, XS# Z p ) # . Theorem 6. 4(iii) of [1] implies that, for any abelian group H , we have a canonical group isomorphism # 2 Zp (H# Z p ) # # H 2 (H, Z p ) This result, combined with Lemma 3.1.6 (2) and (3) yields isomorphisms Z pZ # # H 2 (G v0 , Z p ) and respectively (Z pZ) 3 # # H 2 (G, Z p ) Therefore, the long ....

....since G 0 acts trivially on Z pZ, we have H 0 (G 0 , Z pZ) Z pZ. Therefore, Proposition 3.4.3 shows that the last non trivial morphism in the long exact sequence above is in fact an isomorphism. Now, if we take into account that H 1 (G 0 , Z pZ) # # G 0# Z pZ # # (Z pZ) 2 (see [1]) the long exact sequence above yields the following surjective group morphism. Z pZ) 2 # A (p) K,S0 (I G1 , I G0 )A (p) K,S0 , where (I G1 , I G0 ) I G1 Z p [G] I G0 Z p [G] Now, since G is a p group, Z p [G] is a Noetherian, local ring, of maximal ideal M p,G : I GZ p [G] ....

K. Brown, Cohomology of Groups, GTM 87 (1982), Springer--Verlag.

....by conjugation; let be the resulted Beltrami coecient. The correspondent quasiconformal map will conjugate to another Fuchsian representation. We have therefore a correspondence: pair of points in the Teichm uller space of quasiconformal representations intensively studied by Bers [B]. The map f j L above is schlicht holomorphic. The correspondence quasiconformal representation Schwartzian of f is called the Bers embedding. Its image contains a ball in H 0 (K 2 ) with L 1 norm and is contained in a ball [Gar] The boundary of this image correspond to some ....

....equivariant cohomology 17.1.1. Consider a space X with an action of a cyclic group C p . Let E B be a classifying bration for C p . Recall that the equivariant cohomology H e (X) is de ned by H e (X) H (X E=C p ; Z) Since X E=C p is a bration over B with ber X, one arrives [B] to the rst spectral sequence, converging to H e (X) with E 2 term H i (C p ; H j (X) On the other hand, let Y = X=C p . There is a sheaf F i on Y with a stalk (F i ) y = H i (G y ; Z) Here G y is a stabilizer of any point x 2 X over y. The second spectral sequence, converging to ....

K. Brown, Cohomology of Groups, Springer..

....free cover with the given data (see Section 2.1) and we want to know the closed G submodules of K 0 . If W is transitive, then (for any w 2 W ) the module K 0 is the G module coinduced from the Aut(W=w) module B(w) with the action being given via the canonical homomorphism (cf. Section III.5 of [8]) Now K 0 is compact and so by Pontriagin duality, this problem is equivalent to determining all the G submodules of the direct sum K 0 = M w2W B(w) Care is needed here in writing down the G action. If W is transitive, then (for any w 2 W ) this module is the G module induced from the ....

....G submodules of the direct sum K 0 = M w2W B(w) Care is needed here in writing down the G action. If W is transitive, then (for any w 2 W ) this module is the G module induced from the Aut(W=w) module B(w) with the action being given via the canonical homomorphism (cf. Section III.5 of [8]) The simplest situation is where the fibre groups and binding groups are all cyclic of order p. Then K 0 = F W p , and K 0 is the permutation module F p W (F p is the field of p elements, considered as a trivial G module) The correspondence between submodules of the two modules given by the ....

[Article contains additional citation context not shown here]

K. S. Brown, Cohomology of Groups, Springer GTM 87, Springer Verlag, Berlin 1982.

....of our invariant ## L K,Q( n) in section 2, we therefore treat the two cases n = 0 and n 0 separately in section 3 and 4, respectively. In the final section 5, we briefly indicate how the expected properties of Lichtenbaum s complexes #(r) for r # 2 naturally give rise to invariants in Cl(Z[ 1 2 ][G] and how these invariants relate to# r 1 (L K) and hence to ## L K,Q(1 r) Meanwhile, in [3] the construction of ## L K,M ) has been generalized from abelian to arbitrary Galois extensions L K. The methods we develop in this paper also give the identity (1) in the general case. ....

....inclusion U S ## US for i = 0. We denote by #S the complex # 0 S # # 1 S (which is Z[G] perfect) Remark. The following two exact triangles in D summarize the relationship between # S , #S and R# c (O L,S , Z) # [ 3] # S # R# c (O L,S , Z) # [ 3] # US US [0] # (31) X S# Q[ 2] # # S # #S # (32) Both X S# Q and U S US are uniquely divisible, hence Q[G] modules. As such they are injective, and because Z[G] # Q[G] is flat, they are also injective Z[G] modules. So for many purposes the di#erences between # S , #S and R# c (O L,S , Z) # [ 3] are inessential ....

[Article contains additional citation context not shown here]

K.S. Brown, Cohomology of Groups, Springer GTM 87, 1982.

No context found.

K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982

No context found.

K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982

No context found.

K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982

No context found.

K. Brown. Cohomology of Groups. Springer-Verlag, New York, 1982

No context found.

K.S. Brown, Cohomology of groups, Springer (1982).

No context found.

K. S. Brown, Cohomology of groups, Springer, New York, 1982.

No context found.

K.S. Brown, Cohomology of Groups, Springer, New York 1982.

No context found.

K.S. Brown, Cohomology of Groups, Springer-Verlag, 1982.

No context found.

K S Brown, Cohomology of Groups, Springer-Verlag, 1982.

No context found.

K. S. Brown, Cohomology of groups, Springer, New York, 1982.

No context found.

K. Brown, Cohomology of Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1982.

No context found.

K. S. Brown. Cohomology of groups, Springer-Verlag (1982).

No context found.

K. Brown, Cohomology of Groups, Springer--Verlag, Berlin, Heidelberg, New York, 1982.

No context found.

K.S. Brown. Cohomology of Groups. GTM 87. Springer-Verlag, New York, 1982.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC