Third homology of perfect central extensions
Behrooz Mirzaii, Fatemeh Yeganeh Mokari, David M. Carbajal Ordinola
aa r X i v : . [ m a t h . K T ] J u l THIRD HOMOLOGY OF PERFECT CENTRALEXTENSIONS
B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
Abstract.
For a central perfect extension of groups A G ։ Q , we study the maps H ( A, Z ) → H ( G, Z ) and H ( G, Z ) → H ( Q, Z ) provided that A ⊆ G ′ . First we show that the imageof H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) is 2-torsion where ρ : A × G → G is the usual product map. When BQ + is an H -space, we also study the kernel of the surjective homomorphism H ( G, Z ) → H ( Q, Z ). Introduction
Homologies and cohomologies are important invariants that one canassign to a given group. Unfortunately, in many important cases these(co)homology groups are too complicated to be computed explicitly.Therefore in many cases results allowing to compare the homologygroups for different groups become quite important.In this article, we study such homomorphism for the third homologygroups of a perfect central extension. A central extension A G ։ Q is called perfect if G is a perfect group, i.e. if G = [ G, G ]. The aimof the current paper is to study the maps H ( A, Z ) → H ( G, Z ) and H ( G, Z ) → H ( Q, Z ) for such extensions provided that A ⊆ G ′ .The interest to this problem comes from two sources. First fromalgebraic K -theory and the study of K -groups of a ring where varioustype of universal central extensions [1] appears. Second from algebraictopology and homology of groups that many often one has to deal withdifferent types of spectral sequences that usually are difficult to dealwith.In Section 1 we give a quick overview of Whitehead’s quadratic func-tor which plays an important role in this article.In Section 2 we show that if A is a central subgroup of a group G such that A ⊆ G ′ , e.g. G a perfect group, then the image of the naturalmap H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) in 2-torsion, where ρ : A × G → G is the usual product map. Inparticular if A G ։ Q is a universal central extension, then theimage of H ( A, Z ) in H ( G, Z ) is 2-torsion.Section 3 has K -theoretic flavor. If A G ։ Q is a perfect centralextension such that K ( Q, + , the plus-construction of the classifyingspace of Q , is an H -space, then we prove that there is the exact sequence A/ → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) → H ( Q, Z ) → . Moreover we prove that with this extra condition, the map H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) is trivial. In particular if the extension isuniversal then the image of H ( A, Z ) in H ( G, Z ) is trivial.Finally in Section 4 we prove cohomological version of these results.If A is a central subgroup of a group G such that A ⊆ G ′ , we show thatthe map H ( G, Z ) → H ( A, Z ) in trivial. Moreover if A G ։ Q isa perfect central extension such that K ( Q, + is an H -space, then weget the exact sequence0 → Ext Z ( A, Z ) → H ( Q, Z ) → H ( G, Z ) ρ ∗ −→ ( A ⊗ Z H ( G, Z )) ∗ , where for an abelian group M , M ∗ is its dual group Hom( M, Z ). Notations. If A → A ′ is a homomorphism of abelian groups, by A ′ /A we mean coker( A → A ′ ). For a group A and a prime p , p ∞ A is the p -power torsion subgroup of A .1. Whitehead’s quadratic functor
Let A G ։ Q be a perfect central extension. By Theorem 2.1the image of the map H ( A, Z ) → H ( G, Z ) is 2-torsion. To study thisimage further and also to study the map H ( G, Z ) → H ( Q, Z ), the useof tools and techniques from algebraic topology seems to be necessary.Standard classifying space theory gives a (homotopy theoretic) fi-bration of Eilenberg-MacLane spaces K ( A, → K ( G, → K ( Q, K ( G, → K ( Q, → K ( A, . By studying the Serre spectral sequence associated to this fibrationwe obtain the exact sequence(1.1) H ( Q, Z ) → H ( K ( A, , Z ) → H ( G, Z ) /ρ ∗ (cid:0) A ⊗ Z H ( G, Z ) (cid:1) → H ( Q, Z ) → . The group H ( K ( A, , Z ) plays very important role in this article. Ithas interesting properties and has been studied extensively [11], [5].A function θ : A → B of (additive) abelian groups is called a qua-dratic map if HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 3 (1) for any a ∈ A , θ ( a ) = θ ( − a ),(2) the function A × A → B with ( a, b ) θ ( a + b ) − θ ( a ) − θ ( b ) isbilinear.For any abelian group A , there is a universal quadratic map γ : A → Γ( A )such that for any quadratic map θ : A → B , there is a unique grouphomomorphism Θ : Γ( A ) → B such that Θ ◦ γ = θ . It is easy to seethat Γ is a functor from the category of abelian groups to itself.The functions φ : A → A/ ψ : A → A ⊗ Z A , given by φ ( a ) = a and ψ ( a ) = a ⊗ a respectively, are quadratic maps. Thus we get thecanonical homomorphismsΦ : Γ( A ) → A/ , γ ( a ) a and Ψ : Γ( A ) → A ⊗ Z A, γ ( a ) a ⊗ a. Clearly Φ is surjective. Moreover coker(Ψ) = A ∧ A ≃ H ( A, Z ) andhence we have the exact sequence(1.2) Γ( A ) Ψ −→ A ⊗ Z A → H ( A, Z ) → . Furthermore we have the bilinear pairing[ , ] : A ⊗ Z A → Γ( A ) , [ a, b ] := γ ( a + b ) − γ ( a ) − γ ( b ) . It is easy to see that for any a, b, c ∈ A , [ a, b ] = [ b, a ], Φ[ a, b ] = 0,Ψ[ a, b ] = a ⊗ b + b ⊗ a and [ a + b, c ] = [ a, c ] + [ b, c ]. Using (1) and thislast equation, for any a, b, c ∈ A , we obtain(a) γ ( a ) = γ ( − a ),(b) γ ( a + b + c ) − γ ( a + b ) − γ ( a + c ) − γ ( b + c ) + γ ( a ) + γ ( b ) + γ ( c ) = 0.Using these properties we can construct Γ( A ).Let A be the free abelian group generated by the symbols w ( a ), a ∈ A . Set Γ( A ) := A / R , where R denotes the relations (a) and (b)with w replaced by γ . Now γ : A → Γ( A ) is given by a w ( a ).It is easy to show that [ a, a ] = 2 γ ( a ). Thus the compositeΓ( A ) Ψ → A ⊗ Z A [ , ] −→ Γ( A )coincide with multiplication by 2. Moreover one sees easily that thecomposite A ⊗ Z A [ , ] −→ Γ( A ) Ψ → A ⊗ Z A sends a ⊗ b to a ⊗ b + b ⊗ a .It is known that the sequence(1.3) A ⊗ Z A [ , ] −→ Γ( A ) Φ → A/ → B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
Proposition 1.1.
For any abelian group A , Γ( A ) ≃ H ( K ( A, , Z ) .Proof. See [5, Theorem 21.1] (cid:3)
For topological proofs of the exact sequences (1.2) and (1.3) one maystudying the Serre spectral sequences associated to the fibration K ( A, → K ( { } , → K ( A, A ≃ A ։ { } and the path space fibrationΩ K ( A, → P K ( A, → K ( A, , respectively. Observe that Ω K ( A, n ) = K ( A, n + 1) and H n +2 ( K ( A, n ) , Z ) ≃ A/ K ( A,
2) is an H -sapce [12, Theorem 7.11, Chap. V] and A ⊗ Z A → H ( K ( A, , Z )is induced by the product structure of the H -space.2. Third homology over central subgroups
Let A be a central subgroup of G such that A ⊆ G ′ . The condition A ⊆ G ′ is equivalent to the triviality of the homomorphism of homologygroups H ( A, Z ) → H ( G, Z ). Let n be a nonzero natural number.From the commutative diagram(2.1) A × A AA × G G, µρ where µ and ρ are the usual product maps, we obtain the commutativediagram H n − ( A, Z ) ⊗ Z H ( A, Z ) H n ( A, Z ) H n − ( A, Z ) ⊗ Z H ( G, Z ) H n ( G, Z ) . =0 This shows that the composite V n Z A → H n ( A, Z ) → H n ( G, Z )is trivial. Since H n ( A, Z ) / V n Z A is torsion [4, Theorem 6.4, Chap. V],the image of H n ( A, Z ) in H n ( G, Z ) is a torsion group. In particu-lar H ( A, Z ) → H ( G, Z ) and H ( A, Z ) → H ( G, Z ) are trivial map.Moreover if A is torsion free, then H n ( A, Z ) → H n ( G, Z ) is trivial forany n ≥ HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 5
Theorem 2.1.
Let A be a central subgroup of G such that A ⊆ G ′ .Then the image of the natural map H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) is -torsion. In particular if A G ։ Q is a universal central exten-sion, the image of H ( A, Z ) in H ( G, Z ) is -torsion.Proof. It is well-known that the sequence0 → V Z A → H ( A, Z ) → Tor Z ( A, A ) Σ → , is exact [9, Lemma 5.5], where Σ = { id , − σ } . The homomorphism onthe right side of the exact sequence is obtained from the composition H ( A, Z ) ∆ ∗ −→ H ( A × A, Z ) → Tor Z ( A, A ) , where ∆ is the diagonal map A → A × A , a ( a, a ). Moreover theaction of σ on Tor Z ( A, A ) is induced by the involution ι : A × A → A × A ,( a, b ) ( b, a ).From the diagram (2.1), we obtain the commutative diagram e H ( A × A, Z ) H ( A, Z ) e H ( A × G, Z ) H ( G, Z ) , µ ∗ ρ ∗ where e H ( A × A, Z ) := ker( H ( A × A, Z ) ( p ∗ ,p ∗ ) −−−−→ H ( A, Z ) ⊕ H ( A, Z )) , ˜ H ( A × G, Z ) := ker( H ( A × G, Z ) ( p ∗ ,p ∗ ) −−−−→ H ( A, Z ) ⊕ H ( G, Z )) . As we have seen, the condition A ⊆ G ′ implies that the composite V Z A → H ( A, Z ) → H ( G ) B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA is trivial. This fact together with the K¨unneth formula for e H ( A × A, Z )gives us the commutative diagramTor Z ( A, A ) Tor Z ( A, A ) Σ ˜ H ( A × A ) / L i =1 H i ( A, Z ) ⊗ Z H − i ( A, Z ) H ( A, Z ) / V Z A ˜ H ( A × G ) / L i =1 H i ( A, Z ) ⊗ Z H − i ( G, Z ) H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z ))Tor Z ( A, H ( G, Z )) . ¯ µ ∗ α ≃ µ ∗ f inc ∗ β ≃ inc ∗ ≃ ρ ∗ Note thatim( H ( A, Z ) ⊗ Z H ( G, Z ) → H ( G, Z )) ⊆ im( A ⊗ Z H ( G, Z ) → H ( G, Z ))(see [8, Proposition 4.4, Chap. V]). Since the mapTor Z ( A, A ) = Tor Z ( H ( A, Z ) , A ) → Tor Z ( A, H ( G, Z ))is trivial, we see that ρ ∗ ◦ f inc ∗ ◦ α − is trivial. This shows that thecomposite map inc ∗ ◦ β − ◦ ¯ µ ∗ is trivial. Therefore the image of H ( A, Z )in H ( G, Z ) is equal to the image ofTor Z ( A, A ) Σ / ¯ µ ∗ Tor Z ( A, A ) . By the above arguments, one sees that the homomorphism¯ µ ∗ : Tor Z ( A, A ) → Tor Z ( A, A ) Σ is induced by the composition A × A µ −→ A ∆ −→ A × A .The morphism of extensions A A × A AA A { } , i µ p = where i ( a ) = ( a, p ( a, b ) = b and µ ( a, b ) = ab , induces the mor-phism of fibrations K ( A × A, K ( A, K ( A, K ( A, K ( { } , K ( A, . HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 7
By analysing the Serre spectral sequences associated to this morphismof fibrations, we obtain the exact sequence0 → ker(Ψ) → H ( K ( A, Ψ → A ⊗ Z A → H ( A ) → , where ker(Ψ) ≃ H ( A, Z ) /µ ∗ ( A ⊗ Z H ( A, Z ) ⊕ Tor Z ( A, A )) . Clearly µ ∗ ( A ⊗ Z H ( A, Z )) ⊆ V Z A ⊆ H ( A, Z ). Thereforeker(Ψ) ≃ Tor Z ( A, A ) Σ ε / (∆ A ◦ µ ) ∗ (Tor Z ( A, A )) . But by the facts from the previous section ker(Ψ) is two torsion. Thisproves our claim. (cid:3)
Remark 2.2. (i) If A is a central subgroup of a group G , then thesame argument as in proof of Theorem 2.1 shows that the image of thenatural map H ( A, Z ) → H ( G, Z ) /ρ ∗ ( e H ( A × G, Z ))is two torsion.(ii) In Proposition 3.3, we show that if A G ։ Q is a uni-versal central extension such that BQ + is an H -space, then the map H ( A, Z ) → H ( G, Z ) is trivial.3. Third homology of central extensions over H -groups For any sequence of abelian groups A n , n ≥
2, Berrick and Millerconstructed a perfect group Q such that H n ( Q, Z ) ≃ A n [3, Theorem 1].Let A be an abelian group. By using the result of Berrick and Miller,choose a perfect group Q such that H ( Q, Z ) ≃ A and H ( Q, Z ) = 0.Then if A G ։ Q is the universal central extension of Q , we havethe exact sequence0 → H ( K ( A, , Z ) → H ( G, Z ) → H ( Q, Z ) → . This example shows that in general for an the universal central exten-sion A G ։ Q , the kernel of H ( G, Z ) → H ( Q, Z ) can be verycomplicated.A group is called quasi-perfect if its commutator group is perfect.We say a quasi-perfect group Q is an H - group if K ( Q, + , the plus-construction of K ( Q,
1) with respect to Q ′ = [ Q, Q ] [6], is an H -space.Note that for a group G , K ( G,
1) is an H -space if and only if G isabelian. B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
Example 3.1. (a) A quasi-perfect group Q is called a direct sum group if there is a homomorphism ⊕ : Q × Q → Q , called an internal directsum on Q , such that(i) for g , . . . , g k ∈ Q ′ and g ∈ Q , there is h ∈ Q ′ such that gg i g − = hg i h − for 1 ≤ i ≤ k ,(ii) for any g , . . . , g n ∈ Q , there are c, d ∈ Q such that c ( g i ⊕ c − = d (1 ⊕ g i ) d − = g i .It is known that any direct sum group is an H -group [10, Proposi-tion 1.2]. The stable general linear, orthogonal, symplectic groups andtheir elementary subgroups, all are groups with direct sum. For moreexamples of such groups see [6, 1.3].(b) For any abelian group A , Berrick has constructed a perfect group Q such that K ( Q, + , is homotopy equivalence to K ( A,
2) [2, Corollary1.4]. Thus Q is an H -group. Theorem 3.2.
Let A G ։ Q be a perfect central extension. If Q is an H -group, then we have the exact sequence A/ → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) → H ( Q, Z ) → . Proof.
From the central extension and the fact that Q is perfect wehave the fibration K ( A, → K ( G, + → K ( Q, + [13, Proposition 1], [1, Theorem 6.4]. From this we obtain the fibration K ( G, + → K ( Q, + → K ( A, K ( A,
2) is an H -space [12, Theorem7.11, Chap. V]. Moreover the map K ( Q, + → K ( A,
2) is an H -map[14, Proposirion 2.3.1]. Since the plus construction does not change thehomology, from the Serre spectral sequence of the above fibration weobtain the exact sequence H ( Q, Z ) → H ( K ( A, , Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) → H ( Q, Z ) → . From the commutative diagram, up to homotopy, of H -spaces and H -maps BQ + × BQ + BQ + K ( A, × K ( A, K ( A, , HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 9 we obtain the commutative diagram H ( Q, Z ) ⊗ Z H ( Q, Z ) H ( Q, Z ) A ⊗ Z A H ( K ( A, , Z ) . Since G is perfect, H ( Q, Z ) → A is surjective. This gives us thesurjective map H ( K ( A, , Z ) / im( A ⊗ Z A ) ։ H ( K ( A, , Z ) / im( H ( Q, Z )) . This together with (1.3) gives us the desired exact sequence. (cid:3) If A G ։ Q is a perfect central extension, Theorem 2.1 impliesthat the image of H ( A, Z ) in H ( G, Z ) is 2-torsion. In the followingproposition we go one step further. Proposition 3.3.
Let A G ։ Q be a perfect central extension. If Q is an H -group, then the natural map H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) is trivial. In particular if the extension is universal, then the naturalmap H ( A, Z ) → H ( G, Z ) is trivial.Proof. From the morphism of extensions
A A { } A G Q, we obtain the morphism of Serre fibrations K ( A, K ( { } , K ( A, K ( G, K ( Q, K ( A, , By analyzing the Serre spectral sequences of these fibrations we obtainthe commutative diagramker(Ψ) H ( A, Z ) H ( K ( A, , Z ) H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) → H ( Q, Z ) → , ≃ where H ( A, Z ) is a quotient of H ( A ) and the mapΨ : Γ( A ) = H ( K ( A, , Z ) −→ A ⊗ Z A is discussed in the previous section. Since Γ( A ) / [ A, A ] ≃ A/ H ( A, Z ) A/ H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) . ≃ If Θ := [ , ] : A ⊗ Z A → Γ( A ), then we have seen that the composite A ⊗ Z A Θ −→ Γ( A ) Ψ −→ A ⊗ Z A, takes a ⊗ b to a ⊗ b + b ⊗ a . Thus from the commutative diagram0 ker(Θ) A ⊗ Z A im(Θ) 00 ker(Ψ) Γ( A ) A ⊗ Z A ΘΘ ΨΨ and the exact sequence (1.3) we obtain the exact sequenceker(Ψ) → A/ δ → ( A ⊗ Z A ) σ → H ( A, Z ) → , where ( A ⊗ Z A ) σ := ( A ⊗ Z A ) / h a ⊗ b + b ⊗ a | a, b ∈ A i and δ ( a ) = a ⊗ a .But the sequence0 → A/ → ( A ⊗ Z A ) σ → H ( A, Z ) → → A/ H ( A, Z ) → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z )) is trivial. (cid:3) Example 3.4.
Let A G ։ Q be a perfect central extension and let Q be an H -group. Here we would like to calculate the homomorphism A/ → H ( G, Z ) /ρ ∗ ( A ⊗ Z H ( G, Z ))from Theorem 3.2.The extension A G ։ Q is an epimorphic image of the uni-versal extension of Q , which is unique up to isomorphism. Thus wemay assume that our extension is universal. Therefore H ( G, Z ) = H ( G, Z ) = 0. HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 11
By studying the spectral sequences of the morphism of Serre fibra-tions K ( A, K ( G, K ( Q, K ( G, K ( Q, K ( A, , we obtain the morphsim of exact sequences H ( Q, Z ) Γ( A ) H ( G, Z ) H ( Q, Z ) ker(Ψ) H ( G, Z ) /ρ ∗ H ( A, Z ) . Note that ker(Ψ) = ker( A ⊗ Z A → H ( A, Z )) = h a ⊗ a : a ∈ A i .By Proposition 3.3, H ( G, Z ) = H ( G, Z ) /ρ ∗ H ( A, Z ). Since the map A/ → H ( G, Z ) factors through Γ( A ) /H ( Q, Z ), it is also factorsthrough the group ker(Ψ) /H ( Q, Z ). In fact it factors throughoutker(Ψ) / h a ⊗ b + b ⊗ a | a, b ∈ A i . Thus it is enough to calculate themap ker(Ψ) / h a ⊗ b + b ⊗ a | a, b ∈ A i η −→ H ( G, Z ) . Let Q = F/S be a free presentations of Q . By a theorem of Hopf H ( Q, Z ) ≃ ( S ∩ [ F, F ]) / [ S, F ] [4, Theorem 5.3, Chap. II]. This iso-morphism can be given by the following explicit formulaΛ : ( S ∩ [ F, F ]) / [ S, F ] ≃ −→ H ( Q, Z ) = H ( B • ( Q ) Q ) , (cid:16) g Y i =1 [ a i , b i ] (cid:17) [ S, F ] P gi =1 (cid:0) [¯ s i − | ¯ a i ]+[¯ s i − ¯ a i | ¯ b i ] − [¯ s i ¯ b i | ¯ a i ] − [¯ s i | ¯ b i ] (cid:1) , where s i = [ a , b ] · · · [ a i , b i ] and for x ∈ F we set ¯ x = xS ∈ F/S = Q [4, Exercise 4, §
5, Chap. II]. Note that ¯ s g = 1. Here B • ( Q ) → Z is thebar resolution of Q .Let G = F/R , Q = F/S and A = S/F be free presentations of G , Q and A respectively. Since A is central we have [ S, F ] ⊆ R and thusthe following diagram H ( Q, Z ) ≃ −−−−−−−−−−−−−→ A = S/R Λ տ ր ( S ∩ [ F, F ]) / [ S, F ] . commutes, where ( S ∩ [ F, F ]) / [ S, F ] → S/R = A is given by s [ S, F ] sR . For any a ∈ F , we denote aR ∈ G = F/R by ˆ a and for any s ∈ S ∩ [ F, F ], we denote s [ S, F ] by ˜ s . The Lyndon-Hochschild-Serre spectral sequence E p,q = H p ( Q, H q ( A, Z )) ⇒ H p + q ( G, Z )gives us a filtration of H ( G, Z )0 = F − H ⊆ F H ⊆ F H ⊆ F H ⊆ F H = H ( G, Z ) , such that E ∞ i, − i = F i H /F i − H . Now by an easy analysis of the abovespectral sequence one sees that F H = F H = 0 and the map η isinduced by the composite(3.2) ker(Ψ) → E , ≃ E ∞ , ≃ F H ⊆ H ( G, Z ) . If s g = Q gi =1 [ a i , b i ] ∈ S ∩ [ F, F ], then we need to compute η (Λ(˜ s g ) ⊗ ˆ s g ) ∈ H ( G, Z )under the composition (3.2). By direct calculation, which we delete thedetails here, this element maps to the following element of H ( G, Z ): λ ( s g ) := [ˆ s g | ˆ s − g | ˆ s g ]+ P gi =1 (cid:0) [ˆ a − i | ˆ s − i − | ˆ s g ] − [ˆ a − i | ˆ s g | ˆ s − i − ] − [ˆ a − i | b − i ˆ s − i | ˆ s g ]+[ˆ a − i | ˆ s g | ˆ b − i ˆ s − i ]+[ˆ b − i | ˆ a − i ˆ s − i − | ˆ s g ] − [ˆ b − i | ˆ s g | ˆ a − i ˆ s − i − ] − [ˆ b − i | ˆ s − i | ˆ s g ]+[ˆ b − i | ˆ s g | ˆ s − i ]+[ˆ s g | ˆ a − i | ˆ s − i − ] − [ˆ s g | ˆ a − i | ˆ b − i ˆ s − i ]+[ˆ s g | ˆ b − i | ˆ a − i ˆ s − i − ] − [ˆ s g | ˆ b − i | ˆ s − i ] (cid:1) . Third cohomology of central perfect extensions
In this section we prove the cohomology analogue of 2.1, 3.2 and 3.3.For an abelian group M , let M ∗ be its dual group Hom Z ( M, Z ). Proposition 4.1.
Let A be a central subgroup of G and let A ⊆ G ′ .Then the map H ( G, Z ) → H ( A, Z ) is trivial.Proof. Let i : A → G be the usual inclusion map. We have seen at thebeginning of Section 2, that i ∗ : H ( A, Z ) → H ( G, Z ) is trivial and theimage of i ∗ : H ( A, Z ) → H ( G, Z ) is torsion. Thus i ∗ : H ( G, Z ) ∗ → H ( A, Z ) ∗ is trivial (because it factors through Hom(im( i ∗ ) , Z ) = 0).Now the claim follows from the commutative diagram0 Ext Z ( H ( G, Z ) , Z ) H ( G, Z ) H ( G, Z ) ∗
00 Ext Z ( H ( A, Z ) , Z ) H ( A, Z ) H ( A, Z ) ∗ , where the rows are universal coefficients sequences. (cid:3) HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 13
Proposition 4.2.
Let A G ։ Q be a perfect central extension. If Q is an H -group, then we have the exact sequence → Ext Z ( A, Z ) → H ( Q, Z ) → H ( G, Z ) ρ ∗ −→ ( A ⊗ Z H ( G, Z )) ∗ . In particular if the extension is universal we have the exact sequence → Ext Z ( A, Z ) → H ( Q, Z ) → H ( G, Z ) → . Proof.
By the universal coefficients theorem for the cohomology ofgroups and spaces we have H ( G, Z ) = 0 , H ( G, Z ) ≃ Hom( H ( G, Z ) , Z ) , H ( K ( A, , Z ) = 0 ,H ( K ( A, , Z ) ≃ Hom( A, Z ) , H ( K ( A, , Z ) ≃ Ext Z ( A, Z )and H ( K ( A, , Z ) ≃ Hom(Γ( A ) , Z ) . From the fibration K ( G, + → K ( Q, + → K ( A, , we obtain the Serre spectral sequence E p,q = H p ( K ( A, , H q ( G, Z )) ⇒ H p + q ( Q, Z ) . By a direct analysis of this spectral sequence we obtain the exact se-quence0 → Ext Z ( A, Z ) → H ( Q, Z ) → ker (cid:0) H ( G, Z ) → Hom(
A, H ( G, Z )) (cid:1) → Γ( A ) ∗ . SinceHom(
A, H ( G, Z )) ≃ Hom( A, Hom( H ( G, Z ) , Z )) ≃ ( A ⊗ Z H ( G, Z )) ∗ , we have the exact sequence0 → Ext Z ( A, Z ) → H ( Q, Z ) → ker (cid:0) H ( G, Z ) ρ ∗ −→ ( A ⊗ Z H ( G, Z )) ∗ (cid:1) → Γ( A ) ∗ . Now first let the extension is universal. Then the above extensionfinds the following form0 → Ext Z ( A, Z ) → H ( Q, Z ) → H ( G, Z ) → Γ( A ) ∗ . From the proof of Theorem 3.2 we see that the map Γ( A ) → H ( G, Z )factors throught A/ A ) / [ A, A ]. Thus H ( G, Z ) = H ( G, Z ) ∗ → Γ( A ) ∗ factors through ( A/ ∗ = 0, which implies that it is trivial In general the extension A G ։ Q is an epimorphic image ofa universal central extension of Q , say A G ։ Q , where A ≃ H ( Q, Z ). That is we have a morphism of extensions A G QA G Q. = This gives us the commutative diagram of exact sequences0 Ext Z ( A, Z ) H ( Q, Z ) e H ( G, Z ) Γ( A ) ∗ Z ( A , Z ) H ( Q, Z ) H ( G , Z ) Γ( A ) ∗ . = 0 where e H ( G, Z ) := ker (cid:0) H ( G, Z ) ρ ∗ −→ ( A ⊗ Z H ( G, Z )) ∗ (cid:1) . (Note thatsince A → A is surjective, Γ( A ) → Γ( A ) is surjective too. Thisimplies that the map Γ( A ) ∗ → Γ( A ) ∗ is injective.) Now from theabove diagram we see that the map e H ( G, Z ) → Γ( A ) ∗ is trivial. Thisproves our claim. (cid:3) References [1] Berrick, A. J. An Approach to Algebraic K-Theory. Research Notes in Math.No. 56, Pitman, London, 1982 1, 8[2] Berrick, A. J. Two functors from abelian groups to perfect groups. J. PureAppl. Algebra (1987), no. 1-3, 35–43 8[3] Berrick, A. J., Miller, C. F., III. Strongly torsion generated groups. Math.Proc. Cambridge Philos. Soc. (1992), no. 2, 219–229 7[4] Brown, K. S. Cohomology of Groups. Graduate Texts in Mathematics, 87.Springer-Verlag, New York, 1994 4, 11[5] Eilenberg, S., MacLane, S. On the groups H (Π , n ), II: Methods of computation.Ann. of Math. (1954), no. 1, 49–139 2, 4[6] Loday, J. L. K -th´eorie alg´ebrique et repr´esentations de groupes. Ann. Sci.´Ecole Norm. Sup. (4) (1976), no. 3, 309–377 7, 8[7] May, J. P., Ponto, K. More concise algebraic topology: Localization, comple-tion, and model categories. Chicago Lectures in Mathematics. University ofChicago Press, Chicago, IL, 2012 2, 8[8] Stammbach, U. Homology in group theory. Lecture Notes in Mathematics, Vol.359. Springer-Verlag, Berlin-New York, 1973. 6[9] Suslin, A. A. K of a field and the Bloch group. Proc. Steklov Inst. Math. (1991), no. 4, 217–239 5[10] Wagoner, J. Delooping classifying spaces in algebraic K-theory. Topology (1972), 349–370 8[11] Whitehead, J. H. C. A certain exact sequence. Ann. Math. (1950), 51–1102 HIRD HOMOLOGY OF PERFECT CENTRAL EXTENSIONS 15 [12] Whitehead, G. W. Elements of homotopy theory. Springer-Verlag. 1978 4, 8[13] Wojtkowiak, Z. Central extension and coverings, Publ. Sec. Mat. Univ.Aut´onoma Barcelona29