A note on whitehead's quadratic functor
Behrooz Mirzaii, Fatemeh Yeganeh Mokari, David M. Carbajal Ordinola
aa r X i v : . [ m a t h . K T ] J u l A NOTE ON WHITEHEAD’S QUADRATIC FUNCTOR
B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
Abstract.
For an abelian group A , we give a precise homologicaldescription of the kernel of the natural map Γ( A ) → A ⊗ Z A , γ ( a ) a ⊗ a , where Γ is whitehead’s quadratic functor from thecategory of abelian groups to itself. Introduction
Whitehead’s quadratic functor is an important functor, which firstappeared in the context of algebraic topology. This is a functor fromthe category of abelian groups to itself and usually is denoted by Γ.Most of important aspects of this functor is known and its has beengeneralized in various ways.For an abelian group A , we give a precise homological description ofthe kernel of the natural mapΓ( A ) → A ⊗ Z A, γ ( a ) a ⊗ a which it is known to be 2-torsion. The cokernel of this map is isomor-phism to H ( A, Z ), the second integral homology group of A . .In this short article we give a precise homological description of thekernel of the above map. As our main result we prove that we have theexact sequence0 → H (Σ ε , Tor Z ( ∞ A, ∞ A )) → Γ( A ) → A ⊗ Z A → H ( A, Z ) → , where ∞ A is the 2-power torsion subgroup of A , Σ := { id , σ ε } thesymmetric group with two elements and σ ε being the involution onTor Z ( ∞ A, ∞ A ) induced by the involution A × A → A × A , ( a, b ) ( b, a ).If A → B is a homomorphism of abelian groups, by B/A we meancoker( A → B ). For a group A , n A is the subgroup of n -torsion elementsof A . For prime p , p ∞ A is the p -power torsion subgroup of A .1. Whitehead’s quadratic functor
A function ψ : A → B of (additive) abelian groups is called a qua-dratic map if(1) for any a ∈ A , ψ ( a ) = ψ ( − a ), (2) the function A × A → B with ( a, b ) ψ ( a + b ) − ψ ( a ) − ψ ( b ) isbilinear.For any abelian group A , there is a universal quadratic map γ : A → Γ( A )such that for any quadratic map ψ : A → B , there is a unique grouphomomorphism Ψ : Γ( A ) → B such that Ψ ◦ γ = ψ . 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 and coker(Ψ) = A ∧ A ≃ H ( A, Z ). Furthermorewe 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 freeabelian 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 ).Using this properties one can show that for any nonnegative integer n , we have γ ( na ) = n γ ( a ) . It is known that the sequence A ⊗ Z A [ , ] −→ Γ( A ) Φ → A/ → , ] is generated by the elements of the form a ⊗ b − b ⊗ a , a, b ∈ A . Therefore we have the exact sequence(1.1) 0 → H (Ω , A ⊗ Z A ) [ , ] −→ Γ( A ) Φ → A/ → , where Ω := { id , ω } and ω is the involution ω ( a ⊗ b ) = b ⊗ a on A ⊗ Z A .It is easy to see that the composition A ⊗ Z A [ , ] −→ Γ( A ) Ψ → A ⊗ Z A HITEHEAD’S QUADRATIC FUNCTOR 3 takes a ⊗ b to a ⊗ b + b ⊗ a . Moreover the compositionΓ( A ) Ψ → A ⊗ Z A [ , ] −→ Γ( A )coincide with multiplication by 2. Thus ker(Ψ) is 2-torsion.To give a homological description of the kernel of ψ , we will need thefollowing fact. Proposition 1.1.
For any abelian group A , Γ( A ) ≃ H ( K ( A, , Z ) ,where K ( A, is the Eilenberg-Maclane space of type ( A, .Proof. See [3, Theorem 21.1] (cid:3) Tor-functor and third homology of abelian groups
Let A and B be abelian groups. For any positive integer n there isa natural homomorphism τ n : n A ⊗ Z n B → n Tor Z ( A, B ) . We denote the image of a ⊗ b , under τ n by τ n ( a, b ).For any pair of integers s and n such that n = sm , the maps τ n arerelated by the commutative diagrams n A ⊗ Z s B s A ⊗ Z s B n A ⊗ Z n B, n Tor Z ( A, B ) p m ⊗ id id ⊗ i m τ s τ n s A ⊗ Z n B s A ⊗ Z s B n A ⊗ Z n B, n Tor Z ( A, B ) id ⊗ p m i m ⊗ id τ s τ n in which i m : s A → n A and p m : n A → s A are the inclusion and themap induced by multiplication by m respectively. The commutativityof these diagrams expresses the relations τ n ( a, b ) = τ s ( ma, b ) , for a ∈ n A and b ∈ s B, and τ n ( a ′ , b ′ ) = τ s ( a ′ , mb ′ ) , for a ′ ∈ s A and b ′ ∈ n B. The following proposition is well-known [1, Proposition 3.5].
B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
Proposition 2.1.
The induced map τ : lim I ( n A ⊗ n B ) → Tor Z ( A, B ) ,where I is the inductive system of objects n A ⊗ Z n B determined by theabove diagrams for varying n , is an isomorphism. Let σ : A ⊗ B → B ⊗ A and σ : Tor Z ( A, B ) → Tor Z ( B, A ) beinduced by interchanging the groups A and B . It is well known thatthe diagram n A ⊗ Z n B n B ⊗ Z n A n Tor Z ( A, B ) n Tor Z ( B, A ) σ τ n τ ′ n − σ commutes. By passing to the inductive limit, the same is true for thediagram lim I ( n A ⊗ Z n B ) lim I ( n B ⊗ Z n A )Tor Z ( A, B ) Tor Z ( B, A ) . σ τ τ ′ − σ It is useful to observe that the map σ : Tor Z ( A, B ) → Tor Z ( B, A ) isindeed induced by the involution A ⊗ Z B → B ⊗ Z A given by a ⊗ b b ⊗ a and therefore − σ is induced by the involution a ⊗ b b ⊗ a Let Σ be the symmetric group of order 2. For an abelian group A ,Σ acts on A ⊗ Z A and Tor Z ( A, A ), through σ and σ . Let us denotethe symmetric group by Σ ε , rather than simply by Σ , when it acts onTor Z ( A, A ) as ( σ ε , x )
7→ − σ ( x ) . We need the following well-known lemma on the third homology ofabelian groups [5, Lemma 5.5], [1, Section 6].
Proposition 2.2.
For any abelian group A we have the exact sequence → V Z A → H ( A, Z ) → Tor Z ( A, A ) Σ ε → , where the right side homomorphism is obtained from the composition H ( A, Z ) ∆ A ∗ −→ H ( A × A, Z ) → Tor Z ( A, A ) , ∆ A being the diagonal map A → A × A , a ( a, a ) . HITEHEAD’S QUADRATIC FUNCTOR 5 The kernel of
Ψ : Γ( A ) → A ⊗ A We study the kernel of Ψ : Γ( A ) → A ⊗ Z A . If Θ = [ , ] : A ⊗ Z A → Γ( A ), then from the commutative diagram0 ker(Θ) A ⊗ Z A im(Θ) 00 ker(Ψ) Γ( A ) A ⊗ Z A ΘΘ γγ and exact sequence (1.1) we obtain the exact sequenceker(Ψ) → A/ δ → ( A ⊗ Z A ) Ω → H ( A, Z ) → , where ( A ⊗ Z A ) Ω = ( A ⊗ Z A ) / h a ⊗ b + b ⊗ a | a, b ∈ A i and δ ( a ) = a ⊗ a .But the sequence0 → A/ → ( A ⊗ Z A ) Ω → H ( A, Z ) → → A/ (cid:0) Γ( A ) Ψ −→ A ⊗ Z A (cid:1) ⊆ im (cid:0) A ⊗ Z A [ , ] −→ Γ( A ) (cid:1) . We give a precise description of the kernel of Ψ.
Theorem 3.1.
For any abelian group A , we have the exact sequence → H (Σ ε , Tor Z ( ∞ A, ∞ A )) → Γ( A ) Ψ → A ⊗ Z A → H ( A, Z ) → . Proof. If A B ։ C is an extension of abelian groups, then stan-dard classifying space theory gives a (homotopy theoretic) fibration ofEilenberg-MacLane spaces K ( A, → K ( B, → K ( C, K ( B, → K ( C, → K ( A, . For the group 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, . B. MIRZAII, F. Y. MOKARI, AND D. C. ORDINOLA
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 )) . By Proposition 2.2 we have the exact sequence0 → V Z A → H ( A, Z ) → Tor Z ( A, A ) Σ ε → . Clearly µ ∗ ( A ⊗ Z H ( A, Z )) ⊆ V Z A . Thereforeker(Ψ) ≃ Tor Z ( A, A ) Σ ε / (∆ A ◦ µ ) ∗ (Tor Z ( A, A )) . We prove that the map ∆ ◦ µ : A × A → A × A , which is given by( a, b ) ( ab, ab ), induces the mapid + σ ε : Tor Z ( A, A ) → Tor Z ( A, A ) . By studying the map (∆ ◦ µ ) ∗ : H ( A × A ) → H ( A × A ) using the factthat A ⊗ A ≃ H ( A × A ) / ( H ( A ) ⊕ H ( A )) (the K¨unneth Formula),one sees that ∆ ◦ µ induces the map A ⊗ A → A ⊗ A, a ⊗ b a ⊗ b − b ⊗ a, Thus to study the induced map on Tor Z ( A, A ) by ∆ ◦ µ we should studythe map induced on Tor Z ( A, A ) by the map A ⊗ A → A ⊗ A, a ⊗ b a ⊗ b + b ⊗ a = (id + ι )( a ⊗ b ) , where ι : A ⊗ A → A ⊗ A is given by a ⊗ b b ⊗ a . Let0 → F ∂ −→ F ǫ −→ A → A . Then the sequence0 → F ⊗ F ∂ −→ F ⊗ F ⊕ F ⊗ F ∂ −→ F ⊗ F → Z ( A, A ), where ∂ = ( ∂ ⊗ id F , − id F ⊗ ∂ ), ∂ = id F ⊗ ∂ + ∂ ⊗ id F . The map id + ι : A ⊗ A → A ⊗ A can beextended to the morphism of complexes0 −→ F ⊗ F F ⊗ F ⊕ F ⊗ F F ⊗ F −→ −→ F ⊗ F F ⊗ F ⊕ F ⊗ F F ⊗ F −→ , ∂ f ∂ f f ∂ ∂ HITEHEAD’S QUADRATIC FUNCTOR 7 where f ( x ⊗ y ) := x ⊗ y + y ⊗ x,f ( x ⊗ y, y ′ ⊗ x ′ ) := ( x ⊗ y + x ′ ⊗ y ′ , y ⊗ x + y ′ ⊗ x ′ ) ,f ( x ⊗ y ) := x ⊗ y − y ⊗ x. Since f ( x ⊗ y, y ′ ⊗ x ′ ) = ( x ⊗ y, y ′ ⊗ x ′ ) + ( x ′ ⊗ y ′ , y ⊗ x ) , ∆ ◦ µ induces the map id + σ ε : Tor Z ( A, A ) → Tor Z ( A, A ). Thereforeker(Ψ) ≃ Tor Z ( A, A ) Σ ε / (id + σ ε )(Tor Z ( A, A )) = H (Σ ε , Tor Z ( A, A )) . Finally since Tor Z ( A, A ) = Tor Z ( A T , A T ), A T being the subgroupof torsion elements of A , and since for any torsion abelian group B , B ≃ L p prime p ∞ B , we have the isomorphism H (Σ , Tor Z ( A, A )) ≃ H (Σ , Tor Z ( ∞ A, ∞ A )) . This completes the proof of the theorem. (cid:3)
Corollary 3.2.
For any abelian group A , we have the exact sequence → lim I H (Σ , n A ⊗ Z n A ) → Γ( A ) Ψ → A ⊗ Z A → H ( A, Z ) → . In particular if ∞ A is finite then we have the exact sequence → H (Σ , ∞ A ⊗ Z ∞ A ) → Γ( A ) Ψ → A ⊗ Z A → H ( A, Z ) → . Proof.
This follows from Theorem 3.1 and Proposition 2.1. (cid:3)
References [1] Breen, L.On the functorial homology of abelian groups. Journal of Pure andApplied Algebra (1999) 199–237. 3, 4[2] Brown, K. S. Cohomology of Groups. Graduate Texts in Mathematics, 87.Springer-Verlag, New York, 1994.[3] Eilenberg, S., MacLane, S. On the groups H (Π , n ), II: Methods of computation.Ann. of Math. (1954), no. 1, 49–139. 3[4] 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. 5[5] Suslin, A. A. K of a field and the Bloch group. Proc. Steklov Inst. Math.183