The symmetric action on secondary homotopy groups
Abstract
We show that the symmetric track groups Sym(n), which are extensions of the symmetric groups Sym(n) associated to the second Stiefel-Whitney class, act as crossed modules on the secondary homotopy groups of a pointed space.
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The symmetric action on secondary homotopy groups Hans-Joachim Baues Fernando Muro∗ Abstract We show that the symmetric track groups Sym(n), which are extensions of the symmetric groups Sym(n) associated to the second Stiefel-Whitney class, act as crossed modules on the secondary homotopy groups of a pointed space. Introduction Secondary homotopy operations like Toda brackets [Tod62] or cup-one products [BJM83], [HM93], are defined by pasting tracks, where tracks are homotopy classes of homotopies. Since secondary homotopy operations play a crucial role in homotopy theory it is of importance to develop the algebraic theory of tracks. We do this by introducing secondary homotopy groups of a pointed space X Πn,∗X=Πn,1X∂ →Πn,0X which have the structure of a quadratic pair module, see Section 1. Here ∂is a group homomorphism with cokernel πnXand kernel πn+1Xfor n≥3. We define Πn,∗Xfor n≥2 directly in terms of maps Sn→Xand tracks from such maps to the trivial map. For n≥0 the functor Πn,∗is an additive version of the ∗The second author was partially supported by the MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the MEC postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract. Received by the editors February 2007. Communicated by Y. F´elix. 1991 Mathematics Subject Classification : 55Q25, 55S45. Key words and phrases : secondary homotopy groups, crossed module, square group, cup-one product. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), 733–768
734 H.-J. Baues – F. Muro functor πn,∗studied in [BM08]. The homotopy category of (n−1)-connected (n+1)- types is equivalent via Πn,∗to the homotopy category of quadratic pair modules for n≥3. In this paper we consider the “generalized coefficients” of secondary homotopy groups Πn,∗Xobtained by the action of the symmetric group Sym(n) on Sn= S1∧n · · · ∧S1via permutation of coordinates. For a permutation σ∈Sym(n) the map σ:Sn→Snhas degree sign σ∈ {±1}. The group {±1}also acts on Snby using the topological abelian group structure of S1and suspending n−1 times. This shows that there are tracks σ⇒sign σwhich, by definition, are the elements of the symmetric track group Sym(n). Also these tracks act on Πn,∗X. We clarify this action by showing that the group Sym(n) gives rise to a crossed module which acts as a crossed module on the quadratic pair module Πn,∗X. The symmetric track group is a central extension Z/2֒→Sym(n)δ ։Sym(n) which, as we show, represents the second Stiefel-Whitney class pulled back to Sym(n). The symmetric track group is computed in Section 6. We actually compute a faithful positive pin representation of Sym(n) from which we derive a finite presentation of this group. This group also arose in a different way in the work of Schur [Sch11] and Serre [Ser84]. In [BM07] we describe the smash product operation on secondary homotopy groups Πn,∗X. This operation endows Π∗,∗with the structure of a lax symmetric monoidal functor where the crossed module action of Sym(n) on Πn,∗Xis of crucial importance. This leads to an algebraic approximation of the symmetric monoidal category of spectra by secondary homotopy groups, see [BM06]. As an example we prove a formula for the unstable cup-one product α ⌣1α∈π2n+1S2mof an element α∈πnSmwhere nand mare even. We show that 2(α ⌣1α) = n+m 2(α∧α)(Σ2(n−1)η) where η:S3→S2is the Hopf map. In the very special case when n/2 is odd and m/2 is even then this formula was achieved by totally different methods in [BJM83]. 1 Square groups and quadratic pair modules In this section we describe the algebraic concepts needed for the structure of secondary homotopy groups. Definition 1.1. Asquare group Xis a diagram X= (Xe P ⇆ H Xee) where Xeis a group with an additively written group law, Xee is an abelian group, Pis a homomorphism, His a function such that the cross effect (a|b)H=H(a+b)−H(b)−H(a) is linear in aand b∈Xe, and the following relations are satisfied for all x, y ∈Xee,
The symmetric action on secondary homotopy groups 735 1. (Px|b)H= 0, (a|Py) = 0, 2. P(a|b)H=−a−b+a+b, 3. PHP(x) = P(x) + P(x). These relations imply that the image of Pis central in Xe, and that Xeis a group of nilpotency class 2. The function T=HP −1: Xee −→ Xee is an involution, i. e. a homomorphism with T2= 1. A morphism of square groups f:X→Yis given by homomorphisms fe:Xe−→ Ye, fee :Xee −→ Yee, commuting with Pand H. Let SG be the category of square groups. A square group Xwith Xee = 0 is the same as an abelian group Xe. This yields the full inclusion of categories Ab ⊂SG where Ab is the category of abelian groups. Square groups were introduced in [BP99] to describe quadratic endofunctors of the category Gr of groups. More precisely, any square group Xgives rise to a quadratic functor − ⊗ X:Gr −→ Gr. Given a group Gthe group G⊗Xis generated by the symbols g⊗xand [g, h]⊗z, g, h ∈G,x∈Xe,z∈Xee subject to the relations (g+h)⊗x=g⊗x+h⊗x+ [g, h]⊗H(x), [g, g]⊗z=g⊗P(z), where g⊗xis linear in xand [g, h]⊗zis central and linear in each variable g, h, z. If Xis an abelian group then G⊗X=Gab ⊗Xe, where Gab is the abelianization of a group G. In fact, any quadratic functor F:Gr →Gr which preserves reflexive coequalizers and filtered colimits has the form F=− ⊗ X, see [BP99]. The theory of square groups is discussed in detail in [BJP05]. There is a natural isomorphism Xe∼ = −→ Z⊗X, x 7→ 1⊗x. In particular the homomorphism n:Z→Zinduces a homomorphism n∗:Xe→Xe fitting into the following commutative diagram Z⊗Xn⊗X// OO ∼ = Z⊗X OO ∼ = Xe n∗ //Xe (1.2)
736 H.-J. Baues – F. Muro The homomorphism n∗is explicitly given by the following formula, n∗x=n·x+ n 2!PH(x). Here we set n 2=n(n−1) 2and for any additively written group Gand any n∈Z, g∈G, n·g= g+n ···+g, if n≥0; −g−−n · · · −g, if n < 0. The function n·:G→Gin general is not a homomorphism, but if Gis abelian then n·is a homomorphism. This homomorphism is generalized by n∗in (1.2) for square groups. Definition 1.3. Aquadratic pair module Cis a morphism ∂:C(1) →C(0) between square groups C(0) = (C0 P0 ⇆ H Cee), C(1) = (C1 P ⇆ H1 Cee), such that ∂ee = 1: Cee →Cee is the identity homomorphism. In particular ∂is completely determined by the diagram Cee P }}{ { { { { { { { C1∂//C0 H aaC C C C C C C C (1.4) where ∂=∂e,H1=H∂ and P0=∂P. The homology of a quadratic pair module Cis given by the abelian groups h0C=C0/∂(C1),(1.5) h1C= Ker[∂:C1→C0]. Morphisms of quadratic pair modules f:C→Dare given by group homomorphisms f0:C0→D0,f1:C1→D1,fee :Cee →Dee, commuting with H,Pand ∂ in (1.4) as in the diagram C0 H// f0 Cee fee P//C1 f1 ∂//C0 f0 D0 H//Dee P//D1 ∂//D0 They form a category denoted by qpm. A morphism in qpm is said to be a weak equivalence if it induces isomorphisms in h0and h1. Quadratic pair modules are also the objects of a bigger category wqpm given by weak morphisms. A weak morphism f:C→Dbetween quadratic pair modules is
The symmetric action on secondary homotopy groups 737 given by three homomorphisms f0, f1, fee as above, but we only require the following two diagrams to be commutative Cee T// fee Cee fee Dee T//Dee ⊗2(C0)ab (−|−)H// ⊗2(f0)ab Cee fee P//C1 f1 ∂//C0 f0 ⊗2(D0)ab (−|−)H//Dee P//D1 ∂//D0 Here ⊗2A=A⊗Adenotes the tensor square of an abelian group. Therefore qpm ⊂wqpm is a subcategory with the same objects. Let (Z,·) be the multiplicative (abelian) monoid of the integers Z. Definition 1.6. Any quadratic pair module Cadmits an action of (Z,·) given by the morphisms n∗:C→Cin wqpm,n∈Z, defined by the equations •n∗x=n·x+n 2∂PH(x) for x∈C0, •n∗y=n·y+n 2PH∂(y) for y∈C1, •n∗z=n2zfor z∈Cee. We point out that n∗:C→Cis an example of a weak morphism which is not a morphism in qpm since n∗is not compatible with H. Notice that n∗:C0→C0 and n∗:C1→C1are induced by the square group morphisms n⊗C(0) and n⊗C(1) respectively, see diagram (1.2). We emphasize that this action is always defined for any quadratic pair module Cand it is natural in the following sense, for any morphism f:C→Din qpm and any n∈Z, the equality fn∗=n∗f holds. This property does not hold if fis a weak morphism. The existence of this action should be compared to the fact that abelian groups are Z-modules. The category squad of stable quadratic modules is described in [Bau91, IV.C] and [BM08]. Quadratic modules in general are discussed in [Bau91] and [Ell93], they are special 2-crossed modules in the sense of [Con84]. More precisely, a stable quadratic module Cis a diagram of group homomorphisms ⊗2(C0)ab ω −→ C1 ∂ −→ C0, such that given ci, di∈Ci,i= 0,1, ∂ω(c0⊗d0) = −c0−d0+c0+d0, ω(∂(c1)⊗∂(d1)) = −c1−d1+c1+d1, ω(c0⊗d0+d0⊗c0) = 0. Morphisms f:C→Din squad are given by homomorphisms fi:Ci→Di,i= 0,1, compatible with ωand ∂. There is a forgetful functor from quadratic pair modules and weak morphisms to stable quadratic modules wqpm −→ squad (1.7)
738 H.-J. Baues – F. Muro sending Cas in Definition 1.3 to the stable quadratic module ⊗2(C0)ab P(−|−)H −→ C1 ∂ −→ C0.(1.8) This functor is faithful over the full subcategory of quadratic pair modules such that the cross effect of His an isomorphism (−|−)H:⊗2(C0)ab ∼ =Cee. Atrack category is a groupoid-enriched category, which is also a 2-category where all 2-morphisms (also termed tracks) are vertically invertible. The vertical composition in track categories is denoted by , the vertical inverse of a track αis α⊟, and the trivial track from a morphism fto itself is 0:f⇒f. Remark 1.9.The category Top∗of pointed spaces is known to be a track category with tracks given by homotopy classes of homotopies. This track category has in addition a strict zero object ∗, which is an object such that the morphism groupoids from or to ∗are trivial, i.e. they consist of only one object and one morphism. In particular the zero morphism 0: X→Ybetween two objects X,Yis uniquely defined as the morphism which factor as X→ ∗ → Y. Moreover, in a track category with a strict zero object the following crucial fact holds: The horizontal composition of any track Hand a zero morphism is a trivial track H0 = 0, 0H= 0. The forgetful functor (1.7) can be used to pull-back to wqpm the track category structure on squad introduced in [BM08, 6]. The track structure on squad was already a pull-back along the forgetful functor squad −→ cross (1.10) from stable quadratic modules to crossed modules considered also in [BM08, 6]. Definition 1.11. We recall that a crossed module ∂:M→Nis a group homomorphism such that Nacts on the right of M(the action will be denoted exponentially) and the homomorphism ∂satisfies the following two properties (m, m′∈M, n ∈N): 1. ∂(mn) = −n+∂(m) + n, 2. m∂(m′)=−m′+m+m′. The crossed module associated via (1.7) and (1.10) to a quadratic pair module Cis given by the homomorphism ∂:C1−→ C0, where C0acts on the right of C1by the formula, x∈C1,y∈C0, xy=x+P(∂(x)|y)H.(1.12) Definition 1.13. Atrack α:f⇒gbetween two morphisms f, g :C→Din wqpm is a function α:C0−→ D1 satisfying the equations, x, y ∈C0,z∈C1, 1. α(x+y) = α(x)f0(y)+α(y),
The symmetric action on secondary homotopy groups 739 2. g0(x) = f0(x) + ∂α(x), 3. g1(z) = f1(z) + α∂(z). Tracks in qpm are tracks in wqpm between morphisms in the subcategory qpm ⊂ wqpm. Proposition 1.14. The categories wqpm and qpm are track categories with the tracks in Definition 1.13. This proposition is a direct consequence of [BM08, 6.4]. Vertical and horizontal compositions are defined in the proof of [BM08, 6.4]. The following result shows that the weak action of (Z,·) defined above is also natural with respect to tracks in qpm. Proposition 1.15. Let f, g :C→Dbe morphisms in qpm and let α:g⇒fbe a track as in Definition 1.13. Then the following diagram commutes C0 α// n∗ D1 n∗ C0 α//D1 Given a pointed set Ewith base point ∗ ∈ Ewe denote by hEinil and Z[E] the free group of nilpotency class 2 and the free abelian group generated by Ewith ∗= 0 respectively. More generally Gnil denotes the projection of a group Gto the variety of groups of nilpotency class 2. Definition 1.16. A quadratic pair module Cis said to be 0-free if C0=hEinil, Cee =⊗2Z[E] and His determined by the equalities H(e) = 0 for any e∈Eand (s|t)H=t⊗sfor any s, t ∈ hEinil. Notice that in this case the cross effect yields an isomorphism (−|−)H:⊗2(C0)ab ∼ =Cee. One can similarly define a 0-free stable quadratic module as a stable quadratic module whose lower-dimensional group is free of nilpotency class 2. No further conditions are required in this case. The next lemma shows that 0-free stable quadratic modules are in the image of the forgetful functor in (1.7). Lemma 1.17. Any 0-free stable quadratic module ⊗2Z[E]ω −→ M∂ −→ hEinil gives rise to a 0-free quadratic pair module ⊗2Z[E] P {{wwwwwwwww M∂//hEinil H ddJJJJJJJJJ with P(a⊗b) = ω(b⊗a).
740 H.-J. Baues – F. Muro Later we will need the following technical lemma which measures the lack of compatibility of certain tracks in wqpm with the action of (Z,·). Lemma 1.18. Let Cbe a 0-free quadratic pair module with C0=hEinil, let f:C0→ C0be an endomorphism induced by a pointed map E→E, and let α:C0→C1be a map satisfying α(x+y) = α(x)f(y)+α(y), m∗x=f(x) + ∂α(x), for some m∈Zand any x, y ∈C0. Then the following formula holds for any n∈Z and x∈C0. α(n∗x) = n∗α(x) + m 2! n 2!P(x|x)H. Proof. We first check that the lemma holds for x+yprovided it holds for x, y ∈C0. αn∗(x+y) = α(n∗x+n∗y) =α(n∗x)f(n∗y)+α(n∗y) =n∗α(x) + n∗α(y) + m 2! n 2!P(x|x)H + m 2! n 2!P(y|y)H+P(−f(n∗x) + n∗m∗x|f(n∗y))H =n∗(α(x) + α(y)) + n∗P(−f(x) + m∗x|fy)H + m 2! n 2!P(x+y|x+y)H =n∗(α(x)f(y)+α(y)) + m 2! n 2!P(x+y|x+y)H =n∗α(x+y) + m 2! n 2!P(x+y|x+y)H. Here we use that fis compatible with the action of (Z,·) and that P(x|x)His linear in x. We now check that the lemma holds for −xprovided it holds for x. For this we use that, by the first equation of the statement, α(−y) = −α(y)−f(y). αn∗(−x) = α(−n∗x) =−(αn∗x)−fn∗x =−n∗α(x)− m 2! n 2!P(x|x)H−P(−n∗f(x) + n∗m∗x| − fn∗x)H =−n∗α(x)− m 2! n 2!P(x|x)H−n∗P(−f(x) + m∗x| − fx)H =−n∗(α(x) + P(−f(x) + m∗x| − f(x))H) + m 2! n 2!P(−x| − x)H =−n∗α(x)−f(x)+ m 2! n 2!P(−x| − x)H =n∗α(−x) + m 2! n 2!P(−x| − x)H.
The symmetric action on secondary homotopy groups 741 Now since C0=hEinil we only need to check that the proposition holds for e∈E. But H(e) = 0, so we have n∗e=n·e. The equality α(n·e) = n·α(e) + n 2!P(f(e)|f(e))H+m n 2!P(e|f(e))H follows easily by induction in nfrom the first equation of the statement and the laws of a quadratic pair module. On the other hand n∗α(e) = n·α(e) + n 2!PH(−f(e) + m·e). One can also check by induction that PH(−f(e) + m·e) = P(f(e)|f(e))H+mP(e|f(e))H− m 2!P(e|e)H. Now the proof is finished. Lemma 1.18 holds under the more general condition that C0is generated by elements x∈C0with H(x) = 0 and Hf(x) = 0. 2 Homotopy groups and secondary homotopy groups Let Top∗be the category of (compactly generated) pointed spaces. Using classical homotopy groups πnXwe obtain for n≥0 the functor Πn:Top∗−→ Ab with ΠnX= πnX, n ≥2, (π1X)ab, n = 1, Z[π0X], n = 0, (2.1) termed additive homotopy groups. One readily checks that the smash product f∧g:Sn∧Sm−→ X∧Y of maps {f:Sn→X} ∈ πnXand {g:Sm→Y} ∈ πmYinduces a well-defined homomorphism ∧: ΠnX⊗ΠmY−→ Πn+m(X∧Y).(2.2) This homomorphism is symmetric in the sense that the interchange map τX,Y :X∧ Y→Y∧Xyields the equation in Πn+m(Y∧X) (τX,Y )∗(f∧g) = (−1)nmg∧f. (2.3) Here the sign (−1)nm is given by the interchange map τn,m =τSn,Sm:Sn+m−→ Sm+n(2.4) which has degree (−1)nm. Here τn,m also designates the corresponding element of the symmetric group Sym(n+m) which acts from the left on Sn+m, see Section 5 below. We want to generalize the smash product operator (2.2) for additive secondary homotopy groups.
748 H.-J. Baues – F. Muro 1. hx+y, ti=hx, tiδ(t)∗y+hy, ti, 2. ε(t)∗(x) = δ(t)∗(x) + ∂hx, ti, 3. ε(t)∗(z) = δ(t)∗(z) + h∂(z), ti, 4. hx, s ·ti=hδ(s)∗(x), ti+hε(t)∗x, si, 5. the ω-formula: hx, ωi=P(x|x)H. Notice that the ω-formula corresponds to the k-invariant, see [BM08, 8]. Remark 3.6.A sign group Ggives rise to a crossed module δ= (ε, δ): G−→ {±1} × G, where {±1} × Gacts on Gby the formula g(x,h)=¯ h−1g¯ hı ε(g)(x·ε(h) 2). Here g∈G,x∈ {±1},h∈Gand ¯ h∈Gis any element with δ(¯ h) = h. This action is well defined since Gis a central extension of Gby {±1}. Lemma 3.7. The sign group action in Definition 3.5 corresponds to an action of the crossed module δon Cin the sense of Definition 3.4 such that {±1}acts on Cby the action of (Z,·)in Definition 1.6, Gacts by morphisms in qpm, and the ω-formula holds. The correspondence is given by the formula hhx, tii =hε(t)∗x, ti, x ∈C0, t ∈G. The proof of this lemma is straightforward. We just want to point out that Definition 3.4 (5) follows in this case from Definition 3.5 (4), (5), and Lemma 1.18. Remark 3.8.A sign group Gis trivial if Gis a trivial group. Notice that a trivial sign group acts on any quadratic pair module in a unique way. The main examples of sign groups considered in this paper are the symmetric track groups in Section 5, which act on the additive secondary homotopy groups. 4 The action of End(Sn)on Πn,∗X Let Snbe the n-sphere and let End∗(Sn) = ΩnSnbe the topological monoid of maps Sn→Snin Top∗. Then the fundamental groupoid of End∗(Sn), denoted by π0,∗End∗(Sn), is a monoid-groupoid in the sense of Definition 3.1. This monoidgroupoid coincides with the endomorphism monoid-groupoid of Snin the track category Top∗. It is well known that the monoid of path components of End∗(Sn) coincides with the multiplicative monoid (Z,·). We now consider the right action of π0,∗End∗(Sn) on Πn,∗Xfor n≥2. That is, we define for each pointed map f:Sn→Snan induced map in qpm f∗: Πn,∗X−→ Πn,∗Y
The symmetric action on secondary homotopy groups 749 and we define for each track H:f⇒gwith f, g :Sn→Sna track in qpm H∗:f∗⇒g∗. This yields a right action of the monoid-groupoid π0,∗End∗(Sn) on the secondary homotopy group Πn,∗Xin the track category qpm of quadratic pair modules in the sense of Definition 3.2. Theorem 4.1. Let Xbe a pointed space. For any n≥2there is a natural action of the monoid-groupoid π0,∗End∗(Sn)on the quadratic pair module Πn,∗X. The rest of this section is devoted to the proof of this theorem, which is carried out in several steps. The discrete monoid π0,0End∗(Sn), which is the underlying set of the topological monoid End∗(Sn), acts on the right of the pointed discrete set ΩnXof pointed maps Sn→Xby precomposition, i. e. given f:Sn→Snthe induced endomorphism is f∗: ΩnX−→ ΩnX, f∗(g) = gf. This induces a right action of π0,0End∗(Sn) on the free group πn,∗X=hΩnXinil of nilpotency class 2 which will be denoted in the same way. In order to extend this action to Πn,1Xwe consider the submonoid ˜π0,1End∗(Sn)⊂π0,1End∗(Sn) (4.2) of the monoid π0,1End∗(Sn) of morphisms in π0,∗End∗(Sn) given by tracks between self-maps of Snof the form γ:f⇒Σn−1(·)deg f= (·)deg f n.(4.3) Here deg f∈Zdenotes the degree of f:Sn→Snand for k∈Z (·)k:S1−→ S1:z7→ zk is given by the (multiplicative) topological abelian group structure of S1. We need a bracket operation h−,−i: Πn,0Xטπ0,1End∗(Sn)−→ Πn,1X, (4.4) defined as follows. Let x∈πn,0X=hΩnXinil and γ:f⇒(·)deg f nin ˜π0,1End∗(Sn). We choose maps ˜x:S1→ ∨ΩnXS1,ǫ:S1→S1∨S1with (π1˜x)nil(1) = xand (π1ǫ)nil =−a+b∈π1(S1∨S1)nil, the difference between the inclusion of the first and the second factor of S1∨S1. Then hx, γi ∈ Πn,1Xis the element represented by the map S1ǫ//S1∨S1˜x∨˜x//(∨ΩnXS1)∨(∨ΩnXS1)(Σf∗,∨ΩnX(·)deg f)//∨ΩnXS1 and the track Sn˜x//Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 33 Σn−1ǫ //Sn∨Sn (1,1) OO Σn−1(˜x∨˜x) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) 77 o o o o o o o o o o o o o o o o o o o o o o o o o o o o (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X N ]eDDDDDDDDD DDDDDDDDD (∨ΩnXγ,0) T\00000 00000 (4.5)
750 H.-J. Baues – F. Muro Here Nis a nil-track. The main properties of the bracket operation in (4.4) are listed in the following proposition. Proposition 4.6. The bracket h−,−i in (4.4) satisfies the following formulas for any x, y ∈Πn,0Xand γ:f⇒(·)deg f n, δ:g⇒(·)deg g nin ˜π0,1End∗(Sn). 1. hx+y, γi=hx, γif∗y+hy, γi, 2. (deg f)∗x=f∗x+∂hx, γi, 3. hx, γδi=hf∗x, δi+h(deg g)∗x, γi, 4. if ω: 1Sn⇒1Snis a track with 06=Hopf (ω)∈ˆ ⊗2Z=Z/2then hx, ωi= P(x|x)H. Moreover, this bracket operation is natural in X. Proof. With the notation in [BM08, 7.4] we have hx, γi=r(ev(∨ΩnXγ)(Σn−1˜x)) for the track ev(∨ΩnXγ)(Σn−1˜x): ev(Σnf∗)(Σn−1˜x) = ev(∨ΩnXf)(Σn−1˜x)⇒ev(∨ΩnX(·)deg f n)(Σn−1˜x), therefore (1) and (2) follow from [BM08, 7.6 and 7.5 (2)]. It is easy to see that the formula ev(∨ΩnXγδ) = (ev(∨ΩnXγ)(∨ΩnX(·)deg g n))(ev(∨ΩnXδ)(Σnf∗)) holds, therefore (3) follows from [BM08, 7.5 (3)]. If we evaluate hx, −i at ωthen the composite track obtained from (4.5) by going from the lower left Snto the upper right Sn Xhas the same reduced Hopf invariant as the track from Snto Sn Xin (2.8). Indeed the formula for both reduced Hopf invariants is (c) in the proof of Proposition 4.8. Therefore (4) follows. The next result follows from the algebraic properties of the bracket (4.4) which are proved in the previous proposition together with Lemma 1.18. Proposition 4.7. The monoid ˜π0,1End∗(Sn)acts on the right of Πn,1Xby the following formula, n≥2: given x∈Πn,1Xand γ:f⇒(·)deg f n γ∗x= (deg f)∗x− h∂(x), γi. This action satisfies ∂γ∗=f∗∂,γ∗P=P(⊗2f∗ ab), and Hf∗= (⊗2f∗ ab)H, therefore it defines an action of ˜π0,1End∗(Sn)on the right of the quadratic pair module Πn,∗X in the category qpm. This action is natural in X. Proof. The equality Hf∗= (⊗2f∗ ab)Hfollows from the fact that the endomorphism f∗carries generators to generators in hΩnXinil. The equality ∂γ∗=f∗∂follows
The symmetric action on secondary homotopy groups 751 from Proposition 4.6 (2). Let us check γ∗P=P(⊗2f∗ ab). Given a, b ∈ΩnX γ∗P(a⊗b) = (deg f)∗P(a⊗b)− h−a−b+a+b, γi =P(deg f)2(a⊗b)− hb, γi − ha, γi+hb, γi+ha, γi +P(−f∗(a) + (deg f)∗a|f∗b)H−P(−f∗(b) + (deg f)∗b|f∗a)H =P(deg f)2(a⊗b) + P(∂hb, γi|∂ha, γi)H +P(−f∗(a) + (deg f)∗a|f∗b)H−P(−f∗(b) + (deg f)∗b|f∗a)H =−P(deg f)2(a|b)H −P(−f∗(a) + (deg f)∗a| − f∗(b) + (deg f)∗b)H +P(−f∗(a) + (deg f)∗a|f∗b)H+P(f∗a| − f∗(b) + (deg f)∗b)H =−P(f∗(a)|f∗(b))H =P(f∗(b)|f∗(a))H =P(f∗(a)⊗f∗(b)). Here we use Proposition 4.6 (1) and (2) and the fact that H(a) = 0 = H(b). Finally given δ:g⇒(·)deg g n γ∗δ∗(x) = γ∗((deg g)∗x− h∂(x), δi) = (deg f)∗(deg g)∗x−(deg f)∗h∂(x), δi − hg∗∂(x), γi = ((deg f)(deg g))∗x− h(deg f)∗∂(x), δi − hg∗∂(x), γi + deg f 2! deg g 2!P(∂(x)|∂(x))H = (deg fg)∗x− h∂(x), γδi = (γδ)∗(x). Here we use Proposition 4.6 (1), (2) and (3), Lemma 1.18 and the fact that P(∂(x)|∂(x))H=−x−x+x+x= 0. Proposition 4.8. For n≥2the right action of the monoid ˜π0,1End∗(Sn)on the group Πn,1Xgiven by Proposition 4.7 factors through the boundary homomorphism q: ˜π0,1End∗(Sn)։π0,0End∗(Sn), q(γ:f⇒(·)deg f n) = f, that is, the homomorphism γ∗=f∗only depends on the boundary q(γ) = f. Proof. Let γ:f⇒(·)deg f nbe any element in ˜π0,1End∗(Sn) and let δ: (·)deg f n⇒ (·)deg f nbe any track. We know that all elements in q−1(f) are of the form δγ, therefore we only have to check that for any [g, G]∈Πn,1X γ∗[g, G] = (δγ)∗[g, G], or equivalently h∂[g, G], γi=h∂[g, G], δγi.
752 H.-J. Baues – F. Muro By definition of the bracket h−,−i in (4.4), see diagram (4.5), the element h∂[g, G], δγiis represented by the following diagram SnΣn−1g//Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 ;; Σn−1ǫ //Sn∨Sn (1,1) OO Σn−1(g∨g) //Sn X∨Sn X (1,1) OO55 (∨ΩnXf,∨ΩnX(·)deg f n) KK (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X N X`88888888888 88888888888 (∨ΩnXγ,0) ]eD D D D D D D D D D D D (∨ΩnXδ,0)RZ,,,,(a) Let us now pay special attention to the following subdiagram of (a) SnΣn−1g//Sn X∨ΩnX(·)deg f n//Sn X Sn 0 ;; Σn−1ǫ //Sn∨Sn (1,1) OO Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnX(·)deg f n,∨ΩnX(·)deg f n) 55 (∨ΩnXδ,0)RZ,,,, N X`88888888888 88888888888 (b) This is a composite track, termed (b), between (n−1)-fold suspensions. By the elementary properties of the Hopf invariant for tracks [BM08, 3] the reduced Hopf invariant of (b) is trivial provided Hopf (δ) = 0. Again by [BM08, 3] if 0 6=Hopf (δ)∈ ˆ ⊗2Z=Z/2 then the reduced Hopf invariant of (b) is given by the formula below. For the formula we need to fix a notation for the linear expansion of (π1g)ab(1) ∈ π1(∨ΩnXS1)ab =Z[ΩnX] in terms of generators ai∈ΩnXand ni∈Z, (π1g)ab(1) = Pk i=0 niai. Hopf (b) = k X i=0 niaiˆ ⊗ai∈ˆ ⊗2Z[ΩnX].(c) By using once again the elementary properties of the Hopf invariant for tracks described in [BM08, 3] the reader can easily check that for any track Q: (∨ΩnX(·)deg f n)(Σn−1g)⇒(Σn−1g)(·)deg f n
The symmetric action on secondary homotopy groups 753 the following composite track has the same reduced Hopf invariant as (b) Sn Σn−1g 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Sn∨Sn ((·)deg f n,(·)deg f n) 55 ((·)deg f n,(·)deg f n) GG Σn−1(g,g) //Sn X∨ΩnX(·)deg f n//Sn X Sn 0 33 Σn−1ǫ //Sn∨Sn Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (Q,Q) QY******* ******* N V^555555555555 555555555555 (δ,0) X`8 8888 88 888 (d) Since (b) and (d) have the same reduced Hopf invariant then we can replace subdiagram (b) in (a) by (d), obtaining the same element in Πn,1X, namely Sn Σn−1g 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Sn∨Sn ((·)deg f n,(·)deg f n) 55 ((·)deg f n,(·)deg f n) GG Σn−1(g,g) //Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 33 Σn−1ǫ //Sn∨Sn Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) == { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X (Q,Q) QY******* ******* N V^555555555555 555555555555 (δ,0) X`88888 88888 (∨ΩnXγ,0) OW&&&&&& &&&&&& (e)
754 H.-J. Baues – F. Muro Pasting the trivial track G0 (see Remark 1.9) to (e) gives Sn 0 Σn−1g 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Sn∨Sn ((·)deg f n,(·)deg f n) 55 ((·)deg f n,(·)deg f n) GG Σn−1(g,g) //Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 33 Σn−1ǫ //Sn∨Sn Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) == { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X (Q,Q) QY******* ******* N V^555555555555 555555555555 (δ,0) X`8888 8 88888 (∨ΩnXγ,0) OW&&&&&& &&&&&& G-5 b b b b bb b b b b (f) Again by Remark 1.9 we can remove some trivial tracks from (f). Sn 0 Σn−1g 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Sn∨Sn ((·)deg f n,(·)deg f n) GG Σn−1(g,g) //Sn X∨ΩnX(·)deg f n//Sn X ev Sn Σn−1ǫ //Sn∨Sn Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) == { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X (Q,Q) QY******* ******* (∨ΩnXγ,0) OW&&&&&& &&&&&& G-5 b b b b bb b b b b (g) Pasting another trivial track to (g) and factoring some maps and tracks through
The symmetric action on secondary homotopy groups 755 (1,1): Sn∨Sn→Snwe obtain Sn 0 Σn−1g 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Sn (·)deg f n { { { { { { { { { { { { { { { { { { { { { { == { { { { { { { Σn−1g//Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 ;; Σn−1ǫ //Sn∨Sn (1,1) OO Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) == { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X Q S[////////// ////////// (∨ΩnXγ,0) OW&&&&&& &&&&&& G-5 b b b b bb b b b b N X`88888888888 88888888888 (h) Finally removing trivial tracks from (h) gives SnΣn−1g//Sn X∨ΩnX(·)deg f n//Sn X ev Sn 0 ;; Σn−1ǫ //Sn∨Sn (1,1) OO Σn−1(g∨g) //Sn X∨Sn X (1,1) OO (∨ΩnXf,∨ΩnX(·)deg f n) ?? ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (Σnf∗,∨ΩnX(·)deg f n) //Sn Xev //X (∨ΩnXγ,0) NV$$$$$$ $$$$$$ N X`88888888888 88888888888 (i) Notice that this last composite track (i) represents h∂[g, G], γi, see (4.5), hence we are done. The next corollary follows from the two previous propositions. Corollary 4.9. For any pointed space Xand n≥2the monoid π0,0End∗(Sn)acts on the right of the quadratic pair module Πn,∗X. This action is natural in X. Now Theorem 4.1 is a consequence of the next result. Proposition 4.10. The action of π0,0End∗(Sn)on the right of Πn,∗Xgiven by Corollary 4.9 extends to an action of the whole monoid-groupoid π0,∗End∗(Sn),n≥ 2. Proof. A morphism Hin π0,∗End∗(Sn) is a track H:f⇒gbetween maps f, g :Sn→ Sn, in particular deg f= deg g=k∈Z. In order to define a track H∗:f∗⇒g∗
756 H.-J. Baues – F. Muro between the quadratic pair module morphisms f∗, g∗: Πn,∗X−→ Πn,∗X we choose tracks in ˜π0,1End∗(Sn) α:f⇒(·)k n, β:g⇒(·)k n, such that H=β⊟α. By Proposition 4.6 the maps h−, αi,h−, βi: Πn,0X−→ Πn,1X are tracks h−, αi:f∗⇒k∗, h−, βi:g∗⇒k∗, in the category wqpm, therefore we can define H∗as the vertical composition H∗=h−, βi⊟h−, αi, i. e. H∗is the map H∗: Πn,0X−→ Πn,1X defined by H∗(x) = hx, αi − hx, βi. By the proof of Proposition 4.6 and by [BM08, 7.5 (3)] the element H(x) coincides with r(ev(∨ΩnXH)(Σn−1˜x)) for ˜x:S1→ ∨ΩnXS1any map with (π1˜x)nil(1) = xin the sense of [BM08, 7.4]. The reader can now use the properties of the bracket (4.4) described in Proposition 4.6 together with [BM08, 7.5 (3)] to check that this yields a monoid-groupoid action. Later we will consider the quotient monoid ¯π0,1End∗(S2) of ˜π0,1End∗(S2) defined as follows: two elements γ:f⇒(·)deg f 2, ¯γ:g⇒(·)deg g 2in ˜π0,1End∗(S2) represent the same element in ¯π0,1End∗(S2) provided deg f= deg gand 0 = Hopf (¯γγ⊟)∈ˆ ⊗2Z=Z/2. Proposition 4.11. The bracket operation (4.4) factors for n= 2 through the natural projection ˜π0,1End∗(S2)։¯π0,1End∗(S2). h−,−i: Π2,0Xׯπ0,1End∗(S2)−→ Π2,1X. Proof. Two tracks γand ¯γin ˜π0,1End∗(S2) represent the same element in ¯π0,1End∗(S2) if and only if ¯γ=δγfor some δ: (·)k 2⇒(·)k 2with Hopf (δ) = 0, so we only need to check that hx, γi=hx, δγi. The element hx, δγiis represented by diagram (a) in the proof of Proposition 4.8 where we assume that gis a map with (π1g)nil(1) = x. As we mention in that proof diagram (b) is a nil-track in these circumstances, therefore we can drop δfrom (a) and still obtain the same element in Π2,1X. But if we drop δwe obtain hx, γi, hence we are done.
The symmetric action on secondary homotopy groups 757 5 The symmetric action on secondary homotopy groups The permutation of coordinates in Sn=S1∧ · · · ∧ S1induces a left action of the symmetric group Sym(n) on the n-sphere Sn. This action induces a monoid inclusion Sym(n)⊂π0,0End∗(Sn).(5.1) We define the symmetric track group for n≥3 Sym(n)⊂˜π0,1End∗(Sn) as the submonoid of tracks of the from α:σ⇒(·)sign(σ) n, where σ∈Sym(n) and sign(σ)∈ {±1}is the sign of the permutation. Compare the notation in (1.7) and (4.3). The submonoid defined as above for n= 2 will be called the extended symmetric track group Sym(2) ⊂˜π0,1End∗(S2). For n= 2 the symmetric track group Sym(2) is the image of Sym(2) by the natural projection ˜π0,1End∗(S2)։¯π0,1End∗(S2) in Proposition 4.11. Sym(2) ⊂¯π0,1End∗(S2).(5.2) Proposition 5.3. The symmetric track group is indeed a group. Moreover, it fits into a central extension, n≥2, Z/2֒→Sym(n)δ ։Sym(n) with δ(α) = σ, which splits if and only if n= 2 or 3. This proposition follows from Corollary 6.9 and Remarks 6.10 and 6.12 below. For n= 0 and n= 1 we define Sym(n) to be the trivial group, and Sym(n) the trivial sign group. Then the symmetric track group Sym(n) is a sign group (n≥0) {±1}֒→Sym(n)δ ։Sym(n)sign −→ {±1} as in Definition 3.5. Theorem 5.4. Let Xbe a pointed space. For n≥0the symmetric group Sym(n) acts naturally on the right of the additive secondary homotopy group Πn,∗Xin the category qpm of quadratic pair modules. Moreover, the restriction h−,−i: Πn,0X×Sym(n)−→ Πn,1X of the bracket defined in (4.4) if n≥3and in Proposition 4.11 if n= 2 yields a natural right action of the sign group Sym(n)on Πn,∗Xin the sense of Definition 3.5.
764 H.-J. Baues – F. Muro Remark 6.10.The low-dimensional mod 2 cohomology groups of symmetric groups Sym(n) are as follows, n≥3, H1(Sym(n),Z/2) = Z/2χ⊕Z/2i∗w1,for n= 3; Z/2i∗w1,for n > 3; H2(Sym(n),Z/2) = Z/2i∗w2 1,for n= 3; Z/2i∗w2 1⊕Z/2i∗w2,for n > 3. Here we write wj∈Hj(BO(n),Z/2) for the jth Stiefel-Whitney class, j= 1,2. The pull-back i∗w1corresponds to the sign homomorphism i∗w1= sign: Sym(n)−→ {±1}∼ =Z/2, The pull-back of the second Stiefel-Whitney class is trivial for n= 3, therefore Sym(3) is a split extension of Sym(3) by Z/2, and χ: Sym(3) ։Z/2 is a retraction. The following structure theorem follows from Corollary 6.9. Theorem 6.11. The symmetric track group Sym(n)is the subgroup of Pin+(n) formed by the units x∈C+(n)such that for any 1≤i≤nthere exists 1≤σ(i)≤n with −xeix−1=eσ(i). The boundary homomorphism δ: Sym(n)։Sym(n)sends xabove to the permutation δ(x) = σ. The group Sym(n)has a presentation given by generators ω,ti,1≤i≤n−1, and relations t2 1= 1 for 1≤i≤n−1, (titi+1)3= 1 for 1≤i≤n−2, ω2= 1, tiω=ωtifor 1≤i≤n−1, titj=ωtjtifor 1≤i < j −1≤n−1; with ω7→ −1and ti7→ 1 √2(ei−ei+1). In particular δ(ω) = 0 and δ(ti) = (i i + 1). This is a group considered by Schur in [Sch11] and by Serre in [Ser84]. Remark 6.12.In case n= 2 we have O(2) = {±1}⋉S1with {±1}acting on S1exponentially, e O(2) = {±1}⋉ R with {±1}acting on Rmultiplicatively, and the projection q:e O(2) ։O(2) defined as in (6.4) is the identity in {±1}and the exponential map in the second coordinate R։S1:x7→ exp(2πix). In particular we have an abelian extension Z֒→e O(2) q ։O(2). The induced action of O(2) on Zis given by the determinant det: O(2) ։{±1}. By Remark 6.2 the extended symmetric track group Sym(2) is the pull-back of i: Sym(2) ⊂O(2) along q, therefore we have an abelian extension Z֒→Sym(2) q ։Sym(2),(6.13)
The symmetric action on secondary homotopy groups 765 where Sym(2) acts on Zby the unique isomorphism Sym(2) ∼ ={±1}. Now the symmetric track group Sym(2) can be identified with the push-forward of the extension (6.13) along the natural projection Z։Z/2, therefore we get a central extension Z/2֒→Sym(2) q ։Sym(2).(6.14) The cohomology group H2(Sym(2),Z) = 0 is trivial, so (6.13) is a split extension and Sym(2) ∼ =Sym(2) ⋉ Z is a semidirect product. Moreover (6.14) is also split since it is the push-forward of (6.13), therefore Sym(2) ∼ =Sym(2) ×Z/2 is a product. 7 An application to the cup-one product Let n≥m > 1 be even integers. The cup-one product operation πnSm−→ π2n+1S2m:α7→ α ⌣1α is defined in the following way, compare [HM93, 2.2.1]. Let kbe any positive integer and let τk∈Sym(2k) be the permutation exchanging the first and the second block of kelements in {1,...,2k}. If kis even then sign τk= 1. We choose for any even integer k > 1 a track ˆτk:τk⇒1S2kin Sym(2k). Consider the following diagram in the track category Top∗of pointed spaces where a:Sn→Smrepresents α. S2na∧a// τn 1S2n (( S2m τm 1S2m vv S2na∧a//S2m ˆτ⊟ n+3ˆτm+3 (7.1) By pasting this diagram we obtain a self-track of a∧a (ˆτm(a∧a))((a∧a)ˆτ⊟ n): a∧a⇒a∧a. (7.2) The set of self-tracks a∧a⇒a∧ais the automorphism group of the map a∧a in the track category Top∗. The element α ⌣1α∈π2n+1S2mis given by the track (7.2) via the well-known Barcus-Barratt-Rutter isomorphism Aut(a∧a)∼ =π2n+1S2m, see [BB58], [Rut67] and also [Bau91, VI.3.12] and [BJ01] for further details. The following theorem generalizes [BJM83, 6.5]. Theorem 7.3. The formula 2(α ⌣1α) = n+m 2(α∧α)(Σ2(n−1)η) holds, where η:S3→S2is the Hopf map. The proof of Theorem 7.3 is based on the following lemma. Lemma 7.4. The following formula holds in Sym(2k) ˆτ2 k=ω(k 2).
766 H.-J. Baues – F. Muro Proof. Here we use the representation of Sym(2k) in Pin+(2k) given in Theorem 6.11 and the relations (1) and (2) in the definition of the Clifford algebra C+(2k), see Definition 6.6. The permutation τkcan be expressed as a product of transpositions as follows τk= (1 k)(2 k+ 1) ···(k−1 2k−1)(k2k). The element 1 √2(ei−ei+k)∈S2k−1⊂Pin+(2k) acts on R2k(with coordinates ei, 1 ≤i≤2k) by reflection along the hyperplane orthogonal to 1 √2(ei−ei+k), see Definition 6.6. This hyperplane is ei=ei+k, therefore the action of 1 √2(ei−ei+k) on R2kinterchanges the coordinates in eiand ei+kand preserves all the other ones. Now by Theorem 6.11 1 √2(ei−ei+k) lies in the positive pin representation of Sym(2k) and δ1 √2(ei−ei+k)= (i i +k), so ˆτk=±1 2k 2 (e1−ek+1)(e2−ek+2)···(ek−1−e2k−1)(ek−e2k). The following equalities hold in the Clifford algebra C+(2k), see the defining relations in Definition 6.6, i6=j,i6=j+k,i+k6=j,k > 0, (ei−ei+k)2=e2 i−eiei+k−ei+kei+e2 i+k = 1 −eiei+k+eiei+k+ 1 = 2, (ei−ei+k)(ej−ej+k) = eiej−eiej+k−ei+kej+ei+kej+k =−ejei+ej+kei+ejei+k−ej+kei+k =−(ej−ej+k)(ei−ei+k). Hence we observe that ˆτ2 k=1 2k 2 1 2k 2 (−1)k−12(−1)k−22···(−1)12(−1)02 = (−1)k−1(−1)k−2···(−1)1(−1)0 = (−1)(k 2). The proof is now finished. Proof of Theorem 7.3. The element 2(α ⌣1α) corresponds to the pasting of the following diagram S2na∧a// τn 1S2n (( S2m τm 1S2m vv S2na∧a// τn 1S2n (( S2m τm 1S2m vv S2na∧a//S2m ˆτ⊟ n+3ˆτm+3 ˆτ⊟ n+3ˆτm+3
The symmetric action on secondary homotopy groups 767 By Lemma 7.4 and using that nand mare even this composite track coincides with S2n 1S2n ## 1S2n <<S2na∧a//S2m 1S2m ## 1S2m << S2m ωn 2 ωm 2 therefore 2(α ⌣1α) corresponds to the self-track ((ωm 2)(a∧a))((a∧a)(ωn 2)).(a) The self-track ωm 2(a∧a) corresponds to the homotopy class m 2(Σ2(m−1)η)(Σ(α∧α)).(b) Since Σ(α∧α) = ±(Σm+1α)(Σn+1α) which is a composite of two triple suspensions (b) is (α∧α)m 2(Σ2(n−1)η).(c) Moreover, the self-track (a∧a)(ωn 2) corresponds to (α∧α)n 2(Σ2(n−1)η),(d) so the self-track (a) corresponds to the sum of (c) and (d) (α∧α)n+m 2(Σ2(n−1)η). References [Bau91] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Walter de Gruyter, Berlin, 1991. [BB58] W. D. Barcus and M. G. Barratt, On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc. 88 (1958), 57–74. [BJ01] H.-J. Baues and M. Jibladze, Suspension and loop objects and representability of tracks, Georgian Math. J. 8(2001), no. 4, 683–696. [BJM83] M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, The Kervaire invariant problem, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) (Providence, RI), Contemp. Math., vol. 19, Amer. Math. Soc., 1983, pp. 9–22. [BJP05] H.-J. Baues, M. Jibladze, and T. Pirashvili, Quadratic algebra of square groups, Advances in Math. 217 (2008), 1236-1300. [BM06] H.-J. Baues and F. Muro, The algebra of secondary homotopy operations in ring spectra, Preprint of the Max-Planck-Institut f¨ur Mathematik MPIM2006-137, arXiv:math.AT/0610523, 2006.
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