Secondary homotopy groups

arXiv:math/0604029v1 [math.AT] 3 Apr 2006 SECONDARY HOMOTOPY GROUPS HANS-JOACHIM BAUES AND FERNANDO MURO Abstract. Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n−1)-connected (n+ 1)-types for n≥0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secondary operations termed Toda brackets. Such bracket operations are defined by pasting tracks where a track is a homotopy class of homotopies. Since Toda brackets play a crucial role in homotopy theory it seems feasible to investigate the algebraic nature of tracks. Therefore we shift focus from homotopy groups πnX to secondary homotopy groups πn,∗X= (πn,1X∂ −→ πn,0X) defined in this paper. Here ∂is a homomorphism of groups with Coker∂=πnX and Ker ∂=πn+1X. The groups πn,0Xand πn,1Xare defined directly by use of continuous maps f:Sn→Xand tracks of such maps to the trivial map, so that πn,∗Xis actually a functor in X. For n≥2 the definition involves the new concept of Hopf invariant for tracks. We show that the homomorphism ∂has additional algebraic structure, namely π1,∗Xis a crossed module, π2,∗Xis a reduced quadratic module and πn,∗X,n≥3, is a stable quadratic module. Crossed modules were introduced by J. H. C. Whitehead in [Whi49] and, in fact, for a CW-complex Xour secondary homotopy group π1,∗Xis weakly equivalent to the crossed module π2(X, X1)−→ π1X1 studied by [Whi49]. Similarly πn,∗Xfor n≥2 is weakly equivalent to the quadratic modules obtained in [Bau91] in terms of the cell structure of Xwhich can also be derived from the Kan loop simplicial group associated to X, see for example [Con84] and [BCC93]. The topological and functorial definition of secondary homotopy groups πn,∗X is crucial to understand new properties of these concepts in the literature. For 1991 Mathematics Subject Classification. 18D05, 55Q25, 55S45. Key words and phrases. secondary homotopy groups, track category, crossed module, reduced (stable) quadratic module, Hopf invariant of tracks. The second author was partially supported by the project MTM2004-01865 and the MEC postdoctoral fellowship EX2004-0616. 1 2 HANS-JOACHIM BAUES AND FERNANDO MURO example, we are able to determine the algebraic properties of the loop and suspension operators on secondary homotopy groups. As main new results, we describe the fiber sequence for secondary homotopy groups, and we show that secondary homotopy groups form a track functor on the track category of pointed spaces. In a sequel of this paper we determine the algebraic nature of smash product operations on secondary homotopy groups. For the (stable) secondary homotopy groups of spectra this leads to an algebraic invariant approximating the smash product of spectra. The computation of the algebra of secondary cohomology operations in [Bau] shows examples where secondary homotopy groups can be algebraically determined successfully. It is the aim of the authors to generalize the theory of [Bau], concerning the Eilenberg-MacLane spectrum, for general spectra. Moreover, we will discuss in a sequel of this paper generalized Whitehead products for secondary homotopy groups. In fact, J. H. C. Whitehead introduced in [Whi41] Whitehead products as an additional algebraic structure of homotopy groups. We may consider the secondary homotopy groups together with their algebraic properties also as such an enriching structure. The required “quadratic algebra” associated to properties of secondary homotopy groups is studied in [BJP05]. 1. Tracks between maps We consider the category Top∗of compactly generated pointed spaces X= (X, ∗) and pointed maps f:X→Y. For any (unpointed) space Xwe define X+=X⊔ {∗} as the same space with an outer base-point ∗. The smash product of two pointed spaces is defined by X∧Y= (X×Y)/(X× ∗ ∪ ∗ × Y). It is strictly associative. Homotopies IX →Yare defined by using the reduced cylinder IX =I+∧X, where I= [0,1] is the unit interval, with structure maps (1.1) X∨Xi −→ IX p −→ X. Here ∨is the symbol for the coproduct, iis the inclusion of the boundary and p is the projection. Given two maps f, g :X→Yatrack H:f⇒gis a homotopy class of homotopies IX →Y, from fto g, relative to the boundary. By abuse of language we denote a homotopy and the represented track by the same symbol. In diagrams tracks will be denoted as follows. (1.2) X f && g 88Y. H  SECONDARY HOMOTOPY GROUPS 3 The trivial track 0 f:f⇒fis represented by fp:IX →Yand the inverse of a track H:f⇒gis H⊟:g⇒f. The vertical composition of tracks X f  g// h BBY H  K  is defined by pasting homotopies representing Hand Kand is denoted by X f ## h ;;Y KH  One can also compose horizontally a track as in diagram (1.2) with maps k:W→X and l:Y→Zto obtain tracks Hk:fk ⇒gk and lH :lf ⇒lg in the obvious way. If we have a diagram like X g 88 f &&Y g′ 88 f′ &&Z H H′  the equality (g′H)(H′f) = (H′g)(f′H) holds and this element is the horizontal composition of Hand H′denoted by juxtaposition X f′f ## g′g ;;Z H′H  Tracks endow Top∗with the structure of a groupoid-enriched category, which we call a track category. The track category Top∗has a strict zero object, the one-point space ∗. In particular zero maps are defined. Such a track category has the crucial property that any track composed with a zero map becomes automatically a trivial track. Maps from a coproduct X∨Yin Top∗are given by pairs of maps (f1, f2): X∨ Y→Z. Similarly a track H: (f1, f2)⇒(g1, g2) between maps (f1, f2),(g1, g2): X∨ Y→Zis given by a pair of tracks H= (H1, H2) with Hi:fi⇒gi(i= 1,2). The suspension ΣXis the quotient space IX/(X∨X) = S1∧X. We will use the identifications (1.3) Σ(X∨Y) = ΣX∨ΣY, ΣnS0=Sn, n ≥0. For the definition of homotopy groups we choose a particular co-H-group structure on S1given by maps µ:S1→S1∨S1and ν:S1→S1satisfying the usual properties. We use explicitly these maps in many constructions throughout this 4 HANS-JOACHIM BAUES AND FERNANDO MURO paper, however these constructions do not depend on this choice since the maps µ and νare unique up to a canonical track. The loop space functor Ω is the right-adjoint of the suspension Σ. The adjoint of a map f: ΣX→Yis denoted by ad(f): X→ΩY. The adjoint of the identity map 1: ΣX→ΣXis a natural inclusion (1.4) ad(1): X ֒→ΩΣX. As a pointed set the n-fold loop space ΩnXis the set of pointed maps Sn→X and the base-point corresponds to the trivial map. By using the interchange homeomorphism of the smash product we see that suspensions and cylinders commute up to natural isomorphism in Top∗,IΣX∼ =ΣIX. However one has to be careful with signs because the interchange of factors in S1∧S1=S2induces −1 on the homotopy group π2. 2. Groups of nilpotency degree 2 Consider the forgetful functor from groups to pointed sets Gr −→ Set∗.This functor has a left adjoint h·i:Set∗−→ Gr:A7→ hAi. Here hAiis the quotient of the free group with basis Aby the normal subgroup generated by the base-point ∗ ∈ A. This group is isomorphic to the free group with basis A−{∗}. We denote ∨AS1= ΣA. As usual we identify the fundamental group of ∨AS1with a free group, i. e. π1(∨AS1) = hAi. The free group of nilpotency class 2 (free nil-group for short), generated by the pointed set A, is the quotient hAinil =hAi Γ3hAi where Γ3hAiis the 3rd term of the lower central series of hAi, i. e. the subgroup generated by triple commutators [x, [y, z]] (x, y, z ∈ hAi). In this paper we always write group laws additively, even for non-abelian groups, so that the commutator is [x, y] = −x−y+x+y. The free abelian group Z[A] on a pointed set Ais the abelianization of hAiand of hAinil. If gr,nil and ab are the categories of free groups, free nil-groups and free abelian groups, respectively, then there are obvious nilization and abelianization functors (2.1) gr ab // nil !! C C C C C C C Cab nil ab == { { { { { { { { hAi//  "" F F F F F F F F Z[A] hAinil 7 ;; w w w w w w w w Let hAinil ։Z[A] be the natural projection carrying xto {x}. Since the commutator bracket in hAinil is bilinear the homomorphism ∂:⊗2Z[A]→ hAinil, ∂({x} ⊗ {y}) = [x, y], is well defined. Here the tensor square of an abelian group Ais denoted by ⊗2A= A⊗A. Let T:A⊗B→B⊗Abe the interchange isomorphism T(a⊗b) = b⊗a. The reduced tensor square is the following cokernel ⊗2A1+T −→ ⊗2A¯σ ։ˆ ⊗2A. SECONDARY HOMOTOPY GROUPS 5 We denote ¯σ(a⊗b) = aˆ ⊗b. We define the functor ⊗2 nas ⊗2 n=   ⊗2,if n= 2; ˆ ⊗2,if n≥3. Here we write a⊗b∈ ⊗2 nAwith a⊗b=aˆ ⊗bfor n≥3. Moreover, Γnis the functor Γn=   Γ,if n= 2; − ⊗ Z/2,if n≥3; where −⊗Z/2 is the ordinary tensor product of abelian groups and Γ is Whitehead’s universal quadratic functor, see [Whi50]. There is a natural exact sequence (2.2) ΓnZ[A]֒→ ⊗2 nZ[A]∂ −→ hAinil ։Z[A]. Here the first arrow is induced by the function sending x∈Z[A] to x⊗x∈ ⊗2 nZ[A], see for example [Bau91]. Moreover, these exact sequences fit into a natural commutative diagram ΓZ[A] σ  //⊗2Z[A]∂// ¯σ  hAinil ////Z[A] Z[A]⊗Z/2//ˆ ⊗2Z[A]∂//hAinil ////Z[A] 3. Nil-tracks and Hopf invariants of tracks We now introduce nil-tracks and Hopf invariants of tracks which are needed in the definition of secondary homotopy groups in the next section. Definition 3.1. Let f, g be maps f, g :S1→ ∨AS1where Ais a discrete pointed set, and let Σn−1f, Σn−1g:Sn→ ∨ASn be their (n−1)-fold suspensions, n≥1. A track H: Σn−1f⇒Σn−1g, represented by a homotopy H:ISn→ ∨ASn, is said to be a nil-track if the adjoint ad(H): IS1−→ Ωn−1∨ASn induces a trivial homomorphism 0 = H2ad(H): H2(IS1, S1∨S1)−→ H2(Ωn−1∨ASn,∨AS1). The adjoint of Hsends the boundary of the cylinder IS1into ∨AS1since Hrestricted to the boundary is an (n−1)-fold suspension. Of course for n= 1 all tracks Habove are nil-tracks since H2ad(H) maps to the trivial group. Let f, g be now maps between wedges of 1-spheres f, g:∨BS1→ ∨AS1, and let Σn−1f, Σn−1gbe their (n−1)-fold suspensions. A track H:f⇒gis a nil-track if all restricted tracks Hibare nil-tracks where ib:Sn→ ∨BSnis the inclusion given by b∈B− {∗}. The homology groups involved in the definition of nil-tracks are computable. Indeed, H2(IS1, S1∨S1)∼ =H2(ΣS1) = H2S2=Z. 6 HANS-JOACHIM BAUES AND FERNANDO MURO Moreover, H2(Ωn−1∨ASn)∼ = −→ H2(Ωn−1∨ASn,∨AS1) is an isomorphism, and the Pontrjagin product ⊗2Z[A] = H1(Ωn−1∨ASn)⊗H1(Ωn−1∨ASn)−→ H2(Ωn−1∨ASn) is an isomorphism for n= 2 and induces an isomorphism for n≥2 (3.2) ⊗2 nZ[A]∼ =H2(Ωn−1∨ASn), compare notation in (2.2). Definition 3.3. Let n≥2. Given a track H: Σn−1f⇒Σn−1gfor maps f, g :S1→ ∨AS1the Hopf invariant of His defined as Hopf (H) = (H2ad(H))(1) ∈ ⊗2 nZ[A], where we apply the homology functor H2as in Definition 3.1. In particular, His a nil-track if and only if Hopf (H) = 0. More generally, if H: Σn−1f⇒Σn−1gis a track for maps f, g:∨BS1→ ∨AS1the Hopf invariant of His the homomorphism Hopf (H): Z[B]−→ ⊗2 nZ[A] defined by Hopf (H)(b) = Hopf (Hib), where ib:S1⊂ ∨BS1is the inclusion of the factor corresponding to b∈B− {∗}. Such a track His a nil-track if and only if Hopf (H) = 0. In case n= 1 then Hopf (H) = 0 for any track Has above. Remark 3.4.Any element x∈π3∨AS2determines a track x: 0 ⇒0 for the trivial map 0: S2→ ∨AS2. This track is given by the homotopy IS2→ΣS2=S3x → ∨AS2, where the first map is the obvious projection. The reader can check that −Hopf (x) is the classical Hopf invariant of x. The sign is due to the fact that in order to define the Hopf invariant of xas a track we need to consider the map (1 2): I+∧S1∧S1∼ =S1∧I+∧S1and this map induces −1: S3→S3up to homotopy. The next results are crucial for this paper. Theorem 3.5. Let f, g :∨AS1→ ∨BS1be two maps and n≥1. If a nil-track Nf,g : Σn−1f⇒Σn−1g exists then it is unique. Moreover, Nf,g exists if and only if •π1f=π1g:hAi → hBi, if n= 1; •or (π1f)nil = (π1g)nil :hAinil → hBinil, if n≥2. Furthermore, trivial tracks are nil-tracks and the vertical and horizontal composition of nil-tracks are also nil-tracks. This theorem is a immediate consequence of the following one. Theorem 3.6. Let n≥2and let f, g :∨AS1→ ∨BS1be maps such that for any x∈ hAinil we have (π1g)nil(x) = (π1f)nil(x) + ∂α(x)for some homomorphism α:Z[A]→ ⊗2 nZ[B]. Then there exists a unique track H: Σn−1f⇒Σn−1gwith SECONDARY HOMOTOPY GROUPS 7 Hopf invariant Hopf (H) = αand conversely. Moreover, the Hopf invariant of tracks satisfies the following formulas. Given a diagram ∨ASn Σn−1f  Σn−1g// Σn−1h DD ∨BSn H  K  the equation (1) Hopf (KH) = Hopf (K) + Hopf (H)holds. Furthermore, if we consider the diagram ∨ASnΣn−1k//∨BSn Σn−1g 99 Σn−1f %% ∨CSnΣn−1h//∨DSn H  then (2) Hopf (H(Σn−1k)) = Hopf (H)(π1k)ab, (3) Hopf ((Σn−1h)H) = (⊗2 n(π1h)ab)Hopf (H). In addition given a track H: Σn−1f⇒Σn−1gbetween maps f, g, :∨AS1→ ∨BS1 one gets the following equations. (4) Hopf (ΣH) = 0 if n= 1, (5) Hopf (ΣH) = ¯σHopf (H)if n= 2, (6) Hopf (ΣH) = Hopf (H)if n≥3. This theorem is a simple consequence of the theory developed in [Bau91] that we now recall. Let S(n)⊂Top∗be the full track subcategory of one-point unions of n-spheres and let gr be the category of free groups regarded as a track category with only the trivial tracks. Then there is a track functor π1:S(1) −→ gr given by the fundamental group, π1(∨AS1) = hAi. This track functor is a weak equivalence. This follows easily from [Bau91] VI.3.13 and the fact that wedges of 1-spheres do not have higher-dimensional homotopy groups. For n≥2 we consider the track subcategory ¯ S(n)⊂S(n) of suspended maps. Here objects of ¯ S(n) are one-point unions of 1-spheres ∨AS1, maps f, g:∨A S1→ ∨BS1in ¯ S(n) are maps in Top∗and tracks H:f⇒gin ¯ S(n) are tracks H: Σn−1f⇒Σn−1gin Top∗. The inclusion (3.7) Σn−1:¯ S(n)⊂S(n) is given by the (n−1)-fold suspension on objects and morphisms and it is the identity on tracks. This is actually a weak equivalence of track categories. See [Bau91] VI.4.7. We now consider the algebraic track category nil(n) defined as follows. Objects and morphisms are the same as in nil. A track α:ϕ⇒ψbetween homomorphisms 8 HANS-JOACHIM BAUES AND FERNANDO MURO ϕ, ψ:hAinil → hBinil is a homomorphism α:Z[A]→ ⊗2 nZ[B] such that ϕ(x) + ∂α({x}) = ψ(x) for any x∈ hAinil. The vertical composition is given by addition of abelian group homomorphisms, and for the horizontal composition one uses the abelianization functor ab:nil →ab and the bifunctor Homab(−,⊗2 n): abop ×ab −→ Ab. For any n≥2 there is a weak equivalence of track categories Hopf :¯ S(n)−→ nil(n) defined by ∨AS17→ hAinil,f7→ (π1f)nil and H7→ Hopf (H), where we use the Hopf invariant for tracks. Compare [Bau91] VI.4.7. This weak equivalence is compatible on the left hand side with the suspension functor Σ: ¯ S(n)−→ ¯ S(n+ 1) which is the identity on objects and morphisms and on tracks it is given by the suspension of tracks in Top∗, and on the right hand side with the track functors gr −→ nil(2) −→ nil(n), n ≥3, given by the nilization and the natural projection ¯σ:⊗2։ˆ ⊗2respectively. Here we set ¯ S(1) = S(1) for n= 1. Theorem 3.6, and therefore Theorem 3.5, follows readily from this. 4. Secondary homotopy groups of a pointed space We now introduce secondary homotopy groups which enrich the structure of the classical homotopy groups πnXof a pointed space. Definition 4.1. Let Xbe a pointed space and n≥1. The secondary homotopy group πn,∗Xis the map ∂:πn,1X−→ πn,0X defined as follows. Let πn,0X=   hΩXi, n = 1; hΩnXinil, n ≥2. Here the n-fold loop space, regarded as a discrete pointed set, generates a free (nil- )group. Moreover, πn,1Xis the set of equivalence classes [f, F ] represented by a map f:S1→ ∨ΩnXS1and a track Sn Σn−1f // 0  Sn Xev //X F KS Here the pointed space Sn X=∨ΩnXSn= ΣnΩnX is the n-fold suspension of the n-fold loop space ΩnX, where ΩnXis regarded as a pointed set with the discrete topology. Hence Sn Xis the coproduct of n-spheres indexed by the set of non-trivial maps Sn→X, and ev:Sn X→Xis the obvious evaluation map. Moreover, for the sake of simplicity given a map f:S1→ ∨ΩnXS1 we will denote fev =ev(Σn−1f), so that Fin the previous diagram is a track F:fev ⇒0. The equivalence relation [f, F ] = [g, G] holds provided the nil-track SECONDARY HOMOTOPY GROUPS 9 Nf,g : Σn−1f⇒Σn−1gexists, see Theorem 3.5, and the composite track in the following diagram is the trivial track. Sn 0  0 @@ Σn−1f %% Σn−1g 99 Sn X ev //X F⊟  Nf,g  G   That is F=G(ev Nf,g). The map ∂is defined by the formula ∂[f, F ] =    (π1f)(1), n = 1, (π1f)nil(1), n ≥2, where 1 ∈π1S1=Z. A map g:X→Yin Top∗induces a map πn,∗g:πn,∗X→πn,∗Ygiven by the following commutative diagram. (4.2) πn,1Xπn,1g // ∂  πn,1Y ∂  πn,0Xπn,0g//πn,0Y Here the lower homomorphism πn,0g=hΩngiis induced by the map of pointed sets Ωng: ΩnX→ΩnY. Moreover, an element [f, F]∈πn,1Xis sent by the upper map πn,1gto [(ΣΩng)f, gF], Xg  Sn 0.. Σn−1f //Sn X ev == { { { { { { { { ΣnΩng//Sn Yev //Y F emT T T T T T We also define πn,∗Xfor n= 0 as follows. Definition 4.3. For n= 0 let π0,∗Xbe the fundamental pointed groupoid of the pointed space Xfor which π0,0Xis Xregarded as a discrete pointed set and π0,1Xis the set of tracks between points in X. For this we recall that a pointed groupoid is a small category Gwith a distinguished object ∗ ∈ ObGsuch that all morphisms are isomorphisms. A morphism of pointed groupoids F:G→H is a functor preserving the distinguished object F(∗) = ∗, and the category of pointed groupoids is denoted by grd∗. The morphism Fis a weak equivalence if it induces a bijection between the pointed sets of isomorphism classes of objects Iso(F): Iso(G)∼ =Iso(H) and if F: AutG(x)∼ =AutH(F(x)) is an isomorphism for any object xin G. The fundamental groupoid is a functor π0,∗:Top∗→grd∗in the obvious way. We now study the algebraic structure of secondary homotopy groups πn,∗Xwith n≥1. 16 HANS-JOACHIM BAUES AND FERNANDO MURO Our definition of πn,∗Xabove is a “singular” and hence functorial version of secondary homotopy groups. For many purposes it suffices to consider smaller models of πn,∗Xby choosing a subset of ΩnXwhich generates πnXas an abelian group. Let us make precise this observation. Proposition 4.15. Let Xbe a pointed space. If E0→Xis a pointed map between pointed sets then there is a unique pointed groupoid π0,∗(X, E0)with object set E0 endowed with a full and faithful functor π0,∗(X, E0)−→ π0,∗X given by E0→Xon object sets. This morphism of pointed groupoids is a weak equivalence provided any component of Xhas points in the image of E0. Moreover, a map of pointed sets E1→ΩXinduces a crossed module morphism by the pull-back π1,1(X, E1) ∂  // pull π1,1X ∂  π1,0(X, E1) = hE1i//hΩXi=π1,0X which is a weak equivalence π1,∗(X, E1)∼ −→ π1,∗X provided the loops in the image of E1generate the group π1X. Furthermore, for n≥2the a map of pointed sets En→ΩnXinduces a reduced (stable if n≥3) quadratic module morphism by the pull-back ⊗2(πn,0(X, En))ab =⊗2Z[En] ω  // ∂ %% ⊗2Z[ΩnX] = ⊗2(πn,0X)ab ω  πn,1(X, En) ∂  // pull πn,1X ∂  πn,0(X, En) = hEninil //hΩnXinil =πn,0X which is a weak equivalence πn,∗(X, En)∼ −→ πn,∗X, n ≥2 provided the n-loops in the image of Engenerate the abelian group πnX. This proposition can be used to reduce the number of generators of a secondary homotopy group, as one can check in the following example. Remark 4.16.So far we have not computed any secondary homotopy group. Now, with the help of Proposition 4.15 we give a small model for the secondary homotopy group πn,∗(∨ESn) of a wedge of spheres indexed by the pointed set E. For this we notice that there is a pointed inclusion E⊂Ωn(∨ESn) sending e∈E− {∗} to the inclusion of the corresponding factor of the wedge Sn⊂ ∨ESn. Then we have a weak equivalence πn,∗(∨ESn, E)∼ −→ πn,∗(∨ESn), n ≥1. For n= 1 one easily checks that π1,∗(∨ES1, E) is π1,1(∨ES1, E) = 0 ∂ −→ π1,0(∨ES1, E) = hEi. SECONDARY HOMOTOPY GROUPS 17 For n= 2 the reduced quadratic module π2,∗(∨ES2, E) is given by the following diagram, see (2.2) ⊗2(πn,0(∨ESn, E))ab ω//πn,1(∨ESn, E)∂//πn,0(∨ESn, E) ⊗2Z[E]⊗2Z[E]∂//hEinil This follows from the fact that the next diagram is a pull-back ⊗2Z[E]// %% ⊗2Z[Ω2∨ES2] = ⊗2(π2,0∨ES2)ab ω  ⊗2Z[E] ∂  φ// pull π2,1∨ES2 ∂  hEinil //hΩ2∨ES2inil =π2,0∨ES2 Here the homomorphism φis defined as follows. Given x∈ ⊗2Z[E] the element φ(x) = [φ1(x), φ2(x)] is given by a map φ1(x): S1¯ φ(x) −→ ∨ES1⊂ ∨Ω2∨ES2S1 with (π1¯ φ(x))nil(1) = ∂(x) and the unique track S2 Σφ1(x) // 0 "" S2 ∨ES2ev //∨ES2 φ2(x) HP      with Hopf invariant Hopf (φ2(x)) = −x. Here we use the fact that the composite ev(Σφ1(x)) = Σ¯ φ(x) is a suspension. For n≥3 the stable quadratic module πn,∗(∨ESn, E) is given by the following diagram, see (2.2). ⊗2(πn,0(∨ESn, E))ab ω//πn,1(∨ESn, E)∂//πn,0(∨ESn, E) ⊗2Z[E]¯σ//ˆ ⊗2Z[E]∂//hEinil This can be easily checked as in the case n= 2 by using the Hopf invariant for tracks. 5. Homotopy groups of fibers We first obtain by secondary homotopy groups the classical homotopy groups πnXas in the next result. Proposition 5.1. For all n≥1there is a natural exact sequence of groups πn+1Xι ֒→πn,1X∂ −→ πn,0Xq ։πnX, where qsends a basis element of πn,0X, which is a map f:Sn→X, to its homotopy class in πnX; and ιcarries the homotopy class of f:Sn+1 →Xto the element [0, pf]∈πn,1X, where p:ISn→ΣSn=Sn+1 is the obvious projection. 18 HANS-JOACHIM BAUES AND FERNANDO MURO Proof. Obviously qis surjective. Any element x∈πn,0Xis represented by a map ˜x:S1→ ∨ΩnXS1, i. e. (π1˜x)nil(1) = x. It is immediate to notice that q(x) is the homotopy class of ˜xev :Sn→X. If q(x) = 0 then there exists a track H: ˜xev ⇒0, and the pair [˜x, H]∈πn,1Xsatisfies ∂[˜x, H]−q(x). It is immediate to notice that q∂ = 0 and ∂ι = 0. The injectivity of ιis also easy to check, actually πn+1Xis isomorphic to the subgroup of πn,1Xgiven by the elements which can be represented with a 0 in the first coordinate. Finally suppose that for some [f, F ]∈πn,1Xwe have ∂[f, F] = 0, then the nil-track N: 0 ⇒Σn−1fis defined and [f, F ] = [0, FN], hence we are done.  We now introduce the (algebraic) fiber of a map in cross(n) for n≥1. Definition 5.2. Let f:∂→∂′be a crossed module morphism M ∂  f1 //M′ ∂′  Nf0//N′ We define the fiber Fib(f) as the crossed module Fib(f): Fib1(f)→Fib0(f) where Fib0(f) is the following pull-back Fib0(f) ¯ ∂′  ¯ f0 // pull M′ ∂′  Nf0//N′ Fib1(f) = Mand the homomorphism Fib(f): Fib1(f)→Fib0(f) is induced by (∂, f1): M→N×M′. The action of Fib0(f) on Fib1(f) is the pull-back along ¯ ∂′ of the action of Non M. The axioms of a crossed module are easily verified. There is also a natural crossed module morphism : Fib(f)→∂given by the square Fib1(f) Fib(f)  M ∂  Fib0(f)¯ ∂′//N Let f: (ω, ∂)→(ω′, ∂′) be now a reduced/stable quadratic module morphism ⊗2Nab ω  ⊗2f0 ab //⊗2(N′)ab ω′  M ∂  f1 //M′ ∂′  Nf0//N′ SECONDARY HOMOTOPY GROUPS 19 The fiber Fib(f) is a reduced/stable quadratic module ⊗2(Fib0(f))ab −→ Fib1(f)Fib(f) −→ Fib0(f) where Fib(f): Fib1(f)−→ Fib0(f) is defined as in the crossed module case and the first homomorphism is the composite ⊗2(Fib0(f))ab ⊗2¯ ∂′ ab −→ ⊗2Nab ω −→ M. The natural reduced/stable quadratic module morphism : Fib(f)→(ω, ∂) is also defined as above. Lemma 5.3. Let f:∂→∂′be a morphism of crossed modules, then there is an exact sequence h1Fib(f)h1 ֒→h1∂h1f −→ h1∂′δ −→ h0Fib(f)h0 −→ h0∂h0f −→ h0∂′. This exact sequence is natural in f. Moreover, it is also available for reduced or stable quadratic module morphisms f: (ω, ∂)→(ω′, ∂′). Proof. The homomorphism δis determined by the inclusion M′֒→N×M′:m′7→ (0, m′). The proof of the exactness is a simple exercise.  Theorem 5.4. Let f:X→Ybe a map between pointed spaces and let Ffbe the homotopy fiber of f. Then for all n≥1there is a natural morphism in cross(n) ξ:πn,∗Ff−→ Fib(πn,∗f) which induces an isomorphism (1) πnFf∼ =h0Fib(πn,∗f) and an exact sequence (2) πn+2Y−→ πn+1Ff։h1Fib(πn,∗f), where the first arrow is the boundary homomorphism in the long exact sequence in homotopy. By using the isomorphism (1) above and Proposition 5.1 we can naturally identify the exact sequence in Lemma 5.3 extended on the left by the exact sequence (2) with the following piece of the long exact sequence of homotopy groups πn+2Y→πn+1Ff→πn+1X→πn+1Y→πnFf→πnX→πnY. Proof. Recall that Ffis a pull-back Ff ¯e  ¯ f// pull YI ev0  Xf//Y where YIis the space of based maps ([0,1],1) →(Y, ∗) and ev0is the evaluation at 0 ∈[0,1]. The morphism ξconsists of two morphisms, the upper one is ξ1=πn,1¯e:πn,1Ff→πn,1X= Fib1(πn,∗f). 20 HANS-JOACHIM BAUES AND FERNANDO MURO We now construct the map ξ0:πn,0Ff→Fib0(πn,∗f). For this we consider on the one hand the morphism πn,0¯e:πn,0Ff→πn,0Xinduced by f. On the other hand we define a homomorphism ¯ ξ:πn,0Ff→πn,1Y as follows: an element z∈πn,0Ffis represented by a map ˜z:S1→ ∨ΩnFfS1with (π1˜z)(1) = zif n= 1 or (π1˜z)nil(1) = zif n≥2. The map SnΣn−1˜z −→ Sn Ff ΣnΩn¯ f −→ Sn YI ev −→ YI has an adjoint ad(ev(ΣnΩn¯ f)(Σn−1˜z)): ISn−→ Y, this adjoint represents a track ad(ev(ΣnΩn¯ f)(Σn−1˜z)): ((ΣΩn(f¯e))˜z)ev ⇒0, and ¯ ξ(z) = [(ΣΩn(f¯e))˜z, ad(ev(ΣnΩn¯ f)(Σn−1˜z))] ∈πn,1Y. It is immediate to check that πn,0¯eand ¯ ξdefine a homomorphism to the pull-back ξ0= (πn,0¯e, ¯ ξ): πn,0Ff−→ Fib0(πn,∗f). Now it is easy to check that ξis indeed a morphism in cross(n). By Proposition 5.1 and Lemma 5.3 we obtain from ξa diagram with exact rows πn+2Y//πn+1Ff// ξ∗  πn+1X// ∼ =  πn+1Y ∼ =  //πnFf ξ∗  //πnX ∼ =  //πnY ∼ =  h1Fib(πn,∗f)//h1πn,∗X//h1πn,∗Yδ //h0Fib(πn,∗f)//h0πn,∗X//h0πn,∗Y It is easy to see that this diagram commutes, and hence the theorem follows from the five lemma.  Corollary 5.5. Let f:X→Ybe a map between pointed spaces and let Ffbe the homotopy fiber of f. If πn+2f:πn+2X։πn+2Yis surjective then there is a weak equivalence in cross(n),n≥1, ξ:πn,∗Ff ∼ −→ Fib(πn,∗f). 6. Suspension and loop functors Homotopy groups πnXare objects in the category group(n), n≥0, see (4.12). There are forgetful functors (6.1) φn:group(n)−→ group(n−1) given by φn= 1Ab for n≥3 and by the obvious forgetful functors φ2:Ab −→ Gr, φ1:Gr −→ Set∗. It is a classical result that for any pointed space Xthere are natural isomorphisms n≥0 Ω: πnΩX∼ =φn+1πn+1Xin group(n). The analogue of this isomorphism for secondary homotopy groups is as follows. There are forgetful functors (6.2) φn:cross(n)−→ cross(n−1), SECONDARY HOMOTOPY GROUPS 21 see (4.10), given by φn= 1squad for n≥4 and by the functors (6.3) φ3:squad −→ rquad, φ2:rquad −→ cross, φ1:cross −→ grd∗. The functor φ3in (6.3) is obvious, since stable quadratic modules are special reduced quadratic modules. Given a reduced quadratic module (ω, ∂) we have φ2(ω, ∂) = ∂:M→Nin cross, with the action of Non Mdefined by mn=m+ω({∂m} ⊗ {n}). Finally if ∂:M→Nis a crossed module then the pointed groupoid φ1∂in grd∗ has Nas a set of objects. Moreover the set of all morphisms in φ1∂is the semidirect product N⋉M, which is the group structure on the set N×Mdefined by the formula (n, m) + (n′, m′) = (n+n′, mn′+m′), and the structure maps of the groupoid (identities, source and target) Ni →N⋉Ms ⇒ t N are i(n) = (n, 0), s(n, m) = nand t(n, m) = n+∂m. The composition law ◦is determined by the formula (n+∂m, m′)◦(n, m) = (n, m +m′). The forgetful functors φnin (6.2) clearly commute with h0and h1in (4.13), that is, (6.4) hiφn=φnhi, n ≥1, i = 0,1. Theorem 6.5. There is a natural weak equivalence in cross(n) Ω: πn,∗ΩX−→ φn+1πn+1,∗X, n ≥0, which induces the isomorphism Ω: πnΩX∼ =φn+1πn+1Xin h0and −Ω: πn+1ΩX∼ = φn+2πn+2Xin h1. This weak equivalence is an isomorphism for n≥2. Proof. Let us first consider the case n≥3. We have πn,0ΩX=hΩn+1Xinil =πn+1,0X. We define a group homomorphism πn,1ΩX→πn+1,1Xsending [f, F] with f:S1→ ∨Ωn+1XS1and Sn Σn−1f // 0 Sn ΩXev //ΩX F JR  to [f, ad(F)] where ad(F) is the adjoint track Sn+1 Σnf// 0 !! Sn+1 Xev //X ad(F) MU###### Here we use that ΣSn ΩX=Sn+1 Xand ad(ev(Σn−1f)) = ev(Σnf). 22 HANS-JOACHIM BAUES AND FERNANDO MURO The reader can check that the diagram (a) (πn,0ΩX)ab ⊗(πn,0ΩX)ab ω//πn,1ΩX∂//  πn,0ΩX (πn+1,0X)ab ⊗(πn+1,0X)ab ω//πn+1,1X∂//πn+1,0X commutes, so it is a morphism of stable quadratic modules. Moreover, the following diagram commutes (b) πn+1ΩXι// −Ω∼ =  πn,1ΩX∂//  πn,0ΩXq////πnΩX ∼ =Ω  πn+2X ι//πn+1,1X∂//πn+1,0Xq////πn+1X Here the exact rows are given by Proposition 5.1 and the arrows with ∼ =are (up to sign) the usual isomorphisms of homotopy groups, therefore the central vertical arrow in (a) is an isomorphism by the five lemma. For n= 2 we have π2,0ΩX=hΩ3Xinil =π3,0Xand there is a homomorphism π2,1ΩX→π3,1Xdefined as above. This homomorphism makes commutative diagrams (a) and (b), therefore it defines an isomorphism of reduced quadratic modules. For n= 1 there is an obvious epimorphism π1,0ΩX=hΩ2Xi։hΩ2Xinil = π2,0X. One can also define a homomorphism π1,1ΩX→π2,1Xas above. It is easy to check that the following square defines the desired crossed module morphism (c) π1,1ΩX∂//  π1,0ΩX  π2,1X∂//π2,0X Moreover, the following diagram commutes π2ΩXι// −Ω∼ =  π1,1ΩX∂//  π1,0ΩX  q////π1ΩX ∼ =Ω  π3X ι//π2,1X∂//π2,0Xq////π2X This diagram is analogue to (b) and shows that (c) is a weak equivalence. Now for n= 0 we define the functor π0,∗ΩX→φ1π1,∗X. On objects it is given by the inclusion Obπ0,∗ΩX= ΩX⊂ hΩXi=Obφ1π1,∗X. Given any object f∈ΩXin π0,∗ΩXwe consider the inclusion ¯ f:S1→S1 Xof the factor of the coproduct S1 Xcorresponding to f. Clearly the adjoint ad(¯ fev): S0→ΩXis the inclusion of the point f∈ΩX. If g∈ΩXis another object then a morphism H:f→gin π0,∗ΩXis just a track H:ad(¯ fev)⇒ad(¯gev) in Top∗. The functor sends the morphism Hto the element in π1,0X⋉π1,1X, which is π1,0X×π1,1Xas a set, with (π1¯ f)(1) in the left coordinate and right coordinate given by the map S1µ −→ S1∨S1ν∨1 −→ S1∨S1(¯g, ¯ f) −→ S1 X SECONDARY HOMOTOPY GROUPS 23 and the track S1 Xev  S1µ// 0-- S1∨S1 ν∨1//S1∨S1 (¯ f,¯g) // (¯g,¯g);; w w w w w w w w wS1 Xev //X (ad(H),0) KS N MU"""""" """""" Here Nis a nil-track and ad(H): ¯ fev ⇒¯gev is the adjoint of the track H. We leave to the reader to check that π0,∗ΩX→φ1π1,∗Xis a well-defined functor. One can use again Proposition 5.1 to check that this functor is an equivalence.  The functors φnin (6.1) have left adjoints (6.6) Adn:group(n−1) −→ group(n) given by Adn= 1Ab for n≥3, Ad2:Gr −→ Ab,the abelianization; Ad1:Set∗−→ Gr,taking free group. These adjoints can be used to define for a pointed space Xthe natural suspension morphisms Σ: Adn+1πnX−→ πn+1ΣX as the adjoint of πnXπnad(1) −→ πnΩΣXΩ ∼ =φn+1πn+1ΣX. Here we use the map ad(1): X→ΩΣXwhich is adjoint to the identity in ΣX and the natural isomorphism Ω. Now we generalize the situation for secondary homotopy groups. The functors φnin (6.2) have left adjoints, (6.7) Adn:cross(n−1) −→ cross(n) given by Adn= 1squad if n≥4, (6.8) Ad3:rquad −→ squad, Ad2:cross −→ rquad, Ad1:grd∗−→ cross. Lemma 6.9. The functors in (6.8) preserve 0-free objects and weak equivalences between them. Proof. For Ad1the lemma follows from Lemma 6.12 below. For Ad2and Ad3the lemma follows from the technical fact that the suspension functors between crossed and quadratic complexes described in [Bau91] and [Mur05], which are extensions of Ad2and Ad3, are compatible with the homotopy relation in the category of totally free (i. e. cofibrant) crossed or quadratic complexes. In addition we use that 0-free crossed or quadratic modules are exactly the truncations of totally free crossed or quadratic complexes.  The functor Ad3is the stabilization in [Bau91] IV.C.3. It is defined as follows. Given a reduced quadratic module (ω, ∂) = (⊗2Nab ω −→ M∂ −→ N) 24 HANS-JOACHIM BAUES AND FERNANDO MURO the stabilized stable quadratic module Ad3(ω, ∂) = (⊗2Nab ωΣ −→ MΣ ∂Σ −→ N) is given by the group MΣobtained by quotienting out in Mthe relations ω(a⊗b+b⊗a), a, b ∈Nab, and the homomorphisms ωΣand ∂Σare induced by ωand ∂, respectively, in the obvious way. The functor Ad2in (6.8) is the suspension functor in [Mur05] 3.3. Given a crossed module ∂:M→Nthe reduced quadratic module Ad2∂= (⊗2Nab ω −→ M˜ Σδ −→ Nnil) is given by the group M˜ Σwhich is a quotient of M×(⊗2Nab) by the relations (−m+mn,0) = (0,{∂(m)} ⊗ {n}) = (0,− {n} ⊗ {∂(m)}), for any m∈Mand n∈N; and the homomorphisms δand ωare defined by the following formulas, m∈M,n, n′∈N, δ(m, {n} ⊗ {n′}) = ∂(m) + [n, n′], ω({n} ⊗ {n′}) = (0,{n} ⊗ {n′}). Finally we describe the functor Ad1. Let Gbe a groupoid with object pointed set ObGand morphism set MorG. The crossed module Ad1Gis the quotient of the free crossed module, see [Bau91], generated by the function MorG−→ hObGi (h:U→V)7→ −U+V by the relations u=g+ffor u, f, g ∈MorGwith u=fg, the composition of f and gin G. One readily checks that Ad1is the adjoint of φ1. The functor h0commutes with Adn (6.10) h0Adn=Adnh0, n ≥1. This follows from the definition of Adnabove for n≥2 and from Lemma 6.12 below in case n= 1. For h1the corresponding commutativity law is not true in general, compare Lemma 6.12 below. Theorem 6.11. There are natural morphisms in cross(n+ 1) Σ: Adn+1πn,∗X−→ πn+1,∗ΣX, n ≥0, which induce the classical suspension homomorphism Σ: Adn+1πnX→πn+1ΣX in h0, and for n≥3the homomorphism −Σ: Adn+2πn+1X→πn+2ΣXin h1. Moreover, for n≥3the morphism Σis a weak equivalence provided Xis mconnected and n≤2m−1. It is also a weak equivalence for n= 2 provided X is simply connected, and for n= 1 if Xis connected. Furthermore, Σis always a weak equivalence for n= 0. In the proof of this theorem we will use the following lemma. Lemma 6.12. For any pointed groupoid Gthere are natural isomorphisms (1) h0Ad1G=hIso Gi, (2) h1Ad1G=L x∈Iso(G) (AutG(x))ab ⊗R. SECONDARY HOMOTOPY GROUPS 25 Here Ris the group ring of hIso Gi. Proof. The crossed module Ad1Gdefined above is the truncation N1FBG/d(N2FBG)→N0FBG of the Moore complex N∗FBGof the Milnor construction F BGon the classifying space BGof the pointed groupoid G, see [Kan58] and [GJ99] I.1.4 and V.6. To see this we have on the 0-level N0FBG=hObGi, and on the 1-level the set MorGin Ad1Gis mapped to N1FBG/d(N2FBG) by sending h:U→Vto the coset modulo d(N2FBG) of the element −1U+h∈N1FBG⊂F1BG=hMorGi. This can be checked by computing N1FBG/d(N2FBG) in terms of generators and relations. In order to carry out this computation one uses the Reidemeister-Schreier method, see [MKS66], which simplifies in this particular case since the simplicial identities hold in FBGand the boundaries and degeneracies in this simplicial group are homomorphisms between free groups on pointed sets induced by maps between the generating pointed sets. By the previous observation the kernel and cokernel of Ad1Gare the π1and π2 of the suspension Σ|BG|of the geometric realization |BG|of the classifying space of G, so the lemma follows from elementary facts from homotopy theory. We also remark that given x∈Iso(G) and a representative ˜x∈ObGof xthe group (AutG(˜x))ab does not depend on the choice of ˜x, up to natural isomorphism, therefore we can denote it by (AutG(x))ab. Proof of Theorem 6.11. Consider the morphism πn,∗Xπn,∗ad(1) −→ πn,∗ΩΣXΩ −→ φn+1πn+1,∗ΣX, where ad(1): X→ΩΣXis the adjoint of the identity in ΣXand Ω is given by Proposition 6.5. The morphism in the statement is the adjoint of this one. For n≥3 the range where this morphism is a weak equivalence follows from Proposition 5.1 and the classical suspension theorem for ordinary homotopy groups. For n= 1 the theorem follows from Proposition 8.5 below and [Mur05] 4.8. For n= 2 we use Proposition 8.5 and [Bau91] IV.C. For this we use that we are dealing with 0-free objects and that Adnpreserves weak equivalences between them, see Lemma 6.9. If n= 0 we have Iso(π0,∗X) = π0Xand for any x∈Obπ0,∗X, Autπ0,∗X(x) = π1(X, x). By using elementary homotopy theory one can check that π1ΣX∼ =hπ0Xi and π2ΣX∼ =M x∈π0X (π1(X, x))ab ⊗Zhπ0Xi. Now it is enough to notice that isomorphisms in Lemma 6.12 are compatible with the two isomorphisms above and Proposition 5.1 (in this last case up to sign −1 in kernel).  32 HANS-JOACHIM BAUES AND FERNANDO MURO The track (b) is given by (b) SnΣn−1˜x//Sn X ev //X g  X∨X (f,g) "" F F F F F F F F F (1,1) OO Sn 0 55 Σn−1ε //Sn∨Sn Σn−1(˜x∨˜x) // (1,1) OO Sn X∨Sn X (1,1) OO ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y (H,0) 6> v v vv v v N goVVVVVV VVVVVV Here Nis a nil-track. With the terminology introduced in Definition 7.4 we have H∗(x) = r(H ev(Σn−1˜x)) for the track H ev (Σn−1˜x): ev(ΣnΩnf)(Σn−1˜x) = f ev(Σn−1˜x)⇒g ev(Σn−1˜x=ev(ΣnΩng)(Σn−1˜x. The proof of equation (1) in the definition of tracks in cross(n) follows from Lemma 7.6. Equation (2) follows from Lemma 7.5 (1) or (2). Equation (3) follows from the fact that given [k, K]∈πn,1Xthe following composite tracks coincide. SnΣn−1k//Sn X ev //X g  X∨X (f,g) "" F F F F F F F F F (1,1) OO Sn 0 55 Σn−1ε //Sn∨Sn Σn−1(k∨k) // (1,1) OO Sn X∨Sn X (1,1) OO ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y (H,0) 6> v v vv v v N goVVVVVV VVVVVV Sn 0 && Σn−1k//Sn Xev //X g  X∨X (f,g) "" F F F F F F F F F (1,1) OO Sn 0 55 Σn−1ε //Sn∨Sn Σn−1(k∨k) // (1,1) OO Sn X∨Sn X (1,1) OO ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y (H,0) 6> v v vv v v N goVVVVVV VVVVVV K S[... ... Sn 0 && Σn−1k//Sn Xev //X g  X∨X (f,g) "" F F F F F F F F F (1,1) OO Sn Σn−1ε //Sn∨Sn Σn−1(k∨k) // (1,1) OO Sn X∨Sn X (1,1) OO ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y (H,0) 6> v v vv v v K S[... ... SECONDARY HOMOTOPY GROUPS 33 X g  X∨X (f,g) "" F F F F F F F F F (1,1) OO Sn Σn−1ε //Sn∨Sn 0 ** Σn−1(k∨k) //Sn X∨Sn X ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y (H,0) 6> u u uu u u K∨K [c> >>>> >> > >>>> >> X∨X (f,g) "" F F F F F F F F F Sn Σn−1ε //Sn∨Sn 0 ** Σn−1(k∨k) //Sn X∨Sn X ev∨ev 66 l l l l l l l l l l l l l ΣnΩn(f,g)//Sn Yev //Y K∨K [c> >>>> >> > >>>> >> The vertical composition of tracks fH ⇒gK ⇒his preserved by Lemma 7.5 (3). The proof of the fact that πn,∗preserves horizontal composition is straightforward and it is left to the reader.  Proposition 7.7. The inclusion crossf(n)⊂cross(n)of the full subcategory of 0-free objects induces an equivalence of categories (n≥0) crossf(n)/≃∼ −→ Ho cross(n), where the homotopy category Ho is obtained by inverting weak equivalences. Proof. For n= 0 this result is well-known. For n= 1 this is a consequence of the fact that cross has a model category structure where 0-free objects are the cofibrant objects, see [GM97], and the homotopy relation derived from the cylinders on cofibrant objects is given by the tracks defined above. In a similar way one obtains the result for n≥2.  8. k-Invariants Let K(G, n) be the Eilenberg-MacLane space with πnK(G, n) = G. Following Eilenberg-MacLane’s notation we write Hm(G, n, A) for the m-dimensional cohomology of the space K(G, n) with coefficients in the abelian group A. Here we allow Ato be a G-module in case n= 1. In this case Hm(G, 1, A) = Hm(G, A) is the ordinary cohomology (with local coefficients) of the group G. For any connected CW-complex Xwe write kn(X)∈Hn+2(πnX, n, πn+1X) for the first k-invariant of the (n−1)-connected cover Xhni. Recall that Xhniis the homotopy fiber of the canonical map from Xto its (n−1)-type, X→Pn−1X, where Pn−1Xis a Postnikov section of X. If n= 1 then k1(X) is the usual first k-invariant of a connected CW -complex X, represented by the crossed module ∂:π2(X, X1)−→ π1X1, determined by the skeletal filtration of X, see [MW50]. Otherwise, if n≥2 Hn+2(πnX, n, πn+1X) = Hom(ΓnπnX, πn+1X) 34 HANS-JOACHIM BAUES AND FERNANDO MURO and kn(X): ΓnπnX→πn+1Xis induced by the function η∗:πnX→πn+1Xwhich sends the homotopy class of α:Sn→Xto the homotopy class of α(Σn−2η): Sn+1 → X, where η:S3→S2is the Hopf map. Compare notation in (2.2). The first secondary homotopy group π1,∗Xis a crossed module, see Proposition 4.6. By Proposition 5.1 and [MW50] this crossed module represents an element k(π1,∗X)∈H3(π1X, π2X). In general, any crossed module ∂defines a cohomology class k(∂)∈H3(h0∂, h1∂), see [MW50]. For n≥2 the n-dimensional secondary homotopy group of Xdefines a homomorphism k(πn,∗X): ΓnπnX−→ πn+1X, as follows. Let k(πn,∗X) be the unique homomorphism fitting into the following commutative diagram (8.1) Γn(πn,0X)ab Γnq  //⊗2 n(πn,0X)ab ω  ΓnπnX k(πn,∗X)  πn+1Xι//πn,1X Here the upper horizontal arrow is the injection in (2.2), and ιand qappear in Proposition 5.1. In general any 0-free reduced quadratic module (ω, ∂) defines a homomorphism k(ω, ∂): Γh0(∂, ω)−→ h1(ω, ∂), as in (8.1) and any 0-free stable quadratic module (ω, ∂) defines accordingly a homomorphism k(ω, ∂): h0(ω, ∂)⊗Z/2−→ h1(ω, ∂). Theorem 8.2. For any connected CW -complex Xand any n≥1the equality kn(X) = k(πn,∗X)holds. Proof. We can suppose without loss of generality that the 1-skeleton X1=∨ES1 is just a one-point union of 1-spheres. One can easily check that π1,∗(X, E) in Proposition 4.15 coincides with ∂:π2(X, X1)→π1X1, hence the theorem follows for n= 1. We now prove the theorem for n≥2. Suppose that we have x∈πn,0Xand we choose ˜x:S1→ ∨ΩnXS1with (π1˜x)nil(1) = x. Then ω({x} ⊗ {x})∈πn,1Xis represented by Sn Σn−1β // 0  Sn∨Sn B KS Σn−1(˜x,˜x) //Sn Xev //X SECONDARY HOMOTOPY GROUPS 35 This is the same as Sn Σn−1β // 0  Sn∨Sn B KS (1,1) //Sn Σn−1˜x //Sn Xev //X By Theorem 3.6 Hopf ((1,1)B) = −1∈ ⊗2 nZ=   Z,if n= 2; Z/2,if n≥3. Moreover, (π1(1,1)β)nil = 0, therefore by using the definition of ιin Proposition 5.1, Theorem 3.6, Remark 3.4 and the characterization of η:S3→S2up to homotopy as the unique map with Hopf invariant 1 we get that ω({x}⊗{x}) = ι(q(x)(Σn−2η)), hence we are done.  Let types1 nbe the category of pointed (n−1)-connected CW -complexes Xwith πm(X, x0) = 0 for all m≥n+ 2 and all x0∈X. Proposition 8.3. The functor πn,∗:types1 n→cross(n)induces an equivalence of categories (n≥0) πn,∗: Ho types1 n ∼ −→ Ho cross(n), where the homotopy category Ho is obtained by localizing with respect to weak equivalences. For the proof of Proposition 8.3 we recall the following functors. Let CWnbe the category of CW-complexes Xwith trivial (n−1)-skeleton Xn−1=∗and cellular maps. There is a “cellular” functor (8.4) Pn+1σ:CWn/≃ −→ cross(n)/≃. If n= 1 this functor sends a CW-complex Xto the crossed module ∂:π2(X, X1)→π1X1 given by the boundary operator in the long exact sequence of homotopy groups, see [Mac49] and [MW50]. If n≥2 the the reduced (stable if n≥3) quadratic module Pnσ(X) is the truncation of the totally free quadratic complex σ(X) defined in [Bau91] IV.C, ⊗2Cn(X)ω −→ σn+1(X)/d(σn+2(X)) ∂ −→ σn(X), compare [Bau91] IV.10.4 and [Mur05] 4. Proposition 8.5. The functor Pn+1σin (8.4) is naturally isomorphic to πn,∗:CWn/≃ → crossf(n)/≃ for all n≥1. Proof. If Xis (n−1)-reduced then Xn=∨ESnfor some pointed set E. The inclusion of spheres in the wedge Xn⊂Xdetermines a pointed inclusion E⊂ΩnX. One can easily check that Pn+1σ(X) is isomorphic to πn,∗(X, E) in Proposition 4.15. Now the natural isomorphism in the statement is given by the weak equivalence Pn+1σ(X)∼ =πn,∗(X, E)∼ →πn,∗Xin Proposition 4.15. Compare Proposition 7.7.  36 HANS-JOACHIM BAUES AND FERNANDO MURO Proof of 8.3. For n= 0 this is a well-known result. For n≥1 this follows from Proposition 8.5 and the fact that Pn+1σin 8.4 does induce an equivalence of categories Pn+1σ: Ho types1 n→Ho cross(n). This is shown in [Bau91] III.8.2 for n= 1. For n= 2 the proof follows as in the case n= 1, this case is considered even in the non-simply connected case in [Bau91] IV.10.1. The case n≥3 can be easily proved along the lines of the n= 1 and n= 2 cases, i. e. by using [Bau91] III.8.5, III.8.8 and IV.C.14.  Remark 8.6.In the literature there are further algebraic categories equivalent to Ho types1 n. In particular, see for n= 1 [Tak05], for n= 2 see [CC96], and for n≥3 [BCC93]. These algebraic models, by Proposition 8.3, can also be deduced from objects in cross(n). The objects in cross(n) seem to be the “smallest possible” algebraic objects representing the category Ho types1 n. In addition the definition of these other algebraic models is not topological, but simplicial. The difference between our models and the other ones is similar to the difference between classical homotopy groups as homotopy classes of maps Sn→Xand as the homology of the Moore complex of the Kan loop group of the singular simplicial set on X. References [Bau] H.-J. Baues, The algebra of secondary cohomology operations, To appear in Progress in Math. Birkh¨auser. [Bau91] ,Combinatorial Homotopy and 4-Dimensional Complexes, Walter de Gruyter, Berlin, 1991. [BCC93] M. Bullejos, P. Carrasco, and A. M. Cegarra, Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg-Mac Lane’s classification theorem, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 1, 163–189. [BJP05] H.-J. Baues, M. Jibladze, and T. Pirashvili, Quadratic algebra of square groups, Preprint, 2005. [CC96] P. Carrasco and A. M. Cegarra, (Braided) tensor structures on homotopy groupoids and nerves of (braided) categorical groups, Comm. Algebra 24 (1996), no. 13, 3995–4058. [Con84] D. Conduch´e, Modules crois´es g´en´eralis´es de longueur 2, J. Pure Appl. Algebra 34 (1984), no. 2-3, 155–178. [GJ99] P. J. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, no. 174, Birkh¨auser Verlag, Basel, 1999. [GM97] A. R. Garz´on and J. G. Miranda, Homotopy theory for (braided) CAT-groups, Cahiers Topologie et G´eom. Diff´erentielle Cat´egoriques 38 (1997), no. 2, 99–139. [Kan58] D. M. Kan, A combinatorial definition of homotopy groups, The Annals of Mathematics 67 (1958), no. 2, 282–312. [Mac49] S. MacLane, Cohomology theory in abstract groups III. operator homomorphisms of kernels., Ann. of Math. (2) 50 (1949), 736–761. [MKS66] W. Magnus, A. Karras, and D. Solitar, Combinatorial Group Theory, Interscience, New York, 1966. [Mur05] F. Muro, Suspensions of crossed and quadratic complexes, co-h-structures and applications, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3623–3653. [MW50] S. MacLane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. 36 (1950), 41–48. [Tak05] M. Takuo, The homotopy category of certain topological monoidal categories, Proceedings of the International Conference on Homotopy Theory and Related Topics (Seoul) (M. H. Woo, ed.), Korea University, 2005, pp. 31–38. [Tod62] H. Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. [Whi41] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. [Whi49] ,Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496. [Whi50] ,A certain exact sequence, Ann. Math. 52 (1950), 51–110. SECONDARY HOMOTOPY GROUPS 37 Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail address:[email protected], [email protected]