J. Homotopy Relat. Struct. (2016) 11:643–677 DOI 10.1007/s40062-016-0157-8
Higher Toda brackets and Massey products
Hans-Joachim Baues1 · David Blanc2 · Shilpa Gondhali2
Received: 3 March 2015 / Accepted: 4 September 2015 / Published online: 26 November 2016 © Tbilisi Centre for Mathematical Sciences 2016
Abstract We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and Massey products for differential graded algebras, among others.
Keywords Higher order homotopy operation · Higher order cohomology operation · Toda bracket · Massey product · Chain complex · Enriched category · Path object · Monoidal model category
Mathematics Subject Classification Primary 18G55; Secondary 55S20 · 55S30 · 55Q35 · 18D20 1 Introduction
Toda brackets and Massey products have played an important role in homotopy theory ever since they were first defined in [39,60,61]: in applications, such as [2,5,23,35,55],
Dedicated to Ronnie Brown on the occasion of his eightieth birthday.
Communicated by Tim Porter and George Janelidze.
B Shilpa Gondhali [email protected]; [email protected] Hans-Joachim Baues [email protected] David Blanc [email protected]
1 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany 2 Department of Mathematics, University of Haifa, 31905 Haifa, Israel 123 644 H.-J. Baues et al.
Cf
G n CS CSp f g h +1 n p k Sn S S S X
CSn F
Fig. 1 The Toda bracket construction and in a more theoretical vein, as in [1,8,21,29,38,53,56]. There are a number of variants (see, e.g., [3,20,42,47] and [6, Sect. 3.6.4]), as well as higher order versions including [27,28,30,41,43,46,48,49,51,57,63]. In recent years they have appeared in many other areas of mathematics, including symplectic geometry, representation the- ory, deformation theory, topological robotics, number theory, mathematical physics, and algebraic geometry (see [4,10,18,19,26,31,32,52]). Toda brackets were originally defined for diagrams of the form
f g h Sn −→ S p −→ Sk −→ X, (1.1) with g ◦ f and h ◦ g nullhomotopic. If we choose nullhomotopies F : g ◦ f ∼ 0 and G : h ◦ g ∼ 0, they fit into a diagram of cones as in Fig. 1. This yields an element h, g, f in [Sn+1, X] called the Toda bracket.Thevalue we get depends on the choices of nullhomotopies F and G, so it is not uniquely determined. The Toda bracket is thus more properly a certain double coset of k # h#πn+1(S ) + f πp+1(X). If we view [h] as an element in π∗ X, while [g] is seen as a primary homotopy operation acting trivially on [ f ] and [h]◦[g]=0 is a relation among primary opera- tions, we can think of the Toda bracket as a secondary homotopy operation. Similarly, a diagram of the form
f g h X −→ K (G, n) −→ K (G , p) −→ K (G , k) (1.2) with g ◦ f ∼ 0 ∼ h ◦ g defines a secondary cohomology operation in the sense of [2]. On the other hand, the Massey product in cohomology—defined whenever we have three classes α, β, γ ∈ H ∗ X with α · β = 0 = β · γ —is a different type of secondary cohomology operation which does not fit into this paradigm. All three examples have higher order versions, though the precise definitions are not always self-evident or unique (cf. [27,41,63]). Nevertheless, these higher order operations play an important role in homotopy theory—for instance, in enhancing our theoretical understanding of spectral sequences (cf. [9]) and in providing a conceptual full invariant for homotopy types of spaces (see [13,59]). 123 Higher Toda brackets and Massey products 645
The main goal of this note is to explain that higher order Toda brackets and higher Massey products have a uniform description, covering all cases known to the authors (including both the homotopy and cohomology versions). The setting for our general notion of higher Toda brackets is any category C enriched in a suitable monoidal category M. In fact, the minimal context in which higher Toda brackets can be defined is just an enrichment in a monoidal category equipped with a certain structure of “null cubes”, encoded by the existence of an augmented path space functor PX → X satisfying certain properties (abstracted from those enjoyed by the usual path fibration of topological spaces). We call such an M a monoidal path category—see Sect. 2. In this context we can define the notion of a higher order chain complex: that is, one in which the identity ∂∂ = 0 holds only up to a sequence of coherent homotopies (see Sect. 3). This suffices to allow us to define the values of the corresponding higher order Toda bracket (see Sect. 4, where higher Massey products are also discussed). However, in order for these Toda brackets to enjoy the expected properties, such as homotopy invariance, M must be also be a simplicial model category. In this case there is a model category structure on the category M-Cat of categories enriched in M, due to Lurie, Berger and Moerdijk, and others, in which the weak equivalences are Dwyer-Kan equivalences (see Definition 5.7). This is explained in Sect. 5, where we prove: Theorem A Higher Toda brackets are preserved under Dwyer-Kan equivalences. [See Theorem 5.14 below]. We also show that the usual higher Massey products in a differential graded algebra correspond to our definition (see Proposition 5.20). In Sect. 6 we study the case of ordinary Toda brackets for chain complexes, and show their interpretation as secondary Ext-operations. Notation 1.1 The category of sets will be denoted by Set, that of compactly generated topological spaces by Top (cf. [58], and compare [62]), and that of pointed compactly generated spaces by Top ∗. If R is a commutative ring with unit, the category of R-modules will be denoted by ModR (though that of abelian groups will be denoted simply by AbGp). The 0 category of non-negatively graded R-modules will be denoted by grModR , with objects E∗ ={En}n≥0, and so on. The category of Z-graded chain complexes over ModR will be denoted by ChR, with objects A∗, B∗, and so on, where ∂n ∂n−1 ∂n−2 A∗ := ···An −→ An−1 −−→ An−2 −−→ An−3 ··· .
The category of nonnegatively graded chain complexes over ModR will be denoted by 0 : → : −→ ChR . A chain map f A∗ B∗ inducing an isomorphism f∗ HnA∗ HnB∗ for all n is called a quasi-isomorphism. Finally, the category of simplicial sets will be denoted by S, and that of pointed simplicial sets by S∗. 123 646 H.-J. Baues et al.
2 Path functors in monoidal categories
Higher order homotopy operations in a pointed model category C, such as Top ∗, S∗,or ChR, are usually described in terms of higher order homotopies, which can be defined in turn in terms of an enrichment of C in an appropriate monoidal model category M (see, e.g., [12]). We here abstract the minimal properties of such an M needed for the construction of higher operations. Definition 2.1 A monoidal path category is a functorially complete and cocomplete pointed monoidal category M, ⊗, 1, equipped with an path endofunctor P : M → L M and natural transformations pX : PX → X, θ : PX ⊗ Y → P(X ⊗ Y ), and θ R : X ⊗ PY → P(X ⊗ Y ). We require that the following diagrams commute: (a) Constant path combinations:
θ L θ R PX ⊗ Y / P(X ⊗ Y ) X ⊗ PY / P(X ⊗ Y )
pX ⊗IdY pX⊗Y IdX ⊗pY pX⊗Y = = X ⊗ Y / X ⊗ YX⊗ Y / X ⊗ Y (2.1)
(b) Algebra structure (for each k ≥ 1):
k (p ) P X / Pk+1 X Pk X p p (2.2) Pk X Pk−1 X k−1(p ) P X / Pk X Pk−1 X.
(c) Left and right constants:
θ R PX ⊗ PY / P(PX ⊗ Y )
θ L Pθ L (2.3) / P(X ⊗ PY) P2(X ⊗ Y ) Pθ R
(d) From (2.3) we see that there are natural transformations
θ (i, j) : Pi X ⊗ P j Y → Pi+ j (X ⊗ Y )
for any i, j ≥ 0, defined
θ (i, j) := Pi+ j−1(θ L ) ◦···◦ P j (θ L ) ◦ P j−1(θ R) ◦···◦θ R.
These are required to be associative, in the obvious sense. 123 Higher Toda brackets and Massey products 647
(e) If we let Pn X denote the result of applying the functor P : M → M to X n times (with P0 := IdM), we have n + 1 different natural transformations ∂n : n+1 → n = ,..., i P X P X (i 0 n), defined
∂ = ∂n := i (p ). i i P Pn−i X (2.4)
The natural transformations θ (i, j) are required to satisfy the identities:
(i−1, j) i−1 − ( , ) θ ◦ (∂ ⊗ Id) if 0 ≤ k < i ∂n 1 ◦ θ i j = k (2.5) k θ (i, j−1) ◦ ( ⊗∂ j−1) ≤ < Id k−i if i k n
for every 0 ≤ k < i + j = n.
Remark 2.2 The commutativity of (2.2) implies that the natural transformations of (2.4) satisfy the usual simplicial identities
∂n−1 ◦ ∂n = ∂n−1 ◦ ∂n i j j−1 i (2.6) for all 0 ≤ i < j ≤ n (see [64, Sect. 8.6]).
2.1 Paths and cubes
The natural setting where such path categories arise is when a monoidal category M is also simplicial, in the sense of [50, II, Sect. 1]. More specifically, we require the existence of an unpointed path functor (−)I : M → M which behaves like a mapping space from the interval [0, 1], so we have natural transformations
(a) e0, e1 : X I → X (evaluation at the two endpoints), (b) s : X → X I with e0s = e1s = Id (the constant path), and (c) θ L : X I ⊗ Y → (X ⊗ Y )I and θ R : X ⊗ Y I → (X ⊗ Y )I (paths in a product).
These make the following diagrams commute:
θ L θ R I / I I / I X H ⊗ Y (X ⊗ YV ) X ⊗H Y (X ⊗ YV ) ⊗ i ⊗ i ⊗ ⊗ i i s Id eX IdY eX⊗Y s Id s IdX eY eX⊗Y s = = X ⊗ Y / X ⊗ Y X ⊗ Y / X ⊗ Y (2.7) 123 648 H.-J. Baues et al. for i = 0, 1, as well as
k (ei )I θ R k+1 X / k / X I X I X I ⊗ Y I (X I ⊗ Y )I j j L L I e k e k−1 and θ (θ ) X I X I I k / I k−1 ( ⊗ I )I / I 2 X − X X Y (X ⊗ Y ) ( i )I k 1 (θ R )I eX (2.8) for all k ≥ 1 and i, j ∈{0, 1}. We may then define the required (pointed) path functor P : M → M by the functorial pullback diagram:
/ PX X I PB e0 (2.9) / ∗ X.
The commutativity of the right hand square in (2.8) allows us to define either composite 2 to be the natural transformation θ (1,1) : X I ⊗ Y I → (X ⊗ Y )I . We see that θ L induces a natural transformation θ L : PX ⊗ Y → P(X ⊗ Y ), and similarly θ R : X ⊗ PY → P(X ⊗ Y ), making (2.1) commute. Moreover, from (2.7) we see that (2.3) commutes, and that the natural transforma- tions θ (i, j) are associative and satisfy (2.5).
Example 2.3 The motivating example is provided by M = Top ∗, with the monoidal I structure given by the smash product ⊗:=∧, and X := map∗(I, X) the mapping space out of the interval I := [1]+. Thus PX is the usual pointed path space. Here map∗(X, Y ) denotes the set Hom (X, Y ) equipped with the compact-open Top ∗ topology.
Example 2.4 Similarly for S∗, again with the smash product ⊗:=∧and X I := map∗( [1]+, X), where map∗(X, Y ) ∈ S∗ denotes the simplicial mapping space with ( , ) := ( × [ ] , ) map∗ X Y n HomS∗ X n + Y . When X is a Kan complex, we can use Kan’s model for PX, where (PX)n := ( ... : → ) p : → i Ker d1d2 dn+1 Xn+1 X0 , and X PX X is d0 in simplicial dimension i. Example 2.5 Another variant is provided by a suitable category Sp of spectra with strictly associative smash product ∧, such as the S-modules of [17], the symmetric spectra of [25], and the orthogonal spectra of [36]. One again has function spectra ( , ) I mapSp X Y , which can be used to define X and PX. The unit is the sphere spectrum S0.
Example 2.6 For chain complexes of R-modules we have a monoidal structure with ( ⊗ ) := ⊗ the tensor product A∗ B∗ n i+ j=n Ai B j . 123 Higher Toda brackets and Massey products 649
Recall that the function complex Hom(A∗, B∗) is given by Hom(A∗, B∗)n := Hom(Ai , Bi+n), (2.10) i∈Z
∂ (( ) ) := (∂ B − (− )n ∂ A) ( : → ) with n fi i∈Z i+n fi 1 fi−1 i i∈Z for fi Ai Bi+n i∈Z. I Thus for M = ChR we may set X := Hom(C∗( [1]; R), X), and see that PA∗ has
n+1 (PA)n = An ⊕ An+1 with ∂(a, a ) = (∂a,∂a + (−1) a), (2.11)
and pA∗ the projection.
2.2 Cores and elements
In any monoidal path category M, ⊗, 1, P and for any X ∈ M, we can think of HomM(1, X) as the ‘underlying set’ of X, and think of a map f : 1 → X in M as an ‘element’ of X. More generally, we may have a suitable monoidal subcategory I of M, which we call a core, and define a generalized element of X to be any map f : α → X in M with α ∈ I.
Example 2.7 We may always choose I ={1} to consist of the unit of M alone. However, in some cases other natural choices are possible: I := { n}∞ (a) In the three Examples of 2.3, 2.4, and 2.5, we can let S S n=0 consist of all (non-negative dimensional) spheres—this is evidently closed under ⊗=∧. (b) In the category of chain complexes over a ring R (Example 2.6), we let IR := {M(R, n)∗}n∈Z, where M(R, n)∗ is the Moore chain complex with M(R, n)i = R for i = n, and 0 otherwise. Again we see that M(R, p)∗ ⊗ M(R, q)∗ = M(R, p + q)∗,soIR is indeed a monoidal subcategory of (ChR, ⊗R, M(R, 0)∗). We see that a generalized element in a chain complex A∗ is now a map f : M(R, n)∗ → A∗ in ChR—that is, an n-cycle in A∗. I := { (Z/ , )}∞ (c) Other examples are also possible—for example, if M p n n=1 is the collection of mod p Moore spaces, representing mod p homotopy groups (see 0 [45]), then it is not itself a monoidal subcategory of (Top ∗, ∧, S ), since it is not closed under smash products. However, when p is odd, the collection of finite wedges of such Moore spaces is monoidal, by [45, Corollary 6.6].
3 Higher order chain complexes
The structure defined in the previous section suffices to define higher order chain complexes, as in [9]: 123 650 H.-J. Baues et al.
3.1 Categories enriched in monoidal path categories
Let C be a category enriched in a monoidal path category M, ⊗, 1, P, so that for any a, b ∈ Obj C we have a mapping object mapC(a, b) in M, and for any a, b, c ∈ Obj C we have a composition map
μ = μa,b,c : mapC(b, c) ⊗ mapC(a, b) −→ mapC(a, c)
(written in the usual order for a composite), satisfying the standard associativity rules. As in Sect. 2.2, we can think of a morphism f : 1 → mapC(a, b) in M as an ‘element’ of mapC(a, b),orsimplyamap f : a → b. In particular, we have ‘identity maps’ Ida in mapC(a, a) for each a ∈ Obj C, satisfying the usual unit rules. In addition, a morphism F : 1 → PmapC(a, b) is called a nullhomotopy of := p ◦ : → f mapC (a,b) F. Higher order nullhomotopies are defined by maps F 1 i P mapC(a, b). The functoriality of P implies that we can also compose (higher order) nullhomo- topies by means of the composite of
(i, j) i j θ i+ j P mapC(b, c) ⊗ P mapC(a, b) −−→ P [mapC(b, c) ⊗ mapC(a, b)] i+ j μ P i+ j −−−→ P mapC(a, c), (3.1)
i, j i j i+ j which we denote by μ : P mapC(b, c) ⊗ P mapC(a, b) → P mapC(a, c). Again, the maps μ(−,−) are associative. For a general core I ⊆ M (cf. Sect. 2.2), we have generalized elements given by maps f : α → mapC(a, b) for α ∈ I. We use the fact that I is a monoidal subcategory to define the composite of f : α → mapC(a, b) with g : β → mapC(b, c) (β ∈ I) to be the composite in M of
g⊗ f μ β ⊗ α −−→ mapC(b, c) ⊗ mapC(a, b) −→ mapC(a, c), (3.2) and similarly for generalized (higher order) nullhomotopies. From (2.5) we see that: i−1, j i−1 − , μ ◦ (∂ ⊗ Id) if 0 ≤ k < i ∂n 1 ◦ μi j = k (3.3) k μi, j−1 ◦ ( ⊗∂ j−1) ≤ < + Id k−i if i k i j for every 0 ≤ k < i + j = n.
Remark 3.1 If the path structure P comes from a unpointed path structure (−)I as in I Sect. 2.1, a morphism F : 1 → mapC(a, b) in M is called a homotopy F : f0 ∼ f1 := 0 ◦ := 1 ◦ between f0 emapC F and f1 emapC F. I i Higher order homotopies are defined by maps F : 1 → mapC (a, b), and the functoriality of (−)I implies that we can compose (higher order) homotopies by means of the composite of 123 Higher Toda brackets and Massey products 651
i+ j θ(i, j) μI I i I j / I i+ j / I i+ j mapC(b, c) ⊗ mapC(a, b) [mapC(b, c) ⊗ mapC(a, b)] mapC(a, c) ,
i, j I i I j I i+ j which we denote by μ : mapC(b, c) ⊗mapC(a, b) → (mapC(a, c)) . These induce the maps μi, j , as in Sect. 2.1.
Definition 3.2 Assume given a monoidal path category M, ⊗, 1, P with core I in M (cf. Sect. 2.2), and choose an ordered set = (γ1,...,γN ) of N core elements. K = , {{ k }N }n M An n-th order chain complex K F(i) i=k+1 k=0 over (for ) of length N ≥ n + 2 consists of:
(a) A category K enriched over M, with Obj (K ) ={a0,...,aN } and 1 ∗ if i = j map (ai , a j ) = (3.4) K ∗ if i < j.
K will be called the underlying category of the n-th order chain complex K. (b) For each 0 ≤ k ≤ n and i = k + 1,...N, generalized elements
k : γ ⊗···⊗γ → k ( , ) F(i) i−k i P mapK ai ai−k−1
such that
∂ ◦ k = μk−t−1,t ( k−t−1 ⊗ t ) t F(i) F(i−t−1) F(i) (3.5)
for all 0 ≤ t < k. When N = n + 2, we simply call K an n-th order chain complex.
Remark 3.3 Typically we are given a fixed category C enriched in a monoidal path category M, ⊗, 1, P, and the underlying category K for a higher order chain complex K will simply be a finite subcategory of C (usually not full, because of condition (3.4)). Such a K will be called an n-th order chain complex in C.
K = , {{ k }N }n Definition 3.4 Given an n-th order chain complex K F(i) i=k+1 k=0 over M (for ) of length N, and an enriched functor φ : K → L over M (which we may assume to be the identity on objects, with L also satisfying (3.4)), the induced n-th L = , {{ k }N }n M order chain complex L G(i) i=k+1 k=0 over (for the same ) is defined by setting
k := φ( k ) : γ ⊗···⊗γ → k ( , ) G(i) F(i) i−k i P mapL ai ai−k−1 for all 0 ≤ k ≤ n and k < i ≤ N.
n Remark 3.5 Note that we do not assume that we have n-th order nullhomotopies F(i) > n ( , ) for each i n in P mapK ai ai−n−1 satisfying (3.5). 123 652 H.-J. Baues et al.
However, from (3.5) and (3.3) we see that:
∂ ◦ ∂ ◦ k = μk−t−2,t μk−s−t−2,s( k−s−t−2 ⊗ s ) ⊗ t s t F(i) F(i−s−t−2) F(i−t−1) F(i) if s + t < k − 1, and
∂ ◦ ∂ ◦ k = μk−t−1,t−1 k−t−1 ⊗ μk−s−2,s+t−k+1( k−s−2 ⊗ s+t−k+1) s t F(i) F(i−t−1) F(i−s−t+k−2) F(i) if k − 1 ≤ s + t. Thus from the simplicial identity ∂s ◦ ∂t = ∂t−1 ◦ ∂s for 0 ≤ s < t { k } we deduce that the maps F(i) must satisfy: ⎧ + − ⎪μ( r 1 ⊗ t 1 ⊗ s ) < ⎨ F(i−s−t−3) F(i−s−1) F(i) if s t r s t t s r μ F( − − − ) ⊗ F( − − ) ⊗ F( ) = μ(F ⊗ F ⊗ F ) if s ≥ r and t = 0 i s t 2 i t 1 i ⎪ (i−r−s−2) (i−r−1) (i) ⎩μ( s+1 ⊗ r ⊗ t−1) ≥ > , F(i−r−t−3) F(i−t−2) F(i) if s r and t 0 (3.6) where we have simplified the notation using the associativity of μ.
3.2 A cubical description
Higher order chain complexes were originally defined in [9, Sect. 4] in terms of a cubical enrichment, which is well suited to describing higher homotopies. In general, for an (n − 1)-st order chain complex
0 0 0 F(n+1) F(n) F(1) an+1 −−−→ an −−→ an−2 → ··· → a1 −−→ a0, (3.7)
k we may describe the choices of higher homotopies F(i) succinctly by arranging them as the collection of all the cubical faces in the boundary of I n+2 containing a fixed 0 ⊗ 0 ⊗··· 0 ⊗ 0 vertex (which is indexed by F(1) F(2) F(n) F(n+1)). The k-faces are indexed by
k k k k F 1 ⊗···⊗F r ∈ P 1 map (a , a ) ⊗···⊗ P r map (a + , a − ), (i1) (ir ) K i1 0 K n 1 n kr (3.8) r = = j ( + ) = − + = + with j=1 k j k, i j t=1 kt 1 , and r n k 1 (so i1 k1 1 and ir = n + 1). By intersecting the corner of ∂ I n+2 with a transverse hyperplane in Rn+1 we obtain an (n + 1)-simplex σ, whose n-faces correspond to the (n + 1)-facets of the corner, and so on. More precisely, the cone on this simplex (with cone point the chosen vertex v of I n+2) is homeomorphic to I n+2, with each (n +1)-face of the cone obtained from an (n + 1)-facet τ of the corner by identifying the n-corner opposite v in τ to a single n-simplex in the base of the cone. See Fig. 2. 123 Higher Toda brackets and Massey products 653
Fig. 2 Corner of 3-cube and transverse 2-simplex
∂n : n+1 → n This explains why the maps i P X P X of Definition 2.1, which relate the various ⊗-composites appearing as facets of ∂ I n+1, satisfy simplicial, rather than cubical, identities.
Example 3.6 Consider a second order chain complex
∗ KS (3.9)
∗∗KS g◦h◦k KS
h◦k f ◦g # # k / h / g / f / a4 a3 a2 ; a1 aC 0 g◦h ∗ f ◦g◦h
∗
+ = 0 = 0 = in Top ∗, say, in which we have n 1 4 composable maps: F(1) f , F(2) g, and so on, with all adjacent composites nullhomotopic. 1 = ◦ In this case we may choose nullhomotopies as indicated, namely: F(2) f g ( , )I 1 0( ◦ ) =∗ 1( ◦ ) = 1 = ◦ in map∗ a2 a0 (with e f g and e f g fg), F(3) g h in ( , )I 1 1 = ◦ ( , )I 1 ◦ map∗ a3 a1 , and F(4) h k in map∗ a4 a2 —so that in fact f g is in the ( , ) 2 = ◦ ◦ pointed path space P map∗ a2 a0 . Similarly, F(4) f g h is a homotopy of nullhomotopies between h∗( f ◦ g) and f ∗(g ◦ h). The more suggestive notation f ◦g, and so on, is motivated by the cubical Boardman- Vogt W-construction of [14, Sect. 3], as explained in [9, Sect. 5]: we think a k-th order homotopy as a k-cube in the appropriate mapping spaces.
123 654 H.-J. Baues et al.
1 0 0 F(2) ⊗ F(3) ⊗ F(4) = f ◦ g ⊗ h ⊗ k
f ◦ g ⊗ h ◦ k = f ◦ g ◦ h ⊗ k 1 1 = F(2) ⊗ F(4) 2 0 F(3) ⊗ F(4)
0 0 1 f ⊗ g ⊗ h ⊗ k F(1) ⊗ F(2) ⊗ F(4) = f ⊗ g ⊗ h ◦ k
0 1 0 f ⊗ g ◦ h ◦ k f ⊗ g ◦ h ⊗ k = F(1) ⊗ F(3) ⊗ F(4) = 0 2 F(1) ⊗ F(4)
Fig. 3 The cubical corner
If we apply the usual composition map
I 1 I 1 μ : map∗(a2, a1) ⊗ map∗(a4, a2) → map∗(a4, a1) to g ⊗ h ◦ k, we obtain a nullhomotopy of ghk, and similarly for g ◦ h ⊗ k I 1 in map∗(a3, a1) ⊗ map∗(a4, a3). Thus we may ask if these two nullhomotopies are themselves homotopic (relative to ghk): if so, we have a 2-cube g ◦ h ◦ k in I 2 2 map∗(a4, a1) , which in fact lies in P map∗(a4, a1). The “formal” post-composition 2 with f ∈ map∗(a1, a0) yields f ⊗g◦h◦k in map∗(a1, a0)⊗P map∗(a4, a1). Together 2 with the other two formal composites f ◦ g ◦h ⊗k in P map∗(a3, a0)⊗map∗(a4, a3) and f ◦ g ⊗ h ◦ k in P map∗(a2, a0) ⊗ P map∗(a4, a2), it fits into the corner of the 2 = ◦ 3-cube described in Fig. 3 (where we use both notations F(2) f g, and so on, to label facets). All vertices but the central one represent the zero map, and the dotted edges represent the trivial nullhomotopy of the zero map (and similarly for the invisible facets of the cube, representing the trivial second-order homotopy of the trivial nullhomotopy).
Remark 3.7 The cubical formalism may be used to describe the iterated path complex PnA∗ in the category of chain complexes (see Example 2.6): We may use the conventions of Sect. 3.2 to identify the k-faces of the corner of an n-cube I n (adjacent to a fixed vertex v), for 0 < k ≤ n, with the (k − 1)-dimensional σ k ( − ) [ − ] ≤ ≤ n − faces (i) of the standard n 1 -simplex n 1 for 0 i k 1(seeFig.2). n σ n [ − ] ( − ) Thus I itself is labelled (0) (corresponding to n 1 ), with the n n 1 -facets n v σ n−1 = σ n σ n−1 = σ n v of I adjacent to labelled (0) d0 (0), (1) d1 (0), and so on. The vertex σ 0 [ − ] is labelled (0) (not corresponding to any real face of n 1 ).
123 Higher Toda brackets and Massey products 655
Then
[σ k ] ( n ) = (i) , P A j A j+k (3.10) 0≤k≤n ≤ <(n) 0 i k
[σ k ] ∂ Pn A : ( n ) → ( n ) ∈ (i) ∂ A( ) with the differential P A j P A j−1 sending a A j+k to a in [σ k ] [d σ k ] (i) ( n ) ∈ t (i) (− )n+k+t the summand A j+k−1 of P A j−1, and sending a A j+k−1 to 1 a in the [σ k ] (i) summand A j+k−1. ∂n : n → n−1 The structure maps i P A∗ P A∗ are given by the projections onto the summands labelled by the i-th simplicial facet of [n] and its simplicial faces, for 0 ≤ i ≤ n − 1.
Example 3.8 The double path complex P2A∗ is given by
2 (P A) j = A j ⊕ A j+1 ⊕ A j+1 ⊕ A j+2, (3.11) with
+ + ∂(a, b, b , c) = (∂a,∂b + (−1) j 1a,∂b + (−1) j 1a,∂c + (−1) j (b − b )). (3.12)
3 Example 3.9 Similarly, (P A) j is given by
[σ 0 ] [σ 1 ] [σ 1 ] [σ 1 ] [σ 2 ] [σ 2 ] [σ 1 ] [σ 3 ] (0) ⊕ (0) ⊕ (1) ⊕ (2) ⊕ (0) ⊕ (1) ⊕ (2) ⊕ (0) A j A j+1 A j+1 A j+1 A j+2 A j+2 A j+2 A j+3 and
∂(a, b0, b1, b2, c0, c1, c2, d) = (∂a,∂b0 − τa,∂b1 − τa,∂b2 − τa, ∂c0 + τ(b1 − b0), ∂c1 + τ(b2 − b0), ∂c2 + τ(b2 − b1), ∂d − τ(c2 − c1 + c1)) for τ = (−1) j .
4 Higher Toda brackets
We now show how one may define the higher Toda bracket corresponding to a higher order chain complex. First, we need to define the object housing it:
Definition 4.1 In any monoidal path category M, ⊗, 1, P we define the (modified) n-fold loop functor n : M → M to be the limit:
n X := lim Pk X (4.1) 1≤k≤n 123 656 H.-J. Baues et al.
∂k : k → k−1 where the limit is taken over all the natural maps i P X P X of Definition 2.1. By Sect. 3.2, we may think of this as a diagram indexed by the dual of the standard n-simplex. The simplicial identities (2.6) imply that there is a natural map
σ n : n+1 → n , X P X X (4.2)
n n which composes with the structure maps πi : X → P X for the limit to yield the n+1 n n face maps ∂i : P X → P X (i = 0,...,n), since X is the n-th matching object for the restricted augmented simplicial object P• X (cf. [22, Sect. 16.3.7]). For n = 0weset0 X := X.
Example 4.2 By Sect. 3.2, we may think of (4.1) as the limit of a diagram indexed by the dual of the standard n-simplex. Thus 1 X is the pullback in:
/ 1 X PX PB pX (4.3) p X / PX X, indexed by the inclusion of the two vertices into [1], while 2 X is the limit of the diagram:
2 2 2 P XPOO o P X OO o X OOO ooo OOO ooo OO oo OOO oo OOO ooo OO ooo ∂2 OoO ooO ∂2 0 oo OO ∂2 ∂2 oo OO 0 oo OO1 1 oo OO ooo OOO ooo OOO oo ∂2 OO oo ∂2 OO wo 0 ' wo 1 ' PXO PX PX (4.4) OOO ooo OOO ooo OOO ooo OO 1 oo OO ∂ =pX oo ∂1=p OOO 0 ooo ∂1=p 0 X OOO ooo 0 X OO' wooo X
Definition 4.3 Let K be an (n−1)-st order chain complex (of length n+1) enriched in a monoidal path category M, ⊗, 1, P (for a set = (γ1,...,γn+1) of core elements), as in Definition 3.2. If we apply the iterated composition map to each k-face of the form (3.8), we obtain an ‘element’
k k k μ F 1 ⊗···⊗F r : γ ⊗···⊗γ + → P map (a + , a ) (4.5) (i1) (ir ) 1 n 1 K n 1 0
(using the associativity of μ), From (3.5) and (3.3) we see that these elements (4.5) are compatible under the face ∂ : k ( , ) → k−1 ( , ) maps t P mapK an+1 a0 P mapK an+1 a0 , so that they fit together to 123 Higher Toda brackets and Massey products 657 define an element
K:γ ⊗···⊗γ → n−1 ( , ) 1 n+1 mapK an+1 a0 (4.6) which we call the value of the n-th order Toda bracket associated to the chain complex K. K σ n−1 : n → n−1 := If lifts along the map X P X X of (4.2) (for X ( , ) mapK an+1 a0 ), we say that this value of the Toda bracket vanishes. ( − ) K = , {{ k }n+1 }n−1 Remark 4.4 Given an n 1 -st order chain complex K F(i) i=k+1 k=0 over M (for ), any enriched functor φ : K → L over M as in Definition 3.4 takes K to
L:γ ⊗···⊗γ → n−1 ( , ) 1 n+1 mapL an+1 a0 where L is the (n − 1)-st order chain complex induced by φ, by functoriality of the limits in M. Example 4.5 Consider the second order chain complex K of (3.9), where for simplicity we may assume that γ1 = ...γ3 = 1. k In Top ∗ the path functor is given by the usual simplicial structure, so F(i) (for 0 ≤ k ≤ 2 and i = k + 1,...,4) can be thought of as a map from the k-cube I k to the mapping space map∗(ai , ai−k−1), sending the k-corner opposite the chosen vertex v to the basepoint. = μ( 0 ⊗ 2 ) For k 2 we have three composites of the form (3.8), namely: F(1) F(4) ⊗ ◦ ◦ μ( 1 ⊗ 1 ) (denoted by f g h k in the notation of Example 3.6), F(2) F(4) (denoted ◦ ⊗ ◦ μ( 2 ⊗ 0 ) ◦ ◦ ⊗ 2 by f g h k), and F(3) F(4) (denoted by f g h k)—all in P X for X := map∗(a4, a0). ThesewereusedinFig.3 to index the three 2-facets of the corner of I 2 adjacent to v (which is itself indexed by the composite μ( f ⊗ g ⊗ h ⊗ k) of the original maps in (3.9)). ∂2,∂2 : 2 → 1 As illustrated in Fig. 2, the two face maps 0 1 P X P X are shown in the cubical corner of Fig. 3 as the restrictions from the squares to the respective edges out v ∂2(μ( 0 ⊗ 2 )) ⊗ ⊗ ◦ ∂2(μ( 0 ⊗ 2 )) of : thus 0 F(1) F(4) is denoted by f g h k, while 1 F(1) F(4) is denoted by f ⊗ g ◦ h ⊗ k, and so on. As we see from Fig. 3, these simplicial face maps match up (as in Fig. 2) to yield an element in the limit of diagram (4.4), which is simply a map out of the 2-corner of the cube in Fig. 3 composed of the three visible two-facets, and sending the dotted boundary to the basepoint of X. The set of all such maps is precisely 2 X, and the value K of the third order Toda bracket associated to K is the element of 2 X we have just constructed.
4.1 Massey products
Massey products (and their higher order versions) also fit into our setting, although they cannot be defined as ordinary Toda brackets in a model category. This is because a (uni- tal associative) differential graded algebra A∗ over a commutative ground ring R can 123 658 H.-J. Baues et al.
be thought of as a category C with a single object ξ enriched in (ChR, ⊗R, M(R, 0)∗), with HomC(ξ, ξ) := A∗. In this context we choose the core of ChR to be IR as in Example 2.7(b). Thus an (n − 1)-st order chain complex in A∗ consists of:
(a) The sequence of objects—necessarily ai = ξ for all i. 0 : ( , ) → (ξ, ξ) = (b) A sequence of generalized maps F(i) M R mi ∗ HomC for i ,..., + 0 ∈ 1 n 1, which may be identified with an mi -cycle Hi Zmi A∗ (see Example 2.7(b)). 1 ∈ = ,..., + (c) A sequence of generalized nullhomotopies F(i) PA∗ (i 2 n 1), p ( 1 ) = μ( 0 ⊗ 0 ) with A∗ F(i) F(i−1) F(i) . From the description in Example 2.6 we 1 1 ∈ see that F(i) is completely determined by an element Hi Ami +mi−1+1 with ( 1) = 0 · 0 · d Hi Hi−1 Hi (where d is the differential and is the multiplication in A∗). 2 ∈ 2 (d) From Example 3.8 we see that a ‘second-order nullhomotopy’ F(i) P A∗ (i = 3,...,n + 1), which is a ( j + 2)-cycle for j := mi + mi−1 + mi−2,is 2 ∈ determined uniquely by the element Hi A j+2 (the last summand in (3.11)). 2 From the last term in (3.12) we see that F(i) being a cycle means that
( 2) = (− ) j+1 0 · 1 − 1 · 0 . d Hi 1 Hi−2 Hi Hi−1 Hi
(e) In general, for each 1 ≤ k < n and i = k + 1,...n + 1, we have a (generalized) k ∈ k ( + ) := i F(i) P A∗ which is a j k -cycle for j t=i−k mt , with
∂ ◦ k = k−t−1 · t , t F(i) F(i−t−1) F(i) (4.7)
k and from the description in Remark 3.7 we see that again F(i) is completely [σ k ] k (0) determined by the component Hi in the summand A j+k , with
k−1 ( k) = (− )k+ j+1 (− )s s · k−s−1. d Hi 1 1 Hi−k+s Hi s=0
Thus by Definition 4.3 we see that the value of the n-th order Toda bracket asso- n−1 ciated to this (n − 1)-st order chain complex in A∗ is the element in A∗ = k lim1≤k s · n−1−s ∈ = ,..., − , Hs+1 Hn+1 A j+n−1 for s 0 n 1 (4.8) := n where j t=1 mt . 5 Higher Toda brackets in model categories In order to define the values of higher Toda brackets, all we need is a category enriched in a monoidal path category M. However, in applications we want to use such Toda 123 Higher Toda brackets and Massey products 659 brackets, either as obstructions to rectifying diagrams, or as invariants used in com- putations (e.g., of differentials in spectral sequence). For this we need to make an additional Definition 5.1 A path model category is a pointed monoidal model category M, ⊗, 1 in the sense of [24, Ch. 4] which satisfies the conditions of either of [11, Theorems 1.9, 1.10], and which is also a simplicial model category as in [50,II, Sect. 2], equipped with a core I (cf. Sect. 2.2) consisting of cofibrant objects, and a natural transformation K K K ζX,Y,K : X ⊗ Y → (X ⊗ Y ) (5.1) (natural in X, Y ∈ M and K ∈ S). Remark 5.2 By [24, Proposition 4.2.19], a path model category actually has a S∗- model category structure—that is, we have functors (−)K : M → M and (−) ⊗ K : M → M for every pointed simplicial set K ∈ S∗, satisfying the usual axioms. Examples 5.3 In practice we shall be interested only in the following examples: (a) The monoidal structure on Top is cartesian, so we actually have a natural home- =∼ omorphism ζ˜ : X K × Y K −→ (X × Y )K . It is readily verified that in the pointed 0 ˜ K K version Top ∗, ∧, S of Example 2.3,themapζ induces ζ : X ∧ Y → (X ∧ Y )K . (b) The monoidal structure on S is also cartesian, so in the pointed version S∗, ∧, S0 of Example 2.4 we also have an induced map as in (5.1). (c) If we use symmetric spectra as our model for Sp (cf. Example 2.5) we see that K the spectrum X is defined levelwise, so we have (5.1)asforTop ∗. (d) In the category ChR, ⊗, M(R, 0)∗ of chain complexes of R-modules (Example 2.6), the monoidal structure is not cartesian, but the simplicial structure is defined by setting A∗K := Hom(C∗ K, A∗) (where C∗ K is the simplicial chain complex of K ∈ S). The natural transformation (5.1) is induced by the diagonal : K → K × K in S. Note that all of these satisfy the hypotheses of one of [11, Theorems 1.9, 1.10], by [11, Sect. 1.8] and [33, Proposition A.3.2.4–A.3.2.24], so they are in fact path model categories. Remark 5.4 In this case the simplicial structure defines the functor (−)I : M → M, with X I := X [1] (cf. [50, II, Sect. 1]), and PX → X [1] is defined by the pullback k (2.9). We can therefore identify Pk X for each k ≥ 0 with the subobject of X [0,1] consisting of all maps of the k-cube sending the corner opposite a fixed vertex to the basepoint (see Fig. 3). n k Thus X is a subobject of limk map∗([0, 1] , X), which by adjunction may be colim [0,1]k n k identified with X k . Thus X itself is just map∗(colim [0, 1] , X), where the colimit is now taken over all proper faces of [0, 1]n+1, and we identify the corner opposite our chosen vertex of [0, 1]n+1 to a point. This colimit is homeomorphic to 123 660 H.-J. Baues et al. an n-sphere, so n X is homotopy equivalent to the n-fold loop space n X, defined as usual by iterating the functor : M → M given by the pullback / X PX PB pX (5.2) / ∗ X. Remark 5.5 In any path model category M, for any fibrant object X we have an equivalence relation ∼ on the set of morphisms HomM(1, X) (cf. Sect. 2.2), given by: f ∼ g ⇔∃F : 1 → X I such that e0 ◦ F = f and e1 ◦ F = g. We then define the (pointed) set of components π0 X to be the set of equivalence classes in HomM(1, X) under ∼. Now let C be a category enriched in M, and assume the mapping objects mapC(a, b) are fibrant (e.g., if all objects in M are fibrant, as in Top ∗), If we denote π0mapC(a, b) simply by [a, b], from Sect 3.1 we see that μ induces an associative composition on [−, −], so that this serves as the set of morphisms in the homotopy category ho C of the M-enriched category C (with the same objects as C). Definition 5.6 More generally, if I is the core of a path model category M, for any core element γ (which is cofibrant by Definition 5.1) the simplicial enrichment mapM in M allows us to identify [γ, X] with π0 mapM(γ, X) (see [50, II, 2.6]). Thus if C is enriched in M,wemayset [a, b]γ := π0 mapM(γ, mapC(a, b)). for any a, b ∈ C and γ ∈ I. Note that for any γ,δ ∈ I and i ≥ 0, the bifunctor ⊗,themapζX,Y, [i] of (5.1) for X := mapC(b, c) and Y := mapC(a, b), and the composition μ : X ⊗ Y → Z (for Z := mapC(a, c)) induce natural maps of sets (mapM(γ, X) × mapM(δ, Y ))i [i] [i] = HomM(γ, X ) × HomM(δ, Y ) ζ ⊗∗ [i] [i] [i] −→ HomM(γ ⊗ δ, X ⊗ Y ) −→ HomM(γ ⊗ δ,(X ⊗ Y ) ) (μ [i]) [i] −−−−→∗ HomM(γ ⊗ δ, Z ) = (mapM(γ ⊗ δ, Z))i and thus a composition map ν : mapM(γ, X) × mapM(δ, Y ) → mapM(γ ⊗ δ, Z) in S. This induces an associative composition map ν∗ :[b, c]γ ×[a, b]δ →[a, c]γ ⊗δ. (5.3) 123 Higher Toda brackets and Massey products 661 Thus we have an I-graded category denoted by hoI C, called the I-homotopy category of C. Definition 5.7 Assume given a path model category M with core I. We say that a M ( , ) M , ∈ category K enriched in is fibrant if mapK a b is fibrant in for any a b K . γ ∈ I (γ, ( , )) Note that since each is cofibrant, this implies that mapM mapK a b is a fibrant simplicial set, by SM7. An enriched functor φ : K → L between categories K and L enriched in M is a Dwyer-Kan equivalence if , ∈ C φ : ( , ) → (φ( ), φ( )) (a) For all a b , mapK a b mapL a b is a weak equivalence in M. I I (b) The induced functor φ∗ : ho K → ho L is an equivalence of I-graded cate- gories. See [54], and compare [11]. We say that such a Dwyer-Kan equivalence is a trivial fibration if each φ : ( , ) → (φ( ), φ( )) M mapK a b mapL a b is a fibration in . By Definition 5.1 and [11, Theorems 1.9, 1.10] we have: Theorem 5.8 There is a canonical model category structure on the category M-Cat of small categories enriched in any path model category M, in which the trivial fibrations and fibrant categories are defined object-wise, and the weak equivalences are the Dwyer–Kan equivalences. Definition 5.9 Let M be a path model category with core I, and let K(0) = , { 0 }n+1} + K F(i) i=1 be a fixed fibrant 0-th order chain complex of length n 1 over M for ⊆ I. We define LK(0) to be the collection of all possible fibrant (n − 1)-st order chain complexes K (of length n + 1) extending K(0). K ∈ L K:γ ⊗···⊗γ → n−1 ( , ) Each K(0) has a value 1 n+1 mapK an+1 a0 , as in (4.6), which we may identify with a 0-simplex in the corresponding simplicial mapping space K∈ (γ ⊗···⊗γ , n−1 ( , )) . mapM 1 n+1 mapK an+1 a0 0 (5.4) n−1 ( , ) ( − ) By Remark 5.4 mapK an+1 a0 is weakly equivalent to the n 1 -fold loop space on the mapping space mapC(an+1, a0) in M (cf. [50, I, Sect. 2]). Moreover, we have a natural isomorphism L mapM(Y, X ) −→ mapS(L, mapM(Y, X)) (5.5) for any X, Y ∈ M and L ∈ S any finite simplicial set, by [50, II, Sect. 1], so we may identify the path component [K] of this 0-simplex with the corresponding element in π (γ ⊗···⊗γ ,n−1 ( , )) 0 mapM 1 n+1 mapK an+1 a0 ∼ π n−1 (γ ⊗···⊗γ , ( , )) = 0 mapM 1 n+1 mapK an+1 a0 ∼ π (γ ⊗···⊗γ , ( , )) = n−1 mapM 1 n+1 mapK an+1 a0 123 662 H.-J. Baues et al. We call the set K(0) := {[K] ∈ π (γ ⊗···⊗γ , ( , )) : K ∈ L } n−1 mapM 1 n+1 mapK an+1 a0 K(0) the n-th order Toda bracket for K(0). We say that it vanishes if 0 ∈K(0). Of course, K(0) may be empty (if there are no (n − 1)-st order chain complexes K extending K(0)). It vanishes if and only if there is an n-th order chain complex extending K(0). Remark 5.10 When K is a higher chain complex in C = M in a monoidal path category enriched over itself (e.g., for M = Top ∗ or S∗), the homotopy class [K ] may be thought of as an element in the group n−1 [ γ1 ⊗···⊗γn+1 ⊗ an+1, a0]∗ Moreover, [K ] vanishes if and only if it represents the zero element in this group. Lemma 5.11 In any simplicial model category M: (a) Any (trivial) cofibration i : K → LinS induces a (trivial) fibration i∗ : X L →→ X K , as long as X ∈ M is fibrant. (b) Any (trivial) fibration f : X → YinM induces a (trivial) fibration f∗ : X K →→ Y K for any (necessarily cofibrant) K ∈ S. Proof This follows from Axiom SM7 for M, the natural isomorphism (5.5), and SM7 for S itself (cf. [50, II, Sects. 1–3]). Lemma 5.12 If M is a simplicial model category and f : X → Y is a (trivial) fibration between fibrant objects in M, then the induced maps Pk f : Pk X → PkY and k f : k X → kY are (trivial) fibrations for all k ≥ 1. Furthermore, if f : X → Y is a weak equivalence between fibrant and cofibrant objects in M,soare Pk f : Pk X → PkY and k f : k X → kY. n n Proof Let C+ denote the sub-cubical set of the cube boundary ∂ I consisting of all facets adjacent to a fixed corner v (i.e., the cubical star of v in ∂ I n), with ∂Cn its n boundary (the cubical link of v), and similarly C− is the cubical star of the vertex v n n n # Cn ∂Cn diagonally opposite v in I . The cofibration i : ∂C+ → C+ makes i : X + → X + a fibration in M, by (a) of Lemma 5.11 In particular, the pullback square − / n n 1 X X C+ PB i# (5.6) / n ∗ X ∂C+ defining n−1 X (see Definition 4.1 and compare Sect. 3.2) is a homotopy pullback (see [40]). 123 Higher Toda brackets and Massey products 663 Thus if f : X → Y is a (trivial) fibration in M, then the induced map n−1 f : n−1 X → n−1Y is a (trivial) fibration, by (b) of Lemma 5.11 Similarly, if we consider the (pointed) cofibration sequence in S∗: 0 S ={0, ∗} → [1]+ =[0, 1]∪{∗}→ → [1]=[0, 1] (with ∗ as basepoint in the first two, and 0 as the basepoint in the cofiber), we see from the corresponding fibration sequence in M: 0 PX = X [1] → X I →→ X S = X that if f : X → Y is a (trivial) fibration in M,soisPf : PX → PY, by (b) of Lemma 5.11 again (see Remark 5.2 above). Lemma 5.13 If X is a fibrant object in a simplicial model category M, then for each ≥ σ n : n+1 → n n 0 the map X P X Xof(4.2) is a fibration. = 0 = σ 0 p : → Note that for n 0, X X and X is simply X PX X. Proof If we consider the map of cofibration sequences (pushouts to ∗)inS: ∂ n / n / n /∂ n C+_ C+ _ C+ _C+ (5.7) n / / n n C− I n I /C− n n n n we see that the natural map C+/∂C+ → I /C− is an inclusion (cofibration) in S∗, σ n : n+1 → n so the natural map it induces—namely, X P X X—is a fibration by (b) of Lemma 5.11. For n = 0 this follows directly because pX is a pullback in the following diagram: / PX X I PB pX e0e1 (5.8) Id ∗ / X X × X where e0e1 is a fibration since it is induced by the cofibration {0, 1} → [1] in S. Theorem 5.14 Let M be a path model category with core I, and let K(0) = , { 0 }n+1} L(0) = , { 0 }n+1} K F(i) i=1 and L G(i) i=1 be 0-th order chain complexes of length n + 1 over M (for the same ⊆ I) with K and L fibrant, and let φ(0) : K(0) → L(0) be a map of 0-th order chain complexes which is a Dwyer-Kan equivalence. Then the I I resulting equivalence of categories φ∗ : ho K → ho L induces a bijection between K(0) and L(0). 123 664 H.-J. Baues et al. Proof We assume for simplicity that φ is the identity on objects, so we may identify π (γ, ( , )) π (γ, ( , )) [ , ] both 0 mapM mapK a a and 0 mapM mapL a a with a a γ . Simi- π (γ, ( , )) π (γ, ( , )) larly we may identify ∗ mapM mapK a a and ∗ mapM mapL a a . Given an (n − 1)-st order chain complex K extending K(0), φ induces an (n − 1)- st order chain complex L extending L(0), as in Definition 3.4, and takes the value K⊂[ , ] L an+1 a0 γ1⊗···⊗γn+1 to . (a) First assume that φ(0) : K(0) → L(0) is a trivial fibration. To show that the above correspondence is a bijection, let L be an (n − 1)-st order chain complex extending L(0). We show by induction on k ≥ 0 that we have an k-th ( ) ( ) ( ) order chain complex K k extending K 0 , where φ∗K k agrees with L to k-th order (by assumption this holds for k = 0). In the induction step, we have a (k − 1)-st order chain complex K(k−1) such that ( − ) ( ) φ∗K k 1 agrees with L to (k − 1)-st order, which we wish to extend to K k . Thus we have a commuting diagram in which Qi is the pullback as indicated, and the map αK : γi−k ⊗ ··· ⊗ γi → k−1 ( , ) k−1 = ,..., − mapK ai ai−k−1 is induced by the maps F(t) (t 0 k 1), using (3.5) and (4.5). Here p2 is a trivial fibration and p1 is a fibration by base change (using Lemmas ψ : γ ⊗ γ → ξ : k ( , ) → 5.12 and 5.13). The maps i−1 i Qi and P mapK ai ai−k−1 Qi exist by the universal property, and ξ is a weak equivalence by the 2 out of 3 property. Factor ξ as j ξ k ( , ) −→ k ( , ) −→ , P mapK ai ai−k−1 P mapK ai ai−k−1 Qi where j a trivial cofibration and ξ is a trivial fibration. Since γi−k ⊗···⊗γi ∈ I is cofibrant, we have a lifting as indicated in the solid commuting square: / ∗ _ k ( , ) Pi4mapK ai ai−k−1 ψ i i i i i i ξ (5.9) i i γ − ⊗···⊗γ / i k i ψ Qi 123 Higher Toda brackets and Massey products 665 σ k := k−1 ( , ) Since j is a trivial cofibration and X is a fibration (for X mapK ai ai−k−1 ) by Lemma 5.13, we have a lift ζ as indicated in: k k P map (ai , ai−k−1) P map (ai , ai−k−1) K _ h h3 K ζ h h h − j h h σ k 1 (5.10) h h K / / − k ( , ) k 1 ( , − − ), P mapK ai ai−k−1 σ mapK ai ai k 1 σ := ◦ξ k : γ ⊗···⊗γ → k ( , ) for p1 . Thus if we set F(i) i−k i P mapK ai ai−k−1 equal to ζ ◦ ψ, we see that − ∂ ◦ Fk = ∂ ◦ ζ ◦ ψ = π ◦ σ k 1 ◦ ζ ◦ ψ = π ◦ p ◦ξ ◦ ψ t (i) t t K t 1 = π ◦ ◦ ψ = π ◦ α = μk−t−1,t k−t−1 ⊗ t t p1 t K F(i−t−1) F(i) (see Definition 4.1 and (3.5)) for all 0 ≤ t < k. Similarly, kφ ◦ k = kφ ◦ ζ ◦ ψ = ◦ ξ ◦ ζ ◦ ψ = ◦ξ ◦ ◦ ζ ◦ ψ P F(i) P p2 p2 j = ◦ξ ◦ ψ = k . p2 G(i) Thus by induction we see that any (n −1)-st order chain complex L(n−1) extending ( ) ( − ) L 0 lifts along φ to K n 1 , so that φ∗ is surjective. On the other hand, since φ is a trivial fibration in mapM, in particular n−1φ : n−1 ( , ) → n−1 ( , ) M mapK an+1 a0 mapL an+1 a0 is a trivial fibration in , so it induces an isomorphism =∼ πn−1 mapM(γ1 ...γn+1, mapK (an+1, a0)) −→ πn−1 mapM(γ1 ...γn+1, mapL (an+1, a0)) by SM7. Thus if K is some other extension of K(0) to an (n − 1)-st order chain complex, [φ K] = [φ K] π (γ ...γ , ( , )) [K] = and ∗ ∗ in n−1 mapM 1 n+1 mapL an+1 a0 , then [K] π (γ ...γ , ( , )) in n−1 mapM 1 n+1 mapK an+1 a0 . L(n−1) n : γ ⊗ We can see directly that vanishes if and only if it lifts to F(n+1) 0 ···⊗γ → n+1 ( , ) n+1 P mapK an+1 a0 , this happens if and only if the corresponding value K(n−1) vanishes, too. (b) Now assume that φ(0) : K(0) → L(0) is an arbitrary weak equivalence, but that K(0) and L(0) are both fibrant and cofibrant. Factoring φ(0) as a trivial cofibration followed by a trivial fibration, by (a) it suffices to assume that φ(0) is a trivial cofibration. This implies that we have a lifting as indicated in the diagram of M-categories 123 666 H.-J. Baues et al. ( ) ( ) K 0 _ o7 K 0 ρ o o φ o o (5.11) o o L(0) / ∗ using Theorem 5.8. Thus by [24, Proposition 1.2.8], φ is a homotopy equivalence (with strict left inverse ρ). Assume now that (L(0))I is a path object for L(0) in M-Cat (cf. [50, I, Sect. 1]), (0) I (0) equipped with two trivial fibrations d0, d1 : (L ) →→L —for example, we may apply the unpointed path functor (−)I of Sect. 2.1 objectwise to the mapping spaces (0) (0) (0) I of L . A right homotopy φ ◦ ρ ∼ Id is then given by H : L → (L ) , and d0, d1 induce the required bijection by (a). (c) Finally, if φ : K(0) → L(0) is any Dwyer–Kan equivalence, with cofibrant replace- ments ψ : K(0) → K(0) and ξ : L(0) → L(0) in M-Cat (so both ψ and ξ are trivial fibrations), we have a lifting ∗ _ g3 L(0) g g g ρ g g g g g ξ (5.12) g g g g g ψ φ (0) / / (0) / (0) K K L where ρ is a Dwyer–Kan equivalence between fibrant and cofibrant M-categories, so it induces a bijection as required by (b), while ψ and ξ are trivial fibrations in M-Cat, so they induce the required bijections by (a). Since the lower right quadrangle in (5.12) commutes, φ also induces a bijection as required. Definition 5.15 Given a path model category M with core I,letC be a (small) subcat- egory of M-Cat consisting of fibrant 0-th order chain complexes of length N = n + 1 for ⊆ I.If∼ is the equivalence relation on C generated by Dwyer–Kan equiva- lences, let Ho C := C/ ∼. An equivalence class in Ho C will be called a homotopy chain complex for . Example 5.16 Our motivating example is when C is an M-subcategory of a model cate- gory C, whose weak equivalences f : X → Y between fibrant objects are maps induc- inganisomorphism f∗ : π∗ mapM(γ, mapC (Z, X)) → π∗ mapM(γ, mapC (Z, Y )) for every cofibrant Z ∈ C and every γ ∈ I. Examples include those of 2.3–2.6 with I as in Example 2.7. In this case a homotopy chain complex of length n + 1inHo C is represented by a sequence of elements ϕ ∈[ , ] =∼ π (γ , ( , )) ( = ,..., + ) i ai ai−1 γi 0 mapM i mapC ai ai−1 i 1 n 1 (5.13) such that ν (ϕ ,ϕ ) = [ , ] ( = ,... + ), ∗ i−1 i 0inai ai−2 γi−1⊗γi i 2 n 1 in the notation of Definition 5.6. 123 Higher Toda brackets and Massey products 667 In particular, when I ={1}, may be described by a diagram: ϕn+1 ϕn ϕ1 an+1 −−→ an −→ an−2 →···→ a1 −→ a0, (5.14) in ho C such that ϕi−1 ◦ ϕi = 0fori = 2,...n + 1. However, in the context of Massey products (cf. Sect. 4.1), we do not have such a model category C available. In this case, we let C be a set of DGAs over R with a given homology algebra, = IR as in Example 2.7(b), and a homotopy chain complex in Ho C is a quasi-isomorphism class of DGAs in C. Definition 5.17 Given a path model category M with core I, a category C as in Definition 5.15 for ⊆ I, and a homotopy chain complex of length n + 1for , the corresponding n-th order Toda bracket is defined to be K(0) ⊆ π (γ ⊗···⊗γ , ( , )) K(0) n−1 mapM 1 n+1 mapK an+1 a0 for some representative of . By Theorem 5.14 we see: Lemma 5.18 The n-th order Toda bracket is well-defined. Remark 5.19 As usual, for a homotopy chain complex : ∗KS " h / g / f / a3 a2 < a1 a0 (5.15) ∗ of length 3, we can identify the corresponding secondary Toda bracket ⊆ π (γ ⊗ γ ⊗ γ , ( , )) 1 mapM 1 2 3 mapK a3 a0 ∼ π (γ ⊗ γ ⊗ γ , ( , )) = 0 mapM 1 2 3 mapK a3 a0 ∗π (γ ⊗ γ , ( , )) as a double coset of the subgroups h 0 mapM 1 2 mapK a2 a0 and π (γ ⊗ γ , ( , )) (γ ⊗ γ , f∗ 0 mapM 2 3 mapK a3 a1 , where h acts on mapM 1 2 mapK R (a2, a0)) via θ (see Definition 2.1, Sect. 2.1 and (2.9)), and similarly for f .Com- pare [61, Sect. 1]. However, for higher order Toda brackets the indeterminacy is too complicated to describe by a single formula, as it depends on various intermediate choices. 5.1 Massey products in DGAs Since ChR is a model category, we can consider higher Toda brackets for a differential graded algebra A∗, as in Sect. 4.1 (we think of A∗ as a chain complex, rather than a cochain complex, but since we allow arbitrary Z-grading, this is no restriction). 123 668 H.-J. Baues et al. + (γ )n+1 A chain complex of length n 1inhoA∗ consists of a sequence i i=1 of homology classes in H∗A∗, with γi · γi+1 = 0fori = 1,...,n. If we choose an n-th order chain complex (that is, a DGA A∗) realizing , as above, we obtain the n−1 element given by (4.8)in A∗. However, because we are working over ModR n−1 ∼ n−1 we can define the identification A∗ = A∗ using the Dold–Kan equivalence (essentially, by the homotopy addition theorem—cf. [44]), and thus obtain the value n−1 (− )s s · n−1−s ∈ 1 Hs+1 Hn+1 A j+n−1 (5.16) s=0 n−1 ( + − ) := n in A∗, which is readily seen to be a j n 1 –cycle for j t=1 mt . By comparing this formula with the classical definition of the higher Massey product (see, e.g., [59, (V.4)]), we find: Proposition 5.20 The higher Toda brackets in a differential graded algebra A∗ are identical with the usual higher Massey products. 6 Toda brackets for chain complexes 0 We now study Toda brackets in the category ChR of non-negatively graded chain complexes over a hereditary ring R, such as Z. It turns out that in this case even ordinary Toda brackets have a finer “homological” structure, which we describe. 6.1 Chain complexes over hereditary rings Since R is hereditary, if Q0(G) is a functorial free cover of an R-module G,wehave a projective presentation αG r 0 → Q1(G) −→ Q0(G) −→ G → 0, where Q1(G) := Ker(r). We then define the n-th Moore complex M(G, n)∗ for an R-module G to be the chain complex with (M(G, n))n+1 := Q1(G), (M(G, n))n := Q0(G), and 0 otherwise, ∂ = αG : 0 → 0 with n+1 . This yields a functor C∗ grModR ChR with C∗(E∗) := M(En, n)∗. (6.1) n≥0 0 Recall that ChR has a model structure in which quasi-isomorphisms are the weak equivalences, and a chain complexes is cofibrant if and only if it is projective in ∈ 0 each dimension (see [24, Sect. 2.3]). Because R is hereditary, any A∗ ChR is uniquely determined up to weak equivalence by the graded R-module H∗A∗ (cf. [15, Theorem 3.4]). 123 Higher Toda brackets and Massey products 669 0 Therefore, if we enrich grModR over ChR by setting Hom(E∗, F∗) := Hom(C∗(E∗), C∗(F∗)) (see Example 2.6), C∗ becomes an enriched embedding, and in fact: : 0 → 0 Lemma 6.1 The functor C∗ grModR ChR is a Dwyer-Kan equivalence over ChR. Since the right-hand side of (6.1) is a coproduct, we see that Hom(E∗, F∗) naturally splits as a product Hom(M(En, n)∗, M(Fn, n)∗) × Hom(M(En, n)∗, M(Fn+1, n + 1)∗) × P, n≥0 (6.2) where P is a product of similar terms, but with H0 P = 0. Moreover, since ∼ [M(E, n)∗, M(F, n)∗] = HomR(E, F) and [M(E, n)∗, M(F, n + 1)∗] ∼ = ExtR(E, F), (6.3) we see that (6.2) is an enriched version of the Universal Coefficient Theorem for chain complexes, stating that for chain complexes over a hereditary ring R there is a (split) short exact sequence: 0 → ExtR(Hn−1A∗, HnB∗) →[A∗, B∗]→ HomR(HnA∗, HnB∗) → 0 n>0 n≥0 0 (cf. [16, Corollary 10.13]). Note that in our version for grModR , the splitting is natural! Notation 6.1 From (6.2) we see that there are two kinds of indecomposable maps of chain complexes (and their nullhomotopies) (see (5.7)): (a) ‘Hom-type’ maps H( f ) : M(E, n)∗ → M(F, n)∗, determined by 00 : ( ) → ( ) 11 : ( ) → ( ). fn Q0 E Q0 F and fn Q1 E Q1 F ( ) : ( ) ∼ 01 : ( ) → ( ) A nullhomotopy H S H f 0 is given by Sn Q0 E Q1 F ,the 00 ( )→ ( ) factorization of fn through Q1 F Q0 F . If it exists, it is unique. (b) ‘Ext-type’ maps E( f ) : M(E, n)∗ → M(F, n + 1)∗, 10 : ( ) → ( ) ( ) : ( ) ∼ determined by fn Q1 E Q0 F . A nullhomotopy E S E f 0is 00 : ( ) → ( ) 11 : ( ) → ( ) given by Sn Q0 E Q0 F and Sn Q1 E Q1 F . 123 670 H.-J. Baues et al. Q1(Fn+1) 11 Sn 11 01 fn fn H(f): Q1(En) Q1(Fn) E(f): Q1(En) Q0(Fn+1) (5.7) 01 Sn 00 Sn Q0(E ) Q0(F ) Q0(E ) n 00 n n fn 0 6.2 Secondary chain complexes in grModR In light of the above discussion, we see that any secondary chain complex f g h C∗(E∗) −→ C∗(F∗) −→ C∗(G∗) −→ C∗(H∗) (6.4) 0 in the ChR-enriched category grModR is a direct sum of secondary chain complexes of one of the following four elementary forms: H( f )E( f ) M(En, n)∗ −−−−−−−→ M(Fn, n)∗ ⊕ M(Fn+1, n + 1)∗ E(g)⊥H(g) H(h) −−−−−−→ M(Gn+1, n + 1)∗ −−−→ M(Hn+1, n + 1)∗, (6.5) H( f )E( f ) M(En, n)∗ −−−−−−−→ M(Fn, n)∗ ⊕ M(Fn+1, n + 1)∗ E(g)⊥H(g) E(h) −−−−−−→ M(Gn+1, n + 1)∗ −−→ M(Hn+2, n + 2)∗, (6.6) H( f ) M(En, n)∗ −−−→ M(Fn, n)∗ H(g)E(g) E(h)⊥H(h) −−−−−−→ M(Gn, n)∗ ⊕ M(Gn+1, n + 1)∗ −−−−−−→ M(Hn+1, n + 1)∗ (6.7) E( f ) H(g)E(g) M(En, n)∗ −−−→ M(Fn+1, n + 1)∗ −−−−−−→ M(Gn+1, n + 1)∗ E(h)⊥H(h) ⊕M(Gn+2, n + 2)∗ −−−−−−→ M(Hn+2, n + 2)∗ (6.8) Two additional hypothetical forms, namely: H( f ) H(g) H(h) (i) M(En, n)∗ −−−→ M(Fn, n)∗ −−−→ M(Gn, n)∗ −−−→ M(Hn, n)∗ E( f ) E(g) E(h) (ii) M(En, n)∗ −−−→ M(Fn+1, n + 1)∗ −−→ M(Gn+2, n + 2)∗ −−→ M(Hn+3, n + 3)∗ in fact are irrelevant to Toda brackets, for dimensional reasons. Moreover, the four elementary secondary chain complexes may or may not split further into one of the following six atomic forms: 123 Higher Toda brackets and Massey products 671 H( f ) H(g) E(h) (a) M(En, n)∗ −−−→ M(Fn, n)∗ −−−→ M(Gn, n)∗ −−→ M(Hn+1, n + 1)∗ and two similar cases with a single E-term; H( f ) E(g) E(h) (b) M(En, n)∗ −−−→ M(Fn, n)∗ −−→ M(Gn+1, n + 1)∗ −−→ M(Hn+2, n + 2)∗ and two similar cases with a single H-term. 0 6.3 Secondary Toda brackets in grModR 0 By Definition 5.17, a secondary Toda bracket in the ChR-enriched category grModR 0 is associated to a homotopy chain complex of length 3 in ho grModR as in (5.14). This means that we replace the actual chain maps in each of the twelve examples of Sect. 6.2 by their homotopy classes: that is, elements in HomR(E, F) or ExtR(E, F), respectively. The compositions Hom(E, F)⊗Ext(F, G) → Ext(E, G),Ext(E, F)⊗Hom(F, G) → Ext(E, G) simply define the functoriality of Ext, while Ext(E, F)⊗Ext(F, G) → Ext(E, G) vanishes for dimension reasons. Nevertheless, the associated Toda bracket may be non-trivial. Note that in this case, as in the original construction of Toda in [61](seealso[56]), the subset of [E∗, H∗] is actually a double coset of the group ( f ) [F∗, H∗]+h[E∗, G∗], so we can think of , which we usually denote simply by h, g, f , as taking value in the quotient abelian group h, g, f ∈( f ) [F∗, H∗]\[E∗, H∗] / h[E∗, G∗]. (6.9) Thus the elementary examples of Sect. 6.2 may be interpreted as secondary oper- ations in ExtR, defined under certain vanishing assumptions, and with an explicit indeterminacy (which may be less than that indicated in (6.9) in any specific case). For example, in (6.6) (case (e) above), the operation is defined for elements in the pullback of ( , ) ⊗ ( , ) ⊗ ( , ) Hom En Fn Ext Fn Gn+1Q Ext Gn+1 Hn+2 QQQ QQQ QQQ ⊗ compQQQ Id QQQ QQQ QQ( ( , ) ⊗ ( , ) Ext En Gn+16 Ext Gn+1 Hn+2 mmm mmm mmm mmm mmm ⊗ mmm comp Id mmm Ext(En, Fn+1) ⊗ Hom(Fn+1, Gn+1) ⊗ Ext(Gn+1, Hn+2) 123 672 H.-J. Baues et al. and takes value in the quotient group Ext(En, Hn+1)/h Hom(En, Gn), where h Hom(En, Gn) refers to the image of the given element h ∈ Ext(Gn, Hn+1) under precomposition with all elements of Hom(En, Gn). It turns out that cases (a) and (d) are trivial for dimension reasons, but we shall now provide examples of non-triviality for four of the remaining cases. 0 = Example 6.2 Consider the homotopy chain complex in ho grModR given by E0 Z/2, F0 = Z/4, G0 = Z/2, and H1 = Z/2, with the corresponding maps f = 2 ∈ Z/2 = Hom(E0, F0) = Hom(Z/2, Z/4) g = 1 ∈ Z/2 = Hom(F0, G0) = Hom(Z/4, Z/2) h = 2 ∈ Z/2 = Ext(F0, H1) = Ext(Z/4, Z/2). By Lemma 5.18, we may choose any cofibrant chain complexes in ChZ to realize , not necessarily the functorial versions C∗(E∗), and so on. In our case we shall use the following minimal secondary chain complex: D2 = Z 11 T0 =1 D∗ α1 =2 11 11 10 f0 =1 g0 =2 h0 =1 A1 = Z B1 = Z C1 = Z D1 = Z A∗ 01=1 B∗ C∗ α0 =2 S0 α0 =4 α0 =2 A = Z B = Z C = Z 0 00 0 00 0 f0 =2 g0 =1 The Toda bracket is given by: ( ) = Z ______1______/ = Z A 2 _ D2 _ −2 2 −1 / ( A)1 = Z D1 = Z The indeterminacy is given by ( f ) [F∗, H∗]+h[E∗, G∗]= f Hom(F0, H1) + h Hom(E0, G1) = 2 · (Z/2) + 0 = 0. Hence the Toda bracket h, g, f does not vanish. 0 = Example 6.3 Consider the homotopy chain complex in ho grModR given by E0 Z/2, F1 = Z/4, G1 = Z/4, and H2 = Z, with the corresponding maps f = 1 ∈ Z/2 = Ext(Z/2, Z/4), g = 2 ∈ Z/4 = Hom(Z/4, Z/4), and h = 2 ∈ Z/4 = Ext(Z/4, Z). We choose the following associated secondary chain complex: 123 Higher Toda brackets and Massey products 673 11 10 g0 =2 h1 =2 B2 = Z C2 = Z D2 = Z 00 B∗ T =1 C∗ α1 =4 1 α1 =4 A = Z B = Z C = Z 1 10 1 00 1 f0 =1 g0 =2 A∗ α0 =2 00 S0 =1 A0 = Z The Toda bracket is represented as follows: ( ) = Z 1 / = Z A 2 _ D2 −2 ( A)1 = Z which is a generator of Ext(Z/2, Z) = Z/2. The indeterminacy is ( f ) [F∗, H∗]+h[E∗, G∗]= f Hom(F0, H1) + h Hom(E0, G1) = 1 · 0 + 2 · (Z/2) = 0. Hence the Toda bracket h, g, f does not vanish. 0 = Example 6.4 Consider the homotopy chain complex in ho grModR given by E0 Z/8, F1 = Z/4, G1 = Z/4, and H2 = Z, with the corresponding maps f = 1 ∈ Z/4 = Hom(Z/8, Z/4), g = 2 ∈ Z/4 = Ext(Z/4, Z/4), and h = 1 ∈ Z/4 = Ext(Z/4, Z). We may choose the following associated secondary chain complex: 10 h1 =1 C2 = Z D2 = Z 11 S0 =1 C∗ α1 =4 g10=2 A = Z B = Z 0 C = Z 1 11 1 1 f0 =2 A∗ B∗ α0 =8 α0 =4 A = Z B = Z 0 00 0 f0 =1 The Toda bracket is given by: ( ) = Z 1 / = Z A 2 _ D2 −8 ( A)1 = Z 123 674 H.-J. Baues et al. which is a generator of Ext(Z/8, Z) = Z/8. The indeterminacy is ( f ) [F∗, H∗]+h[E∗, G∗]= f Ext(F0, H2) + h Hom(E0, G1). A generator of f Ext(F0, H2) = 1 · Ext(Z/4, Z) = Z/4inExt(E0, H2) is given by ( ) = Z _ _ _ _2_ _ _ _/ ( ) = Z 1 / = Z A 2 _ B 2 _ D2 −8 −4 1 / ( A)1 = Z (B)1 = Z while a generator of h Hom(E0, G1) = 1 · Hom(Z/8, Z/4) = Z/4inExt(E0, H2) is given by ( ) = Z 2 / = Z A 2 _ D2 −8 ( A)1 = Z so the total indeterminacy is the subgroup Z/4 ⊆ Z/8 = Ext(Z/8, Z) = Ext(E0, H2). Since the Toda bracket h, g, f is represented by a generator of this Z/8, it does not vanish. 0 = Example 6.5 Consider the homotopy chain complex in ho grModR given by E0 Z/16, F0 = Z/8, F1 = Z/16, G1 = Z/16, and H1 = Z/16, with the corresponding maps f = 1 ∈ Z/8 = Hom(E0, F0), f ∈ Z/16 = Ext(E0, F1), g = 4 ∈ Z/8 = Ext(F0, G1), g ∈ Z/16 = Hom(F1, G1) and h = 2 ∈ Z/16 = Hom(G1, H1). We may choose the following associated secondary chain complex: 11 11 (g )1 =8 h1 =2 B2 = Z C2 = Z D2 = Z T 01=1 11 1 D∗ S =1 α1 =16 0 B∗ C∗ α1 =16 α1 =16 00 00 (g )1 =8 h1 =2 B1 = Z C1 = Z D1 = Z (f )10=1 0 ⊕ g10=4 A = Z B = Z 0 1 11 1 f0 =2 00 T0 =1 A∗ B∗ α0 =16 α0 =8 A = Z B = Z 0 00 0 f0 =1 123 Higher Toda brackets and Massey products 675 The Toda bracket is given by: h11◦S11−T 01◦( f )10=1 ( ) = Z _ _ _ 1_ _0 _ 1_ _ _0 _ _ _ _/ = Z A 2 _ D2 _ −16 16 − f 00◦T 00=−1 0 0 / ( A)1 = Z D1 = Z which is a generator of Hom(Z/16, Z/16) = Z/16. The indeterminacy is ( f ) [F∗, H∗]+h[E∗, G∗]= f Hom(F0, H1) + h Hom(E0, G1). A generator of f Hom(F0, H1) = 1 · Hom(Z/8, Z/16) = Z/8inHom(E0, H1) is given by f 11=2 − ( ) = Z _ _ _ _0 _ _ _ _/ ( ) = Z _ _ _ _ 1_ _ _ _/ = Z A 2 _ B 2 _ D2 _ −16 −8 16 f 00=1 0 / 2 / ( A)1 = Z (B)1 = Z D1 = Z while a generator of h Hom(E0, G1) = 2 · Hom(Z/16, Z/16) = Z/8in Hom(E0, H1) is given by − ( ) = Z _ _ _ _ _2 _ _ _ _/ = Z A 2 _ D2 _ −16 16 2 / ( A)1 = Z D1 = Z so the Toda bracket h, g, f does not vanish. Remark 6.6 See [7, Sect. 6.12] for a calculation relating Toda brackets in topology with a certain operation in homological algebra. Acknowledgements We wish to thank the referee for many useful comments, and Stefan Schwede for a helpful pointer on symmetric spectra. The research of the second author was supported by Israel Science Foundation Grant No. 47377, and he also wishes to thank the Max Planck Institute for Mathematics for their hospitality during several visits. The third author was supported by a Vatat fellowship during the period of this research. References 1. Adams, J.F.: On the structure and applications of the Steenrod algebra. Commun. Math. Helv. 32, 180–214 (1958) 2. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(1), 20–104 (1960) 3. Alexander, J.C.: Cobordism Massey products. Trans. AMS 166, 197–214 (1972) 4. Babenko, I.K., Ta˘ımanov, I.A.: Massey products in symplectic manifolds. Mat. Sb. 191, 3–44 (2000) 123 676 H.-J. Baues et al. 5. Barratt, M.G., Jones, J.D.S., Mahowald, M.E.: Relations amongst Toda brackets and the Kervaire invariant in dimension 64. J. Lond. Math. Soc. 30, 533–550 (1984) 6. Baues, H.J.: Obstruction Theory on Homotopy Classification of Maps. Lec. Notes Math., vol. 628. Springer, Berlin, New York (1977) 7. Baues, H.-J.: Homotopy type and Homology. Oxford Mathematical Monographs, New York (1996) 8. Baues, H.J.: The Algebra of secondary cohomology operations, Progress in Math., vol. 247. Birkhäuser, USA (2006) 9. Baues, H.-J., Blanc, D.: Higher order derived functors and the Adams spectral sequence. J. Pure Appl. Alg. 219, 199–239 (2015) 10. Benson, D., Krause, H., Schwede, S.: Introduction to realizability of modules over Tate cohomology. In: Representations of algebras and related topics. AMS, Providence, RI (2005) 11. Berger, C., Moerdijk, I.: On the homotopy theory of enriched categories. Q. J. Math. 64, 805–846 (2013) 12. Blanc, D., Johnson, M.W., Turner, J.M.: Higher homotopy operations and cohomology. J. K Theory 5, 167–200 (2010) 13. Blanc, D., Johnson, M.W., Turner, J.M.: Higher homotopy operations and André-Quillen cohomology. Adv. Math. 230, 777–817 (2012) 14. Boardman, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lec. Notes Math., vol. 347. Springer, Berlin, New York (1973) 15. Dold, A.: Homology of symmetric products and other functors of complexes. Ann. Math. 68(2), 54–80 (1958) 16. Dold, A.: Lectures on algebraic topology. Springer, Berlin (1995) 17. Elmendorf, A.D., Kˇriž, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. AMS, Providence, RI (1997) 18. Fuchs, D., Weldon, L.L.: Massey brackets and deformations. J. Pure Appl. Alg. 156, 215–229 (2001) 19. Grant, M.: "Topological complexity of motion planning and Massey products". In: Algebraic topology- old and new, pp. 193–203. Polish Acad. Sci. Inst. Math., Warsaw (2009) 20. Hardie, K.A., Kamps, K.H., Marcum, H.J.: A categorical approach to matrix Toda brackets. Trans. AMS 347, 4625–4649 (1995) 21. Heller, A.: Stable homotopy categories. Bull. AMS 74, 28–63 (1968) 22. Hirschhorn, P.S.: Model Categories and their Localizations. Math. Surveys & Monographs, vol. 99. AMS, Providence, RI (2002) 23. Hopkins, M.J., Wickelgren, K.: Splitting Varieties for Triple Massey Products. J. Pure Appl. Alg. 219, 1304–1319 (2015) 24. Hovey, M.A.: Model Categories. Math. Surveys & Monographs, vol. 63. AMS, Providence, RI (1998) 25. Hovey, M.A., Shipley, B.E., Smith, J.H.: Symmetric spectra. J. AMS 13, 149–208 (2000) 26. Kim, M.: Massey products for elliptic curves of rank 1. J. AMS 23, 725–747 (2010) 27. Klaus, S.: Towers and Pyramids, I. Fund. Math 13(5), 663–683 (2001) 28. Kraines, D.P.: Massey higher products. Trans. AMS 124, 431–449 (1966) 29. Kristensen, L.: On secondary cohomology operations. Math. Scand. 12, 57–82 (1963) 30. Kristensen, L., Madsen, I.H.: On evaluation of higher order cohomology operations. Math. Scand. 20, 114–130 (1967) 31. Laures, G.: Toda brackets and congruences of modular forms. Algebraic Geom. Top. 11, 1893–1914 (2011) 32. Laurence,P., Stredulinsky, E.: A lower bound for the energy of magnetic fields supported in linked tori. C. R. Acad. Sci. Paris I Math. 331, 201–206 (2000) 33. Lurie, J.: Higher Topos Theory. Princeton University Press, Princeton (2009) 34. Mac Lane, S.: Categories for the Working Mathematician. Grad. Texts in Math. vol. 5. Springer, Berlin, New York (1971) 35. Mahowald, M.E., Peterson, F.P.: Secondary operations on the Thom class. Topology 2, 367–377 (1964) 36. Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. Lond. Math. Soc. 82(3), 441–512 (2001) 37. Marcum, H.J., Oda, N.: Long box bracket operations in homotopy theory. Appl. Categorical Struct. 19, 137–173 (2011) 38. Margolis, H.R.: Spectra and the Steenrod Algebra: Modules over the Steenrod Algebra and the Stable Homotopy Category. North-Holland, Amsterdam, New York (1983) 39. Massey, W.S.: A new cohomology invariant of topological spaces. Bull. AMS 57, 74 (1951) 123 Higher Toda brackets and Massey products 677 40. Mather, M.: Pull-backs in homotopy theory. Can. J. Math. 28, 225–263 (1976) 41. Maunder, C.R.F.: Cohomology operations of the N-th kind. Proc. Lond. Math. Soc. 13(2), 125–154 (1963) 42. Mizuno, T.: On a generalization of Massey products. J. Math. Kyoto Univ. 48, 639–659 (2008) 43. Mori, M.: On higher Toda brackets. Bull. Coll. Sci. Univ. Ryukyus 35, 1–4 (1983) 44. Munkres, J.R.: The special homotopy addition theorem. Mich. Math. J, 2, 127–134 (1953/1954) 45. Neisendorfer, J.A.: Primary homotopy theory. Mem. AMS, vol. 25. AMS, Providence, RI (1980) 46. Oguchi,¯ K.: A generalization of secondary composition and applications. J. Fac. Sci. Univ. Tokyo Sect. I(10), 29–79 (1963) 47. Peterson, F.P., Stein, N.: Secondary cohomology operations: two formulas. Am. J. Math. 81, 231–305 (1959) 48. Porter, G.J.: Higher order Whitehead products. Topology 3, 123–165 (1965) 49. Porter, G.J.: Higher products. Trans. AMS 148, 315–345 (1970) 50. Quillen, D.G.: Homotopical Algebra, Lec. Notes Math., vol. 20. Springer, Berlin, New York (1963) 51. Retakh, V.S.: Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras. J. Pure Appl. Alg. 89(1–2), 217–229 (1993) 52. Rizzi, C.: Infinitesimal invariant and Massey products. Man. Math. 127, 235–248 (2008) 53. Sagave, S.: Universal Toda brackets of ring spectra. Trans. AMS 360, 2767–2808 (2008) 54. Schwede, S., Shipley, B.: Equivalences of monoidal model categories. Algebraic Geom. Topol. 3, 287–334 (2003) 55. Sharifi, R.T.: Massey products and ideal class groups. J. Reine Angew. Math. 603, 1–33 (2007) 56. Spanier, E.H.: Secondary operations on mappings and cohomology. Ann. Math. 75(2), 260–282 (1962) 57. Spanier, E.H.: Higher order operations. Trans. AMS 109, 509–539 (1963) 58. Steenrod, N.E.: A convenient category of topological spaces. Mich. Math. J 14, 133–152 (1967) 59. Tanré, D.: Homotopie Rationelle: Modèles de Chen, Quillen, Sullivan. Lec. Notes Math. vol. 1025. Springer, Berlin, New York (1983) 60. Toda, H.: Generalized Whitehead products and homotopy groups of spheres. J. Inst. Polytech. Osaka City Univ. Ser. A Math. 3, 43–82 (1952) 61. Toda, H: Composition methods in the homotopy groups of spheres. Adv. in Math. Study, vol. 49. Princeton University Press, Princeton (1962) 62. Vogt, R.M.: Convenient categories of topological spaces for homotopy theory. Arch. Math. (Basel) 22, 545–555 (1971) 63. Walker, G.: Long Toda brackets. In: Proceedings of the Advanced Study Institute on Algebraic Topol- ogy, vol. III (Aarhus, 1970), pp. 612–631. Aarhus (1970) 64. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Uninveristy Press, Cambridge (1994) 123