Universal Toda brackets of ring spectra

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.


Introduction
In this paper, we study a question about highly structured ring spectra. More specifically, we construct a cohomological invariant γ R of a ring spectrum R, called its universal Toda bracket, and examine which information about R is encoded in γ R .
We use the term ring spectrum for what is called an S-algebra in [14], a symmetric ring spectrum in [19], or an orthogonal ring spectrum in [30]. A ring spectrum R has an associated module category Mod-R, which is a stable model category and has a triangulated homotopy category Ho(Mod-R).
For an object X of Ho(Mod-R), its stable homotopy groups π * (X) form a graded π * (R)-module. One of our aims is to understand the resulting functor π * (−) : Ho(Mod-R) → Mod-π * (R) better. Particularly, we want to examine under which conditions a π * (R)-module M is realizable, that is, arises as the homotopy groups of an R-module spectrum.
There is an obstruction theory associated to this problem, with obstructions κ i (M ) ∈ Ext i,2−i π * (R) (M, M ) for i ≥ 3. The first obstruction κ 3 (M ) is always defined and unique. It vanishes if and only if M is a retract of a realizable module. For i ≥ 4, κ i (M ) is only defined if κ i−1 (M ) vanishes, and there are choices involved. We examine these obstructions and show how they depend on the structure of R.
The obstruction theory is the special case of an obstruction theory for realizability in a triangulated category T described in [6,Appendix A]. In this generality, it can be used to find out whether a module over the graded endomorphism ring T (N, N ) * of a compact object N can be realized as T (N, X) * for some object X of T . An algebraic instance of this problem is to realize a module over the cohomology of a differential graded algebra A as the cohomology of a differential graded A-module.
Because of this analogy between ring spectra and differential graded algebras, the following result is a motivation for our work: for a differential graded algebra A over a field k, Benson, Krause, and Schwede [6] study a class γ A ∈ HH 3,−1 We develop a similar theory for ring spectra. Though the obstruction theory for the realizability problem takes place completely in triangulated categories, the definition of a cohomology class with that property needs information from an underlying 'model'. In the case of the differential graded algebra A, the A ∞structure of H * (A) can be used to define γ A [6,Remark 7.4]. In the case of ring spectra, there is no such A ∞ -structure. The appropriate replacement is to use that choosing representatives in the model category of maps in the homotopy category is in general not associative with respect to the composition. This non-associativity leads to obstructions which assemble to a well defined cohomology class.
The formulation of our main results uses Mac Lane cohomology groups, denoted by HML. We define this cohomology theory for graded rings using the normalized cohomology of categories [5]. Its ungraded version is equivalent to Mac Lane's original definition [22]. This theory is, for various reasons, an appropriate replacement of the Hochschild cohomology used in [6]. One reason is that one can, similar to Hochschild cohomology, evaluate a representing cocycle on a sequence of composable maps. If the sequence of maps is a complex, it makes sense to ask the evaluation to be an element of the Toda bracket of the complex.
One main result is the following special case of Theorem 8.1: Theorem 1.1. Let R be a ring spectrum. Then there exists a well defined cohomology class γ R ∈ HML 3,−1 (π * (R)) which, by evaluation, determines all triple matric Toda brackets of π * (R). For a π * (R)-module M which admits a resolution by finitely generated free π * (R)-modules, the product id M ∪γ R ∈ Ext 3,−1 π * (R) (M, M ) is the first realizability obstruction κ 3 (M ).
The term universal Toda bracket for such a cohomology class, as well as the usage of cohomology of categories, are motivated by Baues' study of universal Toda brackets for subcategories of the homotopy category of topological spaces [2,3]. The recent preprint [4] is concerned with a class similar to the γ R of the last theorem, but studies different properties, namely a relation to "quadratic pair algebras". Theorem 1.1 applies for example to the real K-theory spectrum KO. As KO has non-vanishing triple Toda brackets, γ KO is non-trivial. Moreover, the obstructions determined by γ KO detect the non-realizable π * (KO)-module (π * (KO)) ⊗ Z/2. We discuss in Remark 8.5 how this contradicts a claim of Wolbert [43,Theorems 20 and 21].
The proof of Theorem 1.1 divides into two parts. In Section 5, we give a general construction of the universal Toda bracket of a small subcategory of the homotopy category of a stable model category. Specializing to the subcategory of finitely generated free modules in Ho(Mod-R), this defines γ R . Theorem 4.15 shows how a cohomology class which determines Toda brackets also determines the obstructions κ 3 .
Many examples of ring spectra have the property that their ring of homotopy groups is concentrated in degrees divisible by n for some n ≥ 2. Then all realizability obstructions κ 3 vanish for degree reasons. The first realizability obstruction not vanishing for degree reasons is determined by a higher universal Toda bracket, which we also introduce in Theorem 8.1.
The higher universal Toda bracket of a ring spectrum R becomes particularly nice if π * (R) is a graded Laurent polynomial ring on a central generator of degree n. In this case, the higher universal Toda bracket γ n+2 R can be defined as an element of HML n+2 (π 0 (R)). As in Theorem 1.1, it determines (n + 2)-fold Toda brackets and realizability obstructions κ n+2 . But now there is a better chance to actually identify γ n+2 R , since computations of (ungraded) Mac Lane-cohomology groups are known in relevant cases. For example, the universal Toda bracket γ 4 KU of the complex Ktheory spectrum KU is an element of HML 4 (Z) ∼ = Z/2. We prove in Proposition 8.15 that it is the non-zero element.
The calculation of γ 4 KU is a consequence of a different kind of information detected by universal Toda brackets. The Toda brackets of a ring spectrum R can be considered as higher order information about zero divisors in π * (R) and its matrix rings. It turns out that the universal Toda bracket also knows about the units of R and its matrix rings.
To make the slogan precise, recall that for a ring spectrum R and q ≥ 1, there is a path connected space B GL q R. It is the classifying space of the topological monoid given by the invertible path components of the mapping space Map R (R q , R q ). The algebraic K-theory of R can be built from the spaces B GL q R [40,14].
If π * (R) is concentrated in degrees divisible by n for some n ≥ 1, we know that π k (B GL q R) = 0 for 1 < k < n + 1. The following corollary follows from Corollary 8.11 and Theorem 8.7. It is related to [21], see also [3,Example 4.9, Theorem 3.10].
Corollary 1.2. Let R be a ring spectrum such that π * (R) is a Laurent polynomial ring on a central generator in degree n. The restriction map HML n+2 (π 0 (R)) → H n+2 (π 1 (B GL q R), π n+1 (B GL q R)) sends γ n+2 R to the first k-invariant of B GL q R not vanishing for dimensional reasons.
Moreover, with an additional assumption on HML n+1 (π 0 (R)), we interpret the vanishing of γ n+2 R in terms of algebraic K-theory in Proposition 8.14.
Organization. The main results can be found in Section 8. There we also discuss the examples mentioned in the introduction.
In the second section, we briefly review cohomology of categories and Mac Lane cohomology, including a version for graded rings, and define the cup product used in Theorem 1.1. In the third section, we explain the obstruction theory for realizability in triangulated categories. The fourth section is devoted to (higher) Toda brackets in triangulated categories. We explain how Toda brackets determine realizability obstructions.
The fifth section is the technical backbone of this paper. We give a general construction of the universal Toda bracket in the framework of stable topological model categories. Section 6 features a comparison of different definitions of Toda brackets. In Section 7, we show how universal Toda brackets are related to kinvariants of classifying spaces. The Appendix consists of a brief discussion of topological model categories and provides a technical result needed in Section 5.
Institut Mittag-Leffler in Djursholm, a "Kurzstipendium" of the German Academic Exchange Service, and the GRK 1150 "Homotopy and Cohomology" in Bonn.

Mac Lane cohomology
We review the definition of Mac Lane cohomology via cohomology of categories and introduce a cup pairing between the Mac Lane cohomology of a graded ring Λ and a group of Λ-module homomorphisms. As a reference for Mac Lane cohomology, we recommend the last chapter of Loday's book [28,Chapter 13].
2.1. Cohomology of categories and HML. Let C be a small category. A Cbimodule is a functor D : C op × C → Ab. For a map f : X → Y in C, we denote the abelian group D(X, Y ) by D f . For maps g : X ′ → X, h : Y → Y ′ , and f : X → Y , the C-bimodule structure induces actions g * : D f → D f g and h * : D f → D hf . If A is a ring and C is the category of A-modules, the bifunctor Hom A (−, −) provides an example for a C-bimodule.
To define the cohomology a category C with coefficients in a C-bimodule D we consider the cochain complex C * (C, D) with Here N (C) is the nerve of C, so an element of N n (C) is a sequence (g 1 , . . . , g n ) of n composable maps in C. The abelian group structure on C n (C, D) is given by the pointwise addition in D g . For n > 1, the differential δ : . . , g n ) + (−1) n+1 (g n ) * c(g 1 , . . . , g n−1 ).
Definition 2.2. [5, Definition 1.4] The cohomology H * (C, D) of the category C with coefficients in the C-bimodule D is the cohomology of (C * (C, D), δ).
There is a normalized version of this. A category is pointed if it has a preferred zero object * , i.e., * is both initial and terminal. A zero morphism in a pointed category is a map which factors through the zero object. If C is a pointed category, a C-bimodule D is normalized if D( * , X) = 0 = D(X, * ) holds for all objects X.
For a pointed category C and a normalized C-bimodule D, we consider the subgroup C n (C, D) = {c ∈ C n (C, D)|c(g 1 , . . . , g n ) = 0 if g i is zero for some i} of nor- induces an isomorphism in cohomology. Therefore, we can assume representing cochains to be normalized as soon as we consider the cohomology of a pointed category with coefficients in a normalized bimodule. Cohomology of categories has good naturality properties. If F : C → D is a functor and D a D-bimodule, there is an induced C-bimodule F * D, and F induces maps F * : C * (D, D) → C * (C, F * D) and F * : H * (D, D) → H * (C, F * D). The latter map is an isomorphism if F is an equivalence of categories [5,Theorem 1.11].
For a ring A, we denote the category of finitely generated free right A-modules by F (A). To avoid set theoretic problems, we assume F (A) to be small, i.e., we require it to contain only one element from each isomorphism class of objects. The category F (A) is pointed by the trivial module, and for an A-bimodule M , the functor Hom A (−, − ⊗ A M ) is a normalized F (A)-module. If M equals A, we adopt the convention HML s (A) = HML s (A, A).
Mac Lane cohomology was originally defined by Mac Lane in 1956 [29]. Jibladze and Pirashvili [22] proved the equivalence of Mac Lane's definition to the one we use. Mac Lane cohomology is also isomorphic to Ext-groups in the abelian category F (A) of functors from F (A) to Mod-A (see [22]) and to topological Hochschild cohomology (see [9] for the definition and [31] or [34,Theorem 6.7] for the equivalence). The computation of Mac Lane cohomology is known for many examples, including the cases HML * (F p ) (see [15]) and HML * (Z) (see [16]) we encounter in Section 8.
For later use we prove Proof. By [22,Corollary 3.11], this translates to a statement about the Ext-group Since the constant functor represented by P is projective in F (A), it cancels out in the first variable. As I is reduced, i.e., I(0) = 0, it has a projective resolution by reduced functors. Since there is only the trivial map from a reduced functor to the constant functor M , the M cancels out as well.
2.5. Mac Lane cohomology of graded rings. If Λ is a graded ring, the morphisms between graded Λ-modules M and N form a graded abelian group by setting Hom i Λ (M, N ) = Hom Λ (M, N [i]) = Hom Λ (M, N ) −i . Definition 2.6. A graded ring, a graded abelian group, or a graded module is n-sparse if it is concentrated in degrees divisible by n. A full subcategory C of Mod-Λ is n-split if for each pair of objects M and N in C, the graded abelian group Hom Λ (M, N ) * is n-sparse.
For a graded ring Λ, let F (Λ) be the category of finitely generated free graded right Λ-modules. The objects of F (Λ) are finite sums of shifted copies of the free module of rank 1. If Λ is n-sparse for n ≥ 1, the full subcategory of F (Λ) given by the n-sparse Λ-modules is denoted by F (Λ, n). For n = 1, we have F (Λ) = F (Λ, n). The category F (Λ, n) is an example of an n-split subcategory of Mod-Λ.
Definition 2.7. Let Λ be an n-sparse graded ring, and let M be a graded right Λ-module. The graded n-split Mac Lane cohomology of Λ with coefficients in M is defined by , a t-fold shift of Λ for some t ∈ Z, we adopt the convention HML s,t n−sp (Λ) = HML s n−sp (Λ, Λ[t]). If n = 1, we drop '1−sp' from the notation and write HML s (Λ, M ) or HML s,t (Λ).
A central unit u of degree n in Λ is a homogeneous element u of degree n which is a unit and is central in the graded sense. If Λ has a central unit, − ⊗ Λ0 Λ is an equivalence of categories, and Λ n is isomorphic to Λ 0 as Λ 0 -bimodules. This proves Lemma 2.8. Let Λ be an n-sparse graded ring with a central unit u of degree n. Then the restriction induces an isomorphism HML * ,−n n−sp (Λ) → HML * (Λ 0 ). 2.9. Relation to group cohomology. We review some well known maps from Mac Lane cohomology to group cohomology.
For an object X in a category C, we denote its group of automorphisms by Aut(X). The category with a single object X and Hom(X, X) = Aut(X) is denoted by Aut(X). It comes with a canonical inclusion functor Aut(X) → C. If D is an Aut(X)-bimodule, the automorphism group Aut(X) acts via the conjugation action gx = (g −1 ) * (g * (x)) from the left on the abelian group D(X, X). Proposition 2.10. Let C be a small category, let X be an object of C, and let D be a C-bimodule. The inclusion functor F : Aut(X) → C induces a restriction map from the cohomology of C with coefficients in D to the cohomology of the group Aut(X) with coefficients in the Aut(X)-module D(X, X).
Proof. The first map is the restriction along the inclusion. The second map is analogous to the Mac Lane isomorphism between the Hochschild homology of a group ring and group homology [28,Proposition 7.4.2]. On a cochain c, the isomorphism is given by (ϕ(c))(g 1 , . . . , g n ) = (g −1 n · · · g −1 1 ) * c(g 1 , . . . , g n ). When A is a ring and M is an A-bimodule, we write as usual GL q A for the group of invertible (q × q)-matrices, which acts on the abelian group Mat q M of all (q × q)-matrices with entries in M by conjugation. The last proposition specializes to Mac Lane cohomology for graded and ungraded rings: Corollary 2.11. Let Λ be an n-sparse graded ring, let A be a ring, and let M be an A-bimodule. For q ≥ 1, there are restriction maps HML * ,−n n−sp (Λ) → H * (GL q Λ 0 , Mat q Λ n ) and HML * (A, M ) → H * (GL q A, Mat q M ). If A = Λ 0 and M = Λ n , the first map factors through the second map and the restriction HML * ,−n n−sp (Λ) → HML * (Λ 0 , Λ n ). 2.12. The Cup-product. In the following, Ext-groups are understood in the sense of Yoneda. For a graded ring Λ, shifting of modules gives rise to a bigrading on Ext, that is, Ext s,t (M, N ) = Ext s (M, N [t]). Construction 2.13. Let Λ be an n-sparse graded ring. Let M and N be Λmodules such that M admits a resolution by objects in F (Λ, n). Then there is a well defined map we refer to as the cup product. It is bilinear and natural in the sense that (gf ) ∪ γ = g * (f ∪ γ) holds for composable maps of Λ-modules f and g.
The mysterious sign is built in to cancel out with another sign arising in Lemma 4.12. (This will keep signs out of the statements of the main results.) The bilinearity and the naturality with respect to composition of maps are obvious. In Lemma 2.16 and Lemma 2.17 we show that the Ext-class of f ∪ γ doesn't depend on the choice of the cocycle representing γ and the resolution of M .
Remark 2.14. If E is a graded k-algebra over a field k, the tensor product of a right module with a bimodule has a left derived functor . Our cup-product should be thought of as similar to this. The relation becomes clearer when HML * is defined via Ext-groups in the category F (A) of functors F (A) → Mod-A. We sketch the ungraded case.
The self-extensions Ext * F (A) (I, I) in F (A) of the inclusion functor I are isomorphic to HML * (A) [22]. We can enlarge , we see that this recovers the cup-product. We do not go into the details as we only use the description of the product given above. Proof. This statement becomes trivial with Ext defined via projective resolutions.  Proof. Suppose we are given another resolution . . .
. . , λ s ))α s does not hold. As we are only interested in the induced maps on Ext-groups, it suffices to show that the two maps give rise to maps τ, τ ′ : ker λ ′ s−1 → M [t] which induce the same map Ext s,0 i and the definition of δ, we obtain the equation are maps we don't need to know explicitly. Composing with λ 0 [t] and applying Lemma 2.15 completes the proof.

Realizability in triangulated categories
In this section we give a quick review of the obstruction theory for realizability in triangulated categories described in [6,Appendix A]. The necessary background on triangulated categories can be found in Weibel's book [41].
Let T be a triangulated category, which we always assume to have infinite coproducts. An object N of T is compact if the functor T (N, −) preserves arbitrary coproducts. For objects X and Y of T , we write T (X, Y ) * for the graded abelian group whose degree k part is T (X[k], Y ).
We fix a compact object N in T . Under composition, Λ := T (N, N ) * becomes a graded ring, and T (N, X) * is a right Λ-module for every object X. The resulting functor T (N, −) : T → Mod-Λ from T to graded Λ-modules maps distinguished triangles in T to long exact sequences. Furthermore, it preserves arbitrary coproducts since N is compact, and it commutes with the shift of T and Mod-Λ.
Definition 3.1. In the above context, a Λ-module M is called realizable if there is an object X in T such that T (N, X) * ∼ = M .
The following example for this situation is studied in [6]. Let A be a differential graded algebra over a field k, and let T = D(A) be the derived category of dg A-modules. If N is the free module of rank 1, we have T (N, N ) − * = H * (A). The realizability question amounts to whether a graded module over the cohomology ring H * (A) is the cohomology of a dg A-module. In Section 8, we will address the corresponding question for a ring spectrum R: when is a module over the homotopy groups of R the homotopy of an R-module spectrum?
3.2. Realizability obstructions. Let T , N and Λ be as above. An object of T is called N -free if it is a sum of shifted copies of N . We note that T (N, −) * restricts to an equivalence between the full subcategory of N -free objects in T and the category of free Λ-modules.
such that all arrows of the form 1 o o denote morphisms of degree 1, all triangles are distinguished triangles in T , and each object X i is N -free. Moreover, the maps d j = π j−1 ι j with j ≥ 2 and d 1 = ι 1 are required to induce an exact sequence An N -exact Postnikov system is a collection of distinguished triangles as above which extends infinitely to the left.
Proposition A.19 of [6] shows that a Λ-module M is realizable if there exists an N -exact Postnikov system of M . By realizing the first two steps of a free resolution of M , one can easily see that N -exact 2-Postnikov systems exist for every M . Therefore, the realizability problem can be attacked by extending Postnikov systems stepwise to the left.
By [6, Lemma A.12(iii)], every N -exact k-Postnikov system of M induces an exact sequence   3.6. A criterion for uniqueness of obstruction classes. To compare the obstruction classes of different Postnikov systems, we need Definition 3.7. Let (X j , Y j , α j , ι j , π j , M ) and (X ′ j , Y ′ j , α ′ j , ι ′ j , π ′ j , M ) be two Nexact k-Postnikov systems for M . A morphism between them consists of maps f j : X j → X ′ j and g j : Y j → Y ′ j such that f k−1 d k = d ′ k f k and the following commutativity relations hold for 1 ≤ j ≤ k − 1: More generally, for 1 ≤ l ≤ k, an l-map of N -exact k-Postnikov systems for M is a map of the underlying N -exact l-Postnikov systems.
A map between two N -exact k-Postnikov systems induces a map of the long exact sequences representing the obstruction classes, and this map is id M on the outer terms. So the obstruction classes of two Postnikov systems coincide if there is a map between them. Note that this does not need the relation g k−1 ι k = ι ′ k f k , which therefore wasn't required in Definition 3.7. To produce such maps, we use Lemma 3.8. Suppose we are given an l-map between two N -exact k-Postnikov systems with 1 ≤ l < k. There is an element in Ext l,1−l Λ (M, M ) whose vanishing implies the existence of an (l + 1)-map between the Postnikov systems.
Let us assume for a moment our map of Postnikov systems satisfies g l−1 ι l = ι ′ l f l . Then we could find a g l : . The maps f l+1 and g l would complete the required data of an (l + 1)-map. In The g l−1 satisfies the required relations. In addition, , and the modified l-map extends to an (l+1)-map by the argument above.
Recall that a graded abelian group or a graded ring is n-sparse if it is concentrated in degrees divisible by n.

Toda brackets and realizability
We recall the definition of Toda brackets in triangulated categories and show how they are related to the realizability obstructions of the last section.
The maps Our definition differs from [35,Definition A.1] in that we require the objects X j [j] to be the cones of the maps i j , rather than to be isomorphic to the cones. This does not make a difference since triangles isomorphic to distinguished triangles are distinguished again.
For a map λ 1 : If there exists an n-filtered object X ∈ {λ 1 , . . . , λ n−1 }, each twofold composition λ i λ i+1 has to be zero since it can be written as a composition of maps which contains two consecutive maps in a distinguished triangle.
Though a filtered object consists of similar data as a Postnikov system, we emphasize the difference: a filtered object starts from a fixed complex of maps, while a Postnikov system starts from a module and is assumed to have some underlying resolution. Lemma 4.12 shows how in special cases a filtered object gives rise to a Postnikov system.
We refer to Remark 6.2 for a discussion of other definitions of Toda brackets. For n = 3, this defines the triple Toda bracket λ 1 , λ 2 , λ 3 . The cone of λ 2 serves as the 2-filtered object. The set λ 1 , λ 2 , λ 3 is non-empty iff λ 1 λ 2 = 0 = λ 2 λ 3 . It is easy to check that two elements of λ 1 , λ 2 , λ 3 differ by an element of the set ( , which we refer to as the indeterminacy of the Toda bracket. Remark 4.5. In the situation of (λ 1 , λ 2 , λ 3 ) with λ 1 λ 2 = 0 = λ 2 λ 3 , there are two more equivalent definitions of λ 1 , λ 2 , λ 3 which involve distinguished triangles containing λ 1 or λ 3 instead of λ 2 . By choosing distinguished triangles in the horizontal lines and appropriate extensions, one builds the commutative diagram of Figure 2. Considering the middle line as a filtered object, one sees that ǫ 2 τ 3 ∈ λ 1 , λ 2 , λ 3 in the sense of Definition 4.4 above. Starting with the upper line, one can first choose ǫ 3 . Since ǫ 2 τ 3 is a choice for extending λ 1 ǫ 3 to X 3 [1], this is an equivalent definition not involving C 2 . A third definition uses the distinguished triangle in the lower line. Existence and indeterminacy of higher Toda brackets. A sequence (λ 1 , . . . , λ n ) of composable maps has to satisfy restrictive conditions for its Toda bracket to be non-empty. For example, 0 ∈ λ 2 , . . . , λ n−1 is a necessary condition for the existence of an (n − 1)-filtered object X ∈ {λ 2 , . . . , λ n−1 } [35, Proposition A.5], and the additional requirement λ 1 λ 2 = 0 = λ n−1 λ n will in general not be sufficient for λ 1 , . . . , λ n to be non-empty. We introduce an additional assumption to obtain non-empty Toda brackets with controllable indeterminacy.
This is the analog to Definition 2.6 for triangulated categories. If T has a compact object N for which T (N, N ) * is n-sparse, the subcategory of sums of copies of N which are shifted by integral multiples of n is n-split. Proof. To show the first part, we choose a map α : For the second part, we first observe that in which the first term is trivial for j ≤ l − 2 can be used to show the assertion by induction.
Lemma 4.9. Let U be an n-split subcategory of a triangulated category T . Then a sequence X l Proof. The map from X 0 to the cone of λ 1 : X 1 → X 0 gives the data of a 2-filtered object in {λ 1 }. Inductively, we assume that X ∈ {λ 1 , . . . , λ j−1 } is a j-filtered object with j ≤ n and consider the solid arrow diagram This provides the existence of the dotted arrow β. By Lemma 4.3, the cone of β is a (j + 1)-filtered object in {λ 1 , . . . , λ j }. We prove uniqueness by inductively constructing isomorphisms f j : F j X → F ′ j X compatible with all structure maps. This is trivial for the 1-filtered objects. Assume we are given an isomorphism f j−1 : Proposition 4.10. Let U be an n-split subcategory of a triangulated category T and let be a sequence of maps in U with λ i λ i+1 = 0. Then the Toda bracket λ 1 , . . . , λ n+2 is defined, is non-empty, and has the indeterminacy Proof. An (n+1)-filtered object X ∈ {λ 2 , . . . , λ n+1 } exists and is unique by Lemma 4.9. To construct γ n+2 , we consider the exact sequence The last term is trivial by Lemma 4.8. Hence there is a γ n+2 with σ X γ n+2 = λ n+2 [n].
To obtain γ 0 , we use F 1 X ∼ = − → X 1 and λ 1 to get a map F 1 X → X 0 . It can be extended to F 2 X since λ 1 λ 2 = 0. Inductively, we can extend it to a map γ 0 : X = F n+1 X → X 0 : the obstruction for extending a map F j−1 X → X 0 to F j X lies in T (X j−1 [j − 2], X 0 ), which is trivial for 3 ≤ j ≤ n + 1.
Next we compute the indeterminacy. Since we have an exact sequence we know that two different choices of γ n+2 differ by an element in the image of (i n ) * . Using the same argument as in Lemma 4.8, we see that every map X n+2 [n] → F n X factors through σ ′ X : Therefore, the possible difference is in the image of (σ ′ X ) * , and after composing with any choice for γ 0 we obtain that this part of the indeterminacy is ( To examine the other part of the indeterminacy, we first construct an auxiliary Lemma 4.8, there is an ω : X n+1 [n] → X 0 with ωσ X = γ 0 . If we apply (γ n+2 ) * to ωσ X , we see that this part of the indeterminacy is given by (λ n+2 ) * (T (X n+1 [n], X 0 )). 4.11. Relation to realizability obstructions. In this section we exhibit the link between Toda brackets and realizability obstructions. More precisely, we use the cup product of Construction 2.13 to turn the slogan 'the Toda brackets of the resolution are realizability obstructions' into a theorem. The first step is the relation between filtered objects in the sense of Definition 4.2 and Postnikov systems in the sense of Definition 3.3.
, and the (−1)(λ i ) * form the underlying resolution of the Postnikov system.
Before stating the main theorem of this section, we explain why the Mac Lane cohomology groups of Definition 2.7 provide an appropriate tool for the systematic study of Toda brackets.
Definition 4.13. Let T be a triangulated category with a compact object N such that Λ = T (N, N ) * is n-sparse. F T (N, n) is defined to be the full subcategory of T given by finite sums of copies of N which are shifted by integral multiples of n.
The functor T (N, −) * induces an equivalence between F T (N, n) and the category F (Λ, n). This equivalence induces an isomorphism between the Mac Lanecohomology group HML * ,−n n−sp (Λ) and the normalized cohomology of F T (N, n) with coefficients in T (−, −) n Suppose we are given a sequence of composable maps (λ ′ 1 , . . . , λ ′ n+2 ) in F (Λ, n) with λ i+1 λ i = 0 for all i. We define the Toda bracket λ ′ 1 , . . . , λ ′ n+2 of this sequence of maps in Mod-Λ to be the Toda bracket of the sequence (λ 1 , . . . , λ n+2 ) in F T (N, n) associated to it under the equivalence T (N, −) * . If T is the derived category of a dga A, this defines Toda brackets in the cohomology ring H * (A) via the Toda brackets in the derived category D(A). One can check that this recovers the usual notion of Massey products.
Remark 4.14. In the situation above, the indeterminacy of λ 1 , . . . , λ n+2 is , X 0 )) by Proposition 4.10. Now suppose we are given a normalized cocycle c representing a cohomology class γ ∈ H n+2 (F T (N, n), T (−, −) n ). Then c(λ 1 , . . . , λ n+2 ) ∈ T (X n+2 [n], X 0 ). If we change c by adding a coboundary δ(b), the evaluation on (λ 1 , . . . , λ n+2 ) changes by an element of (λ 1 ) Hence the evaluation of a cohomology class has the same indeterminacy as the (n + 2)-fold Toda bracket. Consequently, it makes sense to ask the evaluation of a cohomology class γ ∈ HML n+2,−n n−sp (Λ) on a complex of n-split Λ-modules (λ ′ 1 , . . . , λ ′ n+2 ) to be the Toda bracket λ ′ 1 , . . . , λ ′ n+2 without having to mention indeterminacies. In other words, the indeterminacy of Toda brackets is built into the cohomology of categories. For n = 3, this observation was used for the study of (triple) universal Toda brackets in [3]. Theorem 4.15. Let T be a triangulated category, and let N be a compact object such that Λ = T (N, N ) * is n-sparse. Let M be a Λ-module admitting a resolution by finitely generated free n-sparse Λ-modules. Let γ ∈ HML n+2,−n n−sp (Λ) be a cohomology class such that the evaluation Proof. We denote the realization of the resolution of M by N -free objects by there is a unique n-filtered object Z ∈ {λ 2 , . . . , λ n }. Since the (n + 1)-fold Toda bracket of (λ 1 , . . . , λ n+1 ) contains only zero for degree reasons, we can find maps α : Z → X 0 and β : and Lemma 4.3 tells us that X is an (n + 1)-filtered object in {λ 2 , . . . , λ n+1 } and that Y is an (n + 1)-filtered object in {λ 1 , . . . , λ n }. The Toda bracket of (λ 1 , . . . , λ n+2 ) is non-empty by Proposition 4.10. It can be defined using the n-filtered object X. Hence there are maps γ 0 : X → X 0 and γ n+2 : Looking at the triangle defining X, we see that γ 0 can be constructed by extending α : Z → X 0 to a map X → X 0 . The relation γ 0 ι = α implies the existence of the map ρ in the following commutative diagram: Here we use that the map X 0 → Y from the distinguished triangle defining Y coincides with the map σ ′ Y which is part of the data of the n-filtered object Y . Applying T (N, −) * to the last diagram, we obtain the following commutative diagram of Λ-modules: The lower sequence starting with M in this diagram represents id M ∪γ up to sign. Inspecting (3.4) and Lemma 4.12, we observe that it, up to signs, represents as well the exact sequence associated to the (n + 1)-Postnikov system obtained from Y . This uses that the map (σ Z [1])ω equals the map p n+1 of the (n + 1)-filtered object Y , and therefore the map (−1) n π n [n] of the associated Postnikov system.
The sign of the latter map cancels with the n factors (−1) by which the maps (λ i ) * differ from the differentials of the resolution induced by the Postnikov system. The remaining sign (−1) of the map σ ′ Y cancels with the sign built into the cup product.
Applications of this theorem will be given in Section 8. We point out that for n = 1, the last theorem also leads to an interpretation of the product of a Λ-module homomorphism f : M → M ′ with γ, provided that M satisfies the hypothesis of the theorem: by [6, Proposition 3.4(iv) and Theorem 3.7] and the naturality of the cup product, f ∪ γ vanishes if and only if f factors through a realizable Λ-module.

Construction of universal Toda brackets
As outlined in the introduction, the characteristic Hochschild cohomology class γ A of a dga A considered in [6] is a motivation for the study of the universal Toda bracket γ R of a ring spectrum R. The class γ A cannot be recovered from the derived category D(A) [6,Example 5.15]. This suggests that the construction of γ R from R will need more input than Ho(Mod-R). It turns out that the stable model structure on the category Mod-R together with the topological enrichment provides the necessary information.
Having the example Mod-R in mind, we construct the universal Toda bracket of an n-split subcategory of a general stable topological model category in this section. The applications to ring spectra and the link to the realizability obstructions discussed above are given in Section 8.
Besides [6], Baues' work on universal triple Toda brackets [2,3] is another motivation for our construction (and its name). He is working mainly in an unstable context, considering subcategories of H-group or H-cogroup objects in the homotopy category of topological spaces, though he points out that these constructions generalize to 'cofibration categories' [2, Remark on p. 271]. We will only work in a stable context, in order to provide the link to triangulated categories. This also avoids certain difficulties in the unstable case arising from maps which are not suspensions (see the correction of [3] in [2, Remark on p. 270]). We also do not use Baues' language of 'linear track extensions', as these seem to be only appropriate for the study of triple universal Toda brackets. Nevertheless, the n = 1 case of Proposition A.1 is basically what Baues encodes in a linear track extension.
A motivation for the actual construction of the representing cocycle is the approach of Blanc and Markl to higher homotopy operations [7]. For a directed category Γ, the authors use the bar resolution W Γ in the sense of Boardman and Vogt [8, III, §1] to define general higher homotopy operations. If Γ is the category generated by n + 2 composable morphisms, this specializes to the higher Toda brackets we would like to construct. In this case, W Γ is just an (n + 1)-dimensional cube. As we are not interested in other indexing categories, we will just use the cubes and do not make use of the bar resolution in our construction.
In what follows, we assume familiarity with model categories. Hovey's book [18] provides a good reference. Other than in Quillen's original treatment of model categories [32], we will follow Hovey in assuming our model categories to have all small limits and colimits as well as functorial factorizations. T op will be the category of compactly generated weak Hausdorff spaces, and T op * will be the pointed version. The reason for working with these categories of spaces is that T op * is a closed symmetric monoidal model category [18,Corollary 4.2.12]. We will often use stable topological model categories that are built on T op * . See Appendix A for a brief review. 5.1. Cube systems. Some notation is needed to state the next definition. Let N (U) be the nerve of a small category U. We write d i : N n (U) → N n−1 (U) for the ith simplicial face map, and d fr i : N n (U) → N i (U) and d ba i : N n (U) → N i (U) for the simplicial 'front face' and the 'back face' maps. In our notation for sequences of composable maps from the preceding sections, this means for example . . . , f i ). We resist from reversing the notation for (f 1 , . . . , f n ) to make these formulas more intuitive here, since this would be inconsistent with our previous convention, which was chosen since (f 1 , . . . , f n ) frequently arose from a projective resolution.
With ǫ, ω ∈ {0, 1}, these maps satisfy the relation ǫ i ω j−1 = ω j ǫ i if 1 ≤ i < j ≤ n. We write sk i I n for the i-skeleton of I n in the obvious CW-structure. When we consider I n with n ≥ 1 as a pointed space, we take (1, . . . , 1) as the basepoint.
For a stable topological model category C, we will work with the set of maps T op(I n , Map C (X, Y )). The enriched composition µ of C induces a composition ..,tp+q) (x)))). The associativity of the enriched composition implies that µ p,q+r (id ×µ q,r ) and µ p+q,r (µ p,q × id) correspond under the coherence isomorphism for associativity of the 3-fold cartesian product in T op.
The zero map is a canonical basepoint for Map C (X, Y ). When a possibly different map g : X → Y is used as the basepoint, we write (Map C (X, Y ), g) for the resulting pointed space.
For ǫ ∈ {0, 1} and 1 ≤ i ≤ p, we have (ǫ i ) * : T op(I p , T ) → T op(I p−1 , T ). If 1 ≤ i ≤ p + q, these restrictions satisfy Definition 5.2. Let U be a small full subcategory of the homotopy category of a stable topological model category C. A cube system for U consists of the following data: for every object X of U, there is a cofibrant and fibrant object Φ(X) of C and an isomorphism ϕ X : X → Φ(X) in Ho(C). We write Φ(U) for the set of all those objects. Furthermore, for 0 ≤ j ≤ n there are maps where ∆ is the diagonal and µ i−1,j−i is explained above. By (iii), b j (f 1 , . . . , f j+1 ) maps the basepoint of I j to b 0 (f 1 · · · f j+1 ).
A 0-cube system chooses maps in the model category representing maps in the homotopy category. In general, it is not possible to arrange these choices such that b 0 (f 1 )b 0 (f 2 ) = b 0 (f 1 f 2 ) holds. Nevertheless, these maps are homotopic, and unraveling (ii) and (iii) shows that a 1-cube system specifies a homotopy b 1 (f 1 , f 2 ) between them . For j ≥ 2, the b j (f 1 , . . . , f j+1 ) encode coherence homotopies between different choices of representatives and coherence homotopies of lower degree. Figure 3 (compare [7, Figure 2.12]) illustrates the case n = 3. In the picture, we write Definition 5.3. In the situation of Definition 5.2, a pre n-cube system for U consists of an (n − 1)-cube system for U and a map such that b n and the b j for j < n satisfy conditions (ii)-(iv) of Definition 5.2. This makes sense since (iii) and (iv) only involve the behavior on sk n−1 I n .
Lemma 5.4. An (n − 1)-cube system for U can be extended to a pre n-cube system. The restriction of b n (f 1 , . . . , f n+1 ) to the subcubes (0 i )(I n−1 ) for 1 < i < n is determined by the underlying (n − 2)-cube system.
Proof. Since sk n−1 I n is the union of the (n − 1)-dimensional subcubes (0 i )(I n−1 ) and (1 i )(I n−1 ), we define the restriction of b n to these subcubes by It remains to check that this is well defined on the intersections. Let 1 ≤ j < k ≤ n. On 1 k (1 j (I n−2 )) = 1 j (1 k−1 (I n−2 )) we have Next we check the compatibility on 1 k (0 j (I n−2 )) = 0 j (1 k−1 (I n−2 )). A somewhat lengthy calculation involving the interchange formula for µ p,q and (1 i ) * mentioned y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y (f1)(f2)(f3)•(f4) w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y y y y y y y y y y y y y y y w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w The case of (0 k−1 ) * (1 j ) * b n and (1 j ) * (0 k ) * b n is similar. The remaining case of (0 k−1 ) * (0 j ) * b n and (0 j ) * (0 k ) * b n follows from the associativity of the maps µ p,q .
Proposition 5.5. Let U be a small n-split subcategory of the homotopy category of a stable topological model category C. Then there exists an n-cube system for U.
Inductively, suppose we have constructed a j-cube system for j < n. By Lemma 5.4, it induces a pre (j + 1)-cube system. In order to extend b j+1 (f 1 , . . . , f j+2 ) from sk j I j+1 to I j+1 , it suffices to know that it represents the trivial homotopy class in π j (Map C (Φ(X j+2 ), Φ(X 0 )), b 0 (f 1 · · · f j+2 )). This group is isomorphic to Ho(C) j by means of the isomorphism σ (f1···fj+2) of Proposition A.1, and hence trivial since U is n-split. 5.6. The universal Toda bracket. We now put the data of a cube system together to get the desired cohomology class. We choose an n-cube system for U which is possible by Proposition 5.5, and extend it to a pre (n + 1)-cube system by Lemma 5.
We show in Lemma 5.8 that c is a cocycle. Lemma 5.9 verifies that its cohomology class does not depend on the choice of the cube system. In Proposition 6.1, we show that the evaluation of c on a complex of maps in U is an element of the Toda bracket of that complex. With the comparison of the indeterminacies in Remark 4.14, it follows that the evaluation of the cohomology class γ U of c on a complex in U yields its Toda bracket. This is why we call the well defined cohomology class γ U the universal Toda bracket of U.
Given a pointed topological space K with abelian fundamental group and a pointed map f : sk n I n+2 → K, the Homotopy Addition Theorem states that Here [(ǫ i ) * f ] is the homotopy class of the restriction of f to (ǫ i ) * (sk n I n+2 ), or the image of the operation of a path to the basepoint of sk n I n+2 on [(ǫ) * f ] if (ǫ i ) * (sk n I n+2 ) doesn't contain the basepoint of sk n I n+2 . This is well defined since two such paths are homotopic if n > 1 and the π 1 -action is trivial for n = 1 as π 1 is assumed to be abelian. Our source for this formulation of the theorem is [10, VII.9.6]. The signs result from specifying an orientation through choosing 'even' and 'odd' faces T + and T − , compare [42]. Proof. We fix a sequence of (n + 3) composable maps (f 1 , . . . , f n+3 ) in U. As in Lemma 5.4, the pre (n + 1)-cube system induces We apply the Homotopy Addition Theorem mentioned above to this map to get in π n (Map C (Φ(X n+3 ), Φ(X 0 )), b 0 (f 1 · · · f n+3 )). For n = 1, this uses that C being stable implies π 1 is abelian. For 1 < i < n + 2, the restrictions (0 i ) * e(f ) extend to maps (0 i ) * (sk n+1 I n+1 ) as b i−1 = b i−1 and b n+2−i = b n+2−i . Hence their homotopy classes vanish. Now we apply the isomorphism σ (f1···fn+3) to the sum and use that σ is additive and invariant under the action of basepoint changing paths. Hence . . , f n+2 ).
Lemma 5.9. The cohomology class γ U of Construction 5.7 does not depend on the choice of a cube system.
Proof. In the first step we assume we are given a second (n − 1)-cube system (Φ, ϕ, b j ) for U. We show that it extends to an n-cube system giving the same cohomology class. For every object X in U, our data specifies an isomorphism Φ(X) → Φ(X) in U. We realize it by g X in C and its inverse by g ′ X . For f 1 : ) be a homotopy between them. With similar arguments as in Proposition 5.5, one can iterate the construction to find maps for j < n with (0 1 ) * h j = (g X0 ) * (g ′ Xj+1 ) * b j and (1 1 ) * h j = b j . For j = n, we use the homotopy extension property of (sk n I n+1 ) \ (1 1 )(I n ) → I n+1 to find an h n whose restriction to (1 1 )(I n ) defines b n . When we form the pre (n + 1)-cube systems associated to the two cube systems, the h j assemble to a homotopy between (g X0 ) * (g ′ Xj+2 ) * b n+1 and b n+1 . Hence σ associates the same cocycle to them. Now suppose we are given two n-cube systems for U. The first part shows that we can assume their underlying (n − 1)-cube systems to coincide. Let T dev = {(1, i)|1 ≤ i ≤ n + 1} ∪ {(0, 1), (0, n + 1)}. Lemma 5.4 shows that the associated pre (n + 1)-cube systems can only deviate on the faces specified by T dev .
Let A be the space obtained from sk n I n+1 by gluing for each (ǫ, i) ∈ T dev one copy of I n ∪ sk n−1 I n I n along the right hand side copy of I n to (ǫ i )(I n ). Let i : sk n I n+1 → A be the canonical inclusion and let i : sk n I n+1 → A be the injection which maps (ǫ i )I n to the left copy of I n in the pushout. Then the two pre cube systems induce a map with i * a = b n+1 and i * a = b n+1 . By the slightly different incarnation of the Homotopy Addition Theorem [10, VII.9.5], the evaluations on (f 1 , . . . , f n+2 ) satisfy where ι (ǫ,i) : I n ∪ sk n−1 I n I n → A is the inclusion which belongs to (ǫ, i) ∈ T dev . The signs arise in the same way as in the last lemma.
As in Lemma 5.8, the orientations of the subcubes imply the signs needed for the coboundary formula.
The last lemma completes the proof of γ U being well defined. For later use, we prove two more lemmas closely related to this construction. Proof. We apply G to the data of an n-cube system for U. This gives almost an n-cube system for W. The only missing part is that the objects G(Φ(X)) are not necessarily fibrant. Similarly as in Lemma 5.9, one can construct a cube system for W such that the resulting cocycle representing γ W becomes, after applying G * , equivalent to that of the cube system for U.
Lemma 5.11. Let C be a stable topological model category and let U be a small n-split subcategory of Ho(C). If γ U is trivial, then the map b n of any n-cube system for U can be changed such that the resulting new n-cube system has the zero cochain as a representing cocycle. In particular, the modified n-cube system can be extended to an (n + 1)-cube system.

Comparing definitions of Toda brackets
This section is devoted to the proof of Proposition 6.1. Let C be a stable topological model category, let U be an n-split subcategory of Ho(C), and let X n+2 Remark 6.2. Triple Toda brackets were introduced by Toda [37,38] to study the stable homotopy groups of spheres. Higher Toda brackets were introduced in the 1960's, and there are different approaches in the literature. One of them is Cohen's definition using filtered objects [11, §2]. We used a variant of this for triangulated categories in Definition 4.4.
Another approach is Spanier's definition of higher Toda brackets [36] using the concept of a carrier. A related concept is Klaus' definition of a pyramid [26, 3.4], which is linked to Spanier's definition by [26,Proposition 3.6]. The perhaps most general approach to Toda brackets and other higher homotopy operations is that of Blanc and Markl [7], who define them as obstructions to realizing homotopy commutative diagrams by strictly commutative ones. Their definition of Toda brackets is related to Spanier's [7,Example 3.12].
In Lemma 6.3 below we will see that the evaluation of the universal Toda bracket can be interpreted as something similar to a pyramid in the sense of Klaus. Proposition 6.1 shows that this is equivalent to the Toda bracket defined via filtered objects in Definition 4.4. We work out the comparison as far as needed for our purposes in some detail since we were not able to find an appropriate reference in the literature which relates the different approaches.
For the rest of the section, we fix a U and (f 1 , . . . , f n+2 ) with f i f i+1 = 0 for 1 ≤ i ≤ n + 1 as in the proposition. We also fix an n-cube system b j of U and write b n+1 for the associated pre (n + 1)-cube system and c for the cocycle defined in Construction 5.7. For simplicity, we denote the objects Φ(X) of C chosen by the cube system also by X . The map b j (f 1 , . . . , f j+1 ) : (I j ) + ∧ X j+1 → X 0 will always be the adjoint of the map b j (f 1 , . . . , f j+1 ).
Depending on our chosen n-cube system and (f 1 , . . . , f n+2 ), we now construct objects F j = F j (f 2 , . . . , f j+1 ) in C for j ≤ n + 1. Set The object F j is the coequalizer of two maps h, k : A j A ′ j → B j to be described next. For this, we think of the copies of I j in B j as the j + 1 subcubes (0 i )(I j ) of I j+1 . The copies of I j−1 in A j are thought of as the (j − 1)-dimensional subcubes (0 s )(0 r )(I j−1 ) of I j+1 , and the copies of I j−1 in A ′ j are thought of as the (j − 1)dimensional subcubes (0 s )(1 r )I j−1 .
The map h is given by (0 r ) + ∧ X s on the copy of (I j−1 ) + ∧ X s in A j indexed by (r, s), and by (1 r ) + ∧ X s in the case of A ′ j . The map k is the trivial map to the basepoint on A ′ j . On the copy of (I j−1 ) + ∧ X s in A j indexed by (r, s), it is given by the product of 0 s−1 and the map b s−r−1 (f r+1 , . . . , f s ) using the last (s − r − 1)-coordinates of I j−1 .
Proof. We define (0 i ) * ξ j to be the composition of (I j ) + ∧ X j+2 → (I j ) + ∧ X i given by the identity smashed with b j+1−i (f i+1 , . . . , f j+2 ) using the last (j + 1 − i) coordinates of the cube and the canonical map (I j ) . . , f j+2 ). Its restriction along 1 k−1 is trivial for k > i since b j+1−i (f i+1 , . . . , f j+2 ) can be replaced by the trivial map b j− i (f i+1 , . . . , f k f k+1 , . . . , f j+2 ) there. Its restriction along 1 k is trivial for k < i as well, since these subcubes are mapped to the part of F j (f 2 , . . . , f j+1 ) which gets collapsed. The same arguments as in Lemma 5.4 show that the maps for different i coincide on the intersections. Therefore, we get an induced map ξ j : ∂I j+1 ∧ X j+2 → F j (f 2 , . . . , f j+1 ).
Next we define ζ j . On the copy (I j ) + ∧ X i of B j indexed by i with 1 ≤ i ≤ j + 1, we take b i −1 (f 1 , . . . , f i ) using the first (i − 1) coordinates of I j . This is compatible with the identifications of the coequalizer.
Lemma 6.6. The object F j+1 := F j+1 (f 2 , . . . , f j+2 ) can be constructed from F j := F j (f 2 , . . . , f j+1 ) as the mapping cylinder of the map from F j to the cone C of the map ξ j : ∂I j+1 ∧ X j+2 → F j . The inclusion of F j into the mapping cylinder therefore gives a map ι j : F j → F j+1 .
Proof. Let I j+1 denote the quotient of I j+1 obtained by collapsing the (1 i )(I j ) with 1 ≤ i ≤ j + 1 to a point. Then there is a canonical map ∂I j+1 → I j+1 , and we can interpret I j+1 as a cone on ∂I j+1 . Hence we can model the mapping cone of ξ j by the pushout of I j+1 ∧ X j+2 ← ∂I j+1 ∧ X j+2 ξj − → F j . To replace the map from F j to the cone by a cofibration, we need a cylinder object for F j . One choice for this is (I 1 ) + ∧ F j , which amounts to adding one additional coordinate to each (I i ) + ∧ X k that occurred in the construction of F j . We choose it to be the last coordinate. Hence the mapping cylinder of F j → C is weakly equivalent to the pushout of The pushout of this diagram is isomorphic to F j+1 as defined above. The case j = 1 can easily be deduced from Figure 4.
Proof. This follows from the last lemma and the definition of the distinguished triangles in the homotopy category of a stable model category [18,Chapter 7].
Proof. The last lemma says that we have a cofibration sequence Hence π j is up to homotopy the map from F j to its quotient obtained by collapsing every subcube (I j ) + ∧ X i of B j indexed by 2 ≤ i ≤ j to the (j − 1)-dimensional subcube along which it is glued to (I j ) + ∧ X j+1 . To examine the homotopy class of π j ξ j , we hence only need to know what ξ j does on the subcube (I j ) ∧ X j+2 indexed by j + 1. As it is defined to be the map b 0 (f j+2 ) on that, we are done. Lemma 6.9. If we consider the F j (f 2 , . . . , f j+1 ) as objects of Ho(C), the sequence * → X 1 gives F n (f 2 , . . . , f n+1 ) the structure of an (n + 1)-filtered object in {f 2 , . . . , f n+1 }.
Proof of Proposition 6.1. As we have seen in Lemma 6.5, the composition ζ n ξ n is the map c n (f 1 , . . . , f n+2 ). Hence it represents the evaluation of c by Lemma 6.3.
Let σ X and σ ′ X denote the structure maps of the filtered object F n (f 2 , . . . , f n+1 ). Lemma 6.9 implies σ X ξ n = f n [n − 2]. The definition of ζ n and the fact that σ ′ X is the composition X 1 Hence ξ n ζ n is an element of f 1 , . . . , f n+2 .

Relation to k-invariants of classifying spaces
We saw that the evaluation of the universal Toda bracket on a complex is the Toda bracket of the complex. Since it may as well be evaluated on arbitrary sequences of maps, it will carry more information than just that about the Toda bracket in general. We will now exhibit how its evaluation on a sequence of automorphisms can be expressed. When we apply our theory to ring spectra in Theorem 8.7, this will give us information about the units of ring spectra (and the units of their matrix rings), rather than only the information about zero divisors encoded in the Toda brackets.
A motivation for this comes from Igusa's results [21] about the first k-invariant of the space B GL ∞ (QΩX + ), which is related to Waldhausen's algebraic K-theory of spaces [40]: Igusa shows that the first k-invariant of a connected space X is determined by a cohomology class k H 1 (ΩX) in the cohomology of the monoid π 0 (ΩX) with coefficients in H 1 (X), where the class k H 1 (ΩX) is constructed from the A 4 -part of the A ∞ -structure of ΩX [21, B, Property 1.1]. This observation is also used in [3, Example 4.9, Theorem 3.10].
We fix a stable topological model category C, an n-split subcategory U of Ho(C) for some n ≥ 1, and an n-cube system defining γ U . We also fix an object X of U and denote the representing cofibrant and fibrant object of C which the cube systems chooses as well by X. Consider the topological space Map C (X, X) which is pointed by the zero map in C. Its homotopy groups are As U is n-split, π i (Map C (X, X), 0) is concentrated in degrees divisible by n.
The enriched composition in the category C equips Map C (X, X) with the structure of a topological monoid, and we refer to the composition as the multiplication. Under the identification above, the composition of maps in Ho(C) corresponds to the multiplication of Map C (X, X).
The set π 0 (Map C (X, X)) inherits a monoid structure from Map C (X, X), and Map C (X, X) × denotes the union of all path components of Map C (X, X) which are invertible with respect to the multiplication on π 0 (Map C (X, X)). Therefore, Map C (X, X) × is a group-like topological monoid.
As the basepoint of Map C (X, X) × we take id X , the unit of the multiplication, since the basepoint 0 of Map C (X, X) is not in Map C (X, X) × . There are isomorphisms for i ≥ 1. The second isomorphism is the restriction to the path component. For the first one, we take the isomorphism σ id X of Proposition A.1 combined with an adjunction.
A topological monoid G has a classifying space BG, defined via the bar construction. It comes with a map ω : G → ΩBG. If the topological monoid G is group-like, that is, the monoid π 0 (G) is a group, then ω is a weak equivalence. In our example we get a space B Map C (X, X) × with The left action of π 1 (B Map C (X, X) × ) on π n+1 (B Map C (X, X) × ) corresponds under this isomorphism to the conjugation action g · λ = (g −1 ) Theorem 7.1. Let C be a stable topological model category, let U be a small n-split subcategory of Ho(C), and let X be a cofibrant and fibrant object of C representing an object in U. Then the restriction map of Proposition 2.10 sends the universal Toda bracket γ U to the first k-invariant of B Map C (X, X) × not vanishing for dimensional reasons.
We need an auxiliary lemma for the proof. Let G be a group-like topological monoid and let ω : G → ΩBG be the map to the group completion. Let ϕ : (S n , pt) → (G, g) be any map. The adjoint of ωϕ is a map from the unreduced suspension S(S n ) to BG. It represents an element in π n+1 (BG). On the other hand, we can choose an h ∈ G such that gh is in the component of 1 G . If v is a path from gh to 1 G , we get [ϕ · h] v ∈ π n (G, 1 G ). This does not depend on v and h, as G being a topological monoid implies the π 1 -action on π n (G) to be trivial. Composing with ω, we get ω * ([ϕ · h] v ) ∈ π n+1 (BG). Lemma 7.2. These two ways to associate an element of π n+1 (BG) to ϕ : S n → G are equivalent.
By the definition of c in Construction 5.7 and the restriction map Θ in Proposition 2.10, the evaluation of Θ(γ U ) on (f 1 , . . . , f n+2 ) is We need to examine the image of the homotopy class of this map under q q q q q q q q q q q q q Figure 5. The square . . .
For the next step we use Lemma 7.2. It says that the image of considered as a map S(sk n I n+1 ) → B Map C (X, X) × . The unreduced suspension of S(sk n I n+1 ) is homotopy equivalent to ∂∆ n+2 , the boundary of an (n + 2)-simplex, and we will now explain the resulting map a f = a(f 1 , . . . , f n+2 ) : We denote the set of vertices of ∂∆ n+2 by {1, . . . , n + 3}. Then a f maps every vertex i of ∂∆ n+2 to the basepoint. The 1-simplex of ∂∆ n+2 containing the two vertices i < j is mapped to B Map C (X, X) × using the path associated to b 0 (f i · · · f j−1 ) via the map ω : Map C (X, X) × → ΩB Map C (X, X) × . Hence every 0-dimensional subcube of sk n I n+1 specifies a path from the initial to the terminal vertex of ∂∆ n+2 . This path runs through the vertex containing i < j if the term b 0 (f i · · · f j−1 ) occurs in the restriction of the cube system to that 0-dimensional subcube.
The case n = 1 is displayed in Figure 5, whose right part also appears in [21, B.2.2]. The situation gets a little bit more involved if n > 1, since an (n + 1)-cube has 2(n + 1) subcubes of dimension n, but the (n + 2)-simplex has only (n + 3) sub (n + 1)-simplices. In this case, the 2(n + 1) − (n + 3) = n − 1 codimension 1 subcubes (0 k )(I n ) of sk n I n+1 with 1 < k < n+1 do not contribute new information to the map defined on the boundary of the (n + 2)-simplex. The reason is that the restriction of the pre (n + 1)-cube system to these subcubes is already determined by the underlying (n − 1)-cube system. We recall that the restriction to these subcube is built from b k−1 (f 1 , . . . , f k ) and b n+1−k (f k+1 , . . . , f n+2 ). Accordingly, it corresponds to the restriction of the map a f : ∂∆ n+2 → B Map C (X, X) × to the two simplices with the vertices {1, . . . , k} and {k + 1, . . . , n + 2}. The maps on all other n-dimensional subcubes induce maps on one of the (n + 1)-simplices of ∂∆ n+2 .
The cochain (f 1 , . . . , f n+2 ) → [a(f 1 , . . . , f n+2 )] is a representing cocycle for the first k-invariant as described by Eilenberg and Mac Lane in [13, §19]. In that reference, the authors also give an equivalence of this definition of a k-invariant to a more commonly used one. 7.3. Coherent vanishing of k-invariants. The last theorem says that the vanishing of γ U implies the vanishing of the first k-invariant of the space B Map C (X, X) × for every cofibrant and fibrant object X of C representing an object of U. For our applications, we need a stronger statement in a special case.
For the rest of this section, we assume that C is a stable topological model category in which all objects are fibrant. Furthermore, we assume the n-split subcategory U of Ho(C) to have a fixed object X 1 such that all other objects of U are finite sums of copies of X 1 . Such a q-fold sum will be denoted by X q .
We choose a cofibrant (and automatically fibrant) object of C representing X 1 and denote it also by X 1 . Let the object X q in U be represented by the q-fold coproduct X 1 ∨ . . . ∨ X 1 of copies of X 1 in C, which we also denote by X q . The difference between objects in Ho(C) and C will be emphasized by writing ∨ for the coproduct in C and ⊕ for the coproduct in Ho(C).
We get maps Map C (X q , X q ) → Map C (X q+1 , X q+1 ) by adding id X 1 on the last summand. The restriction of these maps to the set of invertible path components is multiplicative with respect to the monoid structure. Hence we get a map Here it is convenient to work in a setup with all objects fibrant, since the otherwise necessary fibrant replacement of the sum X q ∨ X 1 would mean that we only get a homotopy class of maps Map C (X q , X q ) → Map C (X q+1 , X q+1 ), rather than an actual map.
Denote the mapping telescope of B Map C (X 1 , The vanishing of the first k-invariant of this space does not follow from the vanishing of the first k-invariant of all spaces B Map C (X q , X q ) × in general, since this vanishing does not have to be compatible with the maps t q . The next lemma provides a sufficient condition for this stronger statement. Then the space B Map ∞ C (X, X) × has a vanishing k-invariant k n+2 , i.e., it has the Eilenberg-Mac Lane space |Bπ 1 (B Map ∞ C (X, X) × )| as a retract up to homotopy. Proof. We will construct a section up to homotopy of the π 1 -isomorphism from As we have seen in the proof of the Theorem 7.1, the b j specify maps from all (j + 1)-simplices of |Bπ 1 (B Map ∞ C (X, X) × )| to B Map C (X q , X q ) × . By the compatibility of the cube system, they assemble to a π 1 -isomorphism s n+1 q defined on the (n + 1)-skeleton of |Bπ 1 (B Map C (X q , X q ) × )| In general, t q s n+1 q ≃ s n+1 q+1 (t q ) * will not hold. But without loss of generality, we can build this condition into the cube system: for all j ≤ n, we require in the inductive construction of the cube system the map b j (f 1 ⊕ X 1 , . . . , f j+1 ⊕ X 1 ) to be (− ∨ X 1 )b j (f 1 , . . . , f j+1 ).
We could extend the s n+1 q to the desired maps s q if we knew that our cube system extends to an (n + 1)-cube system. By Lemma 5.11, we know that γ U = 0 implies that we can change the maps b n (f 1 , . . . , f n+1 ) to achieve this. Unfortunately, the modified b n does not have to be compatible with the t q anymore.
We say that a sequence of (n + 1) composable maps (f 1 , . . . , f n+1 ) in U has filtration k if k is the maximal integer such that there exist maps If we change our compatible n-cube system b j by the cochain e k+1 with the procedure of Lemma 5.11, we get an n-cube system b j k+1 which is compatible on all sequences of filtration up to k, and which extends to an (n + 1)cube system. Since b j k+1 and b j k+2 coincide on the sequences of filtration up to k, this is enough to get the desired map on the telescope.
The hypotheses of the lemma may appear unrealistic at the first glance. Nevertheless, the probably strongest condition (iv) will reduce to the vanishing of a single Mac Lane cohomology group when we apply it in Proposition 8.14. This is much easier to verify as to ensure a coherent vanishing of the k-invariants by dealing with the associated obstructions on the level of group cohomology.

The universal Toda bracket of a ring spectrum
We now apply the results of the preceding sections to ring spectra based on topological spaces. These can be the S-algebras of [14], the symmetric ring spectra of [19] (see [30] for a version based on topological spaces), or the orthogonal ring spectra introduced in [30]. For a ring spectrum R, the module category Mod-R is a topological model category. If C is the underlying category of spectra, − ∧ R : C → Mod-R is a left Quillen functor. Hence π * (R) = [R, R] , and R is compact in Ho(Mod-R).
Recall that π * (R) is n-sparse if it is concentrated in degrees divisible by n.
We call γ n+2 R the universal Toda bracket of R. Theorem 1.1 is the n = 1 case of the last theorem.
Remark 8.2. The restriction to modules with a resolution by finitely generated free π * (R)-modules can be avoided. By [22,§2 and Corollary 3.11], replacing F (π * (R), n) by a larger full small additive subcategory doesn't change the cohomology. We choose such a D so that it consists only of free modules, and that it contains all modules from a given free resolution of M . Then there is a subcategory U in Ho(Mod-R) equivalent to D that gives rise to a γ R for which id M ∪γ R is defined and equals the obstruction. However, there is no small D which works for all M simultaneously. Hence we keep the restriction to the M as stated in the theorem, as this seems to be the most natural choice.
Proof. As 2η = 0 = η2 and π 2 (KO) is 2-torsion, the first two statements hold. The ring spectra map S → KO is a π i -isomorphism for 0 ≤ i ≤ 2, so it suffices to calculate the corresponding Toda bracket for the sphere spectrum. This can be either taken from [38] or computed directly, following [39, Theorem 6.1]: Suppose 0 ∈ 2, η, 2 . This would imply the existence of a 4-cell complex X with 2, η and 2 as attaching maps. We consider H * (X, Z/2). Since Sq 1 detects 2 and Sq 2 detects η, the existence of X implies that Sq 1 Sq 2 Sq 1 acts non-trivially on the bottom dimensional class in H * (X, Z/2). But Sq 1 Sq 2 Sq 1 = Sq 2 Sq 2 , and Sq 2 Sq 2 applied to the bottom class of H * (X, Z/2) is trivial for dimensional reasons.
Proof. Write M for π * (KO)⊗Z/2. There is a distinguished triangle KO ·2 − → KO → C(2) → KO [1] in Ho(Mod-KO) which induces a long exact sequence in homotopy. Since M is the cokernel of multiplication with 2 on π * (KO), there is an injection ι : M → π * (C(2)). As the two copies of π * (KO) in the long exact sequence are free modules of rank one, κ 3 (M ) vanishes if and only if ι split. Hence it is enough to show π 2 (C(2)) ∼ = Z/4.
Remark 8.5. The same argument as in the last lemma shows the corresponding statement about the connective real K-theory spectrum. This contradicts [43,Theorem 20]. The reason is an error in [43, 14.1]. In this construction, the author assumes ku * to be flat as a ko * -module, which does not hold. Accordingly, the generalization [43, Theorem 21] is false as well.
8.6. Universal Toda brackets and k-invariants. Let R q denote a cofibrant and fibrant object of Mod-R representing the free R-module spectrum of rank q. We write GL q R for the space Map Mod-R (R q , R q ) × considered in Section 7.3. This definition of the 'general linear group' of a ring spectrum R is an important ingredient for the construction of the algebraic K-theory of R in the sense of Waldhausen [40], if his definition is interpreted in the modern language of ring spectra [14,VI.7]. We will encounter the algebraic K-theory of ring spectra in Proposition 8.14.
Theorem 8.7. Let R be a ring spectrum such that π * (R) is n-sparse for some n ≥ 1. For q ≥ 1, the restriction map HML n+2,−n n−sp (π * (R)) → H n+2 (π 1 (B GL q R), π n+1 (B GL q R)) sends the universal Toda bracket γ n+2 R of R to the first k-invariant of the space B GL q R not vanishing for dimensional reasons.
Proof. Since B GL q R = B Map Mod-R (R q , R q ) × , this follows from Theorem 7.1 and the description of the restriction map in Corollary 2.11.
Proof. The proof uses the same arguments as that of Theorem 8.1. This time, U has the finite sums of (unshifted) copies of R as objects. It is equivalent to F (π 0 (R)). The isomorphism H n+2 (U, [−, −] Ho(C) n ) ∼ = HML n+2 (π 0 (R), π n (R)) induced by the equivalence enables us to define γ n+2 R in the latter group. Proposition 8.9. Let R be a ring spectrum such that π * (R) is n-sparse. Let R ≥0 be its connective cover and let P n (R ≥0 ) be the first non-trivial Postnikov section of R ≥0 . The restriction HML n+2,−n n−sp (π * (R)) → HML n+2 (π 0 (R), π n (R)) sends the universal Toda bracket of R to the one of P n (R ≥0 ).
Proof. Let U be the subcategory of Ho(Mod-R) given by the finite sums of copies of R which are shifted by integral multiples of n. The class γ n+2 R was defined by applying Construction 5.7 to U. If U 0 is the subcategory of U of finite unshifted copies of R, the map from the graded to the ungraded Mac Lane cohomology is induced by the restriction along the inclusion U 0 → U.
Let U ≥0 be the subcategory of Ho(Mod-R ≥0 ) which is given by the finite sums of unshifted copies of R ≥0 . The left Quillen functor − ∧ R ≥0 R : Mod-R ≥0 → Mod-R induces an equivalence between U ≥0 and U 0 , since the induced map on homotopy groups Mod-π * (R ≥0 ) → Mod-π * (R), M → M ⊗ π * (R ≥0 ) π * (R) restricts to an equivalence between the subcategories of unshifted copies of the free module of rank 1. Lemma 5.10 shows that this equivalence maps the universal Toda bracket of U 0 to the one of U ≥0 .
A similar argument applied to − ∧ R ≥0 P n (R ≥0 ) shows that γ U0 equals the universal Toda bracket of the subcategory of Ho(P n R ≥0 ) given by the finite sums of unshifted copies of P n R ≥0 . By Theorem 8.8, this is γ n+2 PnR ≥0 . We consider the example KO again. The restriction of the universal Toda bracket γ KO to HML 3 (π 0 (KO), π 1 (KO)) ∼ = HML 3 (Z, Z/2) is γ P1KO ≥0 . The latter group is Z/2 [28,Proposition 13.4.23]. We show that the image of γ KO is the non-zero element, thereby proving once more γ KO = 0. Since P 1 KO ≥0 ∼ = P 1 ko ∼ = P 1 S, this is a statement about the sphere spectrum, and computations of Igusa [21] imply Let BH m q be the classifying space of H m q . The map colim m BH m q → B GL q S is a homotopy equivalence by [14,Proposition VI.8.3]. From [21] (compare also [3, (7.6)]), we know that the first k-invariant of BH m q is non-trivial for q ≥ 4 and m ≥ 3. The increasing connectivity of the maps in the colimit system therefore implies that the first k-invariant of B GL q S does not vanish for q ≥ 4. Hence the first k-invariant of B GL q P 1 S is non-trivial as well. By Theorem 8.8, γ P1S has to be non-trivial since the HML 3 (Z, Z/2) → H 3 (π 1 (B GL q P 1 (S)), π 2 (B GL q P 1 (S))) sends it to this k-invariant.
Focusing on a ring spectrum with polynomial homotopy again, Proposition 8.9 implies Corollary 8.11. Let R be a ring spectrum with π * (R) ∼ = (π 0 (R))[u ±1 ] for a central unit u in degree n. The isomorphism HML n+2,−n n−sp (π * (R)) → HML n+2 (π 0 (R)) of Lemma 2.8 sends the universal Toda bracket γ n+2 R to the one of the first non-trivial Postnikov section of its connective cover.
This reduces the computation of γ 4 KU to that of γ 4 P2ku . Remark 8.12. A ring spectrum R with only two homotopy groups π 0 (R) and π n (R) has a first k-invariant in the group Der n+1 (π 0 R, π n R) ∼ = THH n+2 (π 0 R, π n R) [27,12]. Since THH n+2 (π 0 R, π n R) ∼ = HML n+2 (π 0 R, π n R) and the universal Toda brackets coincide with the k-invariant in the examples P 1 S and P 2 ku, we expect the universal Toda brackets of first non-trivial Postnikov sections to coincide with these k-invariants in general. We don't have a proof for this. The difficult point is that these two groups are only related by a chain of isomorphisms, and we do not know how to identify the k-invariant or the universal Toda bracket in the intermediate steps.
A proof of this statement would not only be interesting for the computation of universal Toda brackets. It would also relate the first k-invariant of a ring spectrum R with the Toda brackets of R and the first k-invariants of the spaces B GL q R in a very explicit way. 8.13. A relation to K-theory of ring spectra. For a connective ring spectrum R, there is a map R → H(π 0 (R)) from R to the Eilenberg-Mac Lane spectrum of π 0 (R) which is the identity on π 0 . In view of the last remark, we expect the map R → H(π 0 (R)) to split in the homotopy category of ring spectra if R has only two non-trivial homotopy groups and a vanishing universal Toda bracket. Though we are not able to prove this statement, the following proposition will provide a weaker result.
We briefly recall the definition of the algebraic K-theory of a ring spectrum R, following [14,VI]. To avoid technical difficulties, we assume our ring spectrum R to be an S-algebra in the sense of [14]. Since all objects in the category of R-modules are fibrant in this case, we obtain maps B GL q R → B GL q+1 R as described in Section 7.3.
Let B GL R be the (homotopy) colimit of the spaces B GL q R with respect to these maps. We apply Quillen's plus construction to the space B GL R to obtain (B GL R) + . For i ≥ 1, algebraic K-groups of R can be defined as K i (R) = π i ((B GL R) + ). We will not need K 0 (R), which has to be defined separately. If R is an Eilenberg-Mac Lane spectrum of a discrete ring A, this definition recovers the algebraic K-groups K * (A) of A in the sense of Quillen [14,VI,Theorem 4.3].
We will later need that the algebraic K-theory construction increases connectivity by 1. Recall that map R → R ′ of ring spectra is n-connected if the induced map π i (R) → π i (R ′ ) is an isomorphism for i < n and an epimorphism for i = n. If R → R ′ is n-connected, the induced map K i (R) → K i (R ′ ) is an isomorphism for i ≤ n and an epimorphism for i = n + 1. This fact is due to the appearance of the bar construction in the definition of K(R) and can be proved in a similar way as the corresponding statement about simplicial rings in [40, Proposition 1.1].
Proposition 8.14. Let R be a ring spectrum with homotopy groups concentrated in degrees 0 and n. Suppose that the universal Toda bracket γ n+2 R of R is trivial and that HML n+1 (π 0 (R), π n (R)) vanishes. Then the map K i (R) → K i (π 0 (R)) induced by R → H(π 0 (R)) splits for all i.
Proof. It is enough to show that B GL R → B GL(H(π 0 (R)) splits up to homotopy, as this property is preserved by the plus construction. This is equivalent to the splitting of the map B GL R → |Bπ 1 (B GL R)|, since both maps are isomorphisms on the fundamental group and map into an Eilenberg-Mac Lane space.
We show this by applying Lemma 7.4 to the category U used in the proof of Theorem 8.8. The first three conditions are obviously satisfied. It remains to show that H n+1 (U, [(−) ⊕ R q , (−) ⊕ R q ] n ) = 0.
The category U is equivalent to F (π 0 (R)). If we set A = π 0 (R) and M = π n (R), this equivalence induces an isomorphism between the last cohomology group and By Lemma 2.4, this is isomorphic to HML n+1 (π 0 (R), π n (R)) ∼ = 0. Proof. By Corollary 8.11 it is enough to prove γ 4 P2ku = 0. We assume γ 4 P2ku = 0 and show that this leads to a contradiction. that m ′ 4 is a Hochschild cocycle, which can be used to define a cohomology class in HH 4,−2 k (H * (A)). This is the candidate for the higher Hochschild class. However, it is not unique in general.
In the case of ring spectra, Lemma 5.11 is the tool for a similar kind of argument. If for example γ R ∈ HML 3,−1 (π * (R)) vanishes, the lemma says that we can find a 1-cube system which extends to a 2-cube system. This cube system can be used to define γ 4 R ∈ HML 4,−2 (π * (R)) without requiring π * (R) to be 2-sparse. As in the algebraic case, there may be different choices for this class γ 4 R . Moreover, the relation to the Toda brackets becomes more involved since the indeterminacy is not as easy to control as in the 2-sparse case. This also affects the obstruction theory, as there is no unique obstruction class.  Z). The data is asked to satisfy the usual associativity and unit conditions. Moreover, the pushout product axiom is required to ensure compatibility of the model structures. Details can be found in [18, 4.2]. A stable topological model category C is a pointed topological model category in which the suspension functor S 1 ∧ − : C → C and the loop functor (−) S 1 : C → C form a Quillen equivalence.
For an object X in a category C, we write (X ↓ C) for the category of objects under X. If C is a model category, (X ↓ C) inherits a model structure in which a map is a cofibration, fibration, or a weak equivalence if the underlying map in C is one [17, Theorem 7.6.5.(1)].
Proposition A.1. Let C be a stable topological model category. For every map g : X → Y between cofibrant and fibrant objects in C, there is an isomorphism σ g : [S n , (Map C (X, Y ), g)] Ho(T op * ) ∼ = − → [S n ∧ X, Y ] Ho(C) .
If h : W → X and f : Y → Z are maps between cofibrant and fibrant objects, the isomorphisms satisfy (f * )(σ g ) = (σ f g )(f * ) and (h * )(σ g ) = (σ hg )(h * ). For a path w from g to g ′ in Map C (X, Y ), the isomorphisms σ g and σ g ′ are compatible with the isomorphism of homotopy groups induced by w, i.e., σ g ′ (−) w = σ g . If g is the zero map, σ g is the adjunction isomorphism.
The proof needs some notation and an auxiliary lemma. If S n is the n-sphere in T op * , we consider S n + as an object of (S 0 ↓ T op * ). The structure map S 0 → S n + sends the basepoint of S 0 to the 'added' basepoint of S n + , and the other point to the 'original' basepoint of S n . If X is an object in a pointed topological model category C, we consider S n + ∧ X as an object in (X ↓ C) via X ∼ = S 0 ∧ X → S n + ∧ X. By the pushout product axiom, S n + ∧ X is cofibrant if X is. If f : X → Y is another object of (X ↓ C), we write [S n + ∧ X, f ] Ho(X↓C) for the set of maps from X → S n + ∧ X to f in the homotopy category of (X ↓ C).
The following lemma is a reformulation of the well known fact that S n + splits as S n ∨ S 0 after suspension. Lemma A.2. Let S n + ∧ S 1 be a space under S 1 via (S 0 → S n + ) ∧ S 1 , and let S n+1 ∨ S 1 be a space under S 1 via the inclusion of the second summand. Then there is a map µ : (S n ∧ S 1 ) ∨ S 1 → S n + ∧ S 1 in (S 1 ↓ T op) which is a homotopy equivalence. If p : S n + → S n is the map which identifies the two basepoints of S n + , is the identity.
Proof. S n is a CW-complex with one 0-cell and one n-cell. The complex S n + ∧ S 1 ∼ = S n × S 1 /(S n × {s 0 }) has a 0-cell, an 1-cell, and an (n + 1)-cell. The attaching map of the (n + 1)-cell of S n + ∧ S 1 is null-homotopic for n ≥ 1. Hence the desired homotopy equivalence exists. If we compose with p ∧ S 1 , we collapse the 1-cell and do not see the effect of the null-homotopy. This verifies the last assertion.
Proof of Proposition A.1. We define a functor G : C → C by G(X) = (X S 1 ) cof , where (−) cof is the functorial cofibrant replacement. The adjunction of suspension and loop gives a natural transformation τ : S 1 ∧ G(X) → id C . Since C is stable, τ X is a weak equivalence if X is fibrant [18, Proposition 1.3.13(b)].
Let σ g be the following chain of isomorphisms: Here (i) is induced by the weak equivalence τ X , (ii) is adding a basepoint, (iii) results from the Quillen adjunction between (S 0 ↓ T op * ) and (X ′ ↓ C) induced by − ∧ X ′ and Map C (X ′ , −) with X ′ = S 1 ∧ G(X), (iv) uses the weak equivalence under S 1 provided by Lemma A.2, (v) results from the Quillen adjunction between (X ′ ↓ C) and C given by Z → Z ∨ X ′ and the forgetful functor (with X ′ = S 1 ∧ G(X)), and (vi) is induced by τ X again. It is easy to see that the construction is natural in X and Y . It is additive since the addition can be defined in terms of the H-cogroup structure of S n both in the source and the target. Let w be a path from g to g ′ in Map C (X, Y ). Following its action through (i)-(iii), it induces a homotopy of maps S n + ∧ S 1 ∧ G(X) → Y which is itself not a map under S 1 . But after applying (iv) and (v), the representing maps become homotopic in Ho(C). If g is the zero map, σ g reduces to the adjunction isomorphism: Composing with the p of Lemma A.2 is inverse to (ii), hence Lemma A.2 shows the assertion.