On Generalized Whitehead Products

A symmetric monoidal pairing is defined among simply connected co-H spaces and this is used to generalize the Whitehead product map S(X ^ Y) -->SX v SY to co-H spaces.


BRAYTON GRAY
Whitehead products have played an important role in unstable homotopy.They were originally introduced [Whi41] as a bilinear pairing of homotopy groups: π m (X) ⊗ π n (X) → π m+n−1 (X) m, n > 1.
This was generalized ([Ark62], [Coh57], [Hil59]) by constructing a map: Precomposition with W defines a function on based homotopy classes: which is bilinear in case A and B are suspensions.
The case where A and B are Moore spaces was central to the work of Cohen, Moore and Neisendorfer ([CMN79]).In [Ani93] and in particular [AG95], this work was generalized.Much of this has since been simplified in [GT10], but further understanding will require a generalization from suspensions to co-H spaces.
The purpose of this work is to carry out and study such a generalization.Let CO be the category of simply connected co-H spaces and co-H maps.We define a functor: and a natural transformation: (1) W : generalizing the Whitehead product map.The existence of G•H generalizes a result of Theriault [The03] who showed that the smash product of two simply connected co-associative co-H spaces is the suspension of a co-H space.We do not need the co-H spaces to be co-associative and require only one of them to be simply connected.*  We call G • H the Theriault product of G and H.We summarize our results in the following theorems.
Theorem 1.There is a functor: CO × CO → CO given by and homotopy equivalences: Theorem 2. There is a natural transformation: which is the Whitehead product map (1) in case G and H are both suspensions.Furthermore, there is a homotopy equivalence: The next theorem concerns the inclusion of the fiber in certain standard fibration sequences [Gra71]: and an iterated Whitehead product as the composition: for some spaces M n , which are not further decomposed.
Theorem 4. Suppose X is finite dimensional and f: SX → G ∨ H then f is the sum of the projections onto G and H and a finite sum of iterated Whitehead products.
Throughout this work we will assume that all spaces are of the homotopy type of a CW complex.All homology and cohomology will be with a field of coefficients.We will often show that a map between simply connected CW complexes is a homotopy equivalence by showing that it induces an isomorphism in homology with an arbitrary field of coefficients, without further comment.
Section 1 will be devoted to some general remarks about telescopes and we will construct the Theriault product in section 2. Theorem 1 will follow from 2.3, 2.5 and 2.7.The functor in Theorem 2 is defined after 3.2 and the equivalence follows from 3.8.The proof of the first part of Theorem 3 occurs just prior to 3.5 and ithe rest follows from 3.5 Theorem 4 follows from 3.7.

1.
In this section we will discuss some general properties of telescopes of a self may e : G → G where G is a co-H space.We do not assume that e is idempotent.We will call e a quasi-idempotent if the induced homomorphism in homology satisfies the equation: where u is a unit.We construct two telescopes:  Corollary 1.2.Suppose G is a simply connected co-H space and A ⊂ G is a retract of G. Let e be the composition: Then T (e) ≃ A, T (1 − e) ≃ G/A, and the identity map of T (e) can be factored: T where ξ and η are inverse homotopy equivalences.
Proof.The telescope T (e) and A are both simply connected and there are maps T (e) → A and A → T (e) making A a retract of T (ξ), and these maps are homotopy equivalences.By the Van Kampen theorem G/A ≃ G ∪ CA is simply connected.Since 1 − e : G → G factors through the projection π : G → G/A, we can factor the identity map up to homotopy The factorization of the identity map of T (e) is obtained by replacing each space by a telescope where the three in the center are constant.

Now consider two maps
Proof.We define maps between the telescopes: The composition is the shift map which is a homotopy equivalence.

2.
In this section we will consider a pair of co-H spaces in which at least one is simply connected.Let G and H be two such co-H spaces with their structure determined by maps ν 1 : G → SΩG and ν 2 : H → SΩH each of which is a right inverse to the respective evaluation maps (See [Gan70]), which we label as ǫ 1 , ǫ 2 .We define self maps of S(ΩG ∧ ΩH) as follows: here we freely move the suspension coordinate to wherever it is needed.Clearly e 1 and e 2 are idempotents but e 1 e 2 is not an idempotent; however it is a quasi-idempotent.In fact by swapping coordinates, one can see that S(e 1 e 2 ) is homotopic to the negative of the composition: Since e is clearly an idempotent S(e 1 e 2 ) • S(e 1 e 2 ) ∼ e 2 = e ∼ −S(e 1 e 2 ), so (e 1 e 2 ) 2 * = −(e 1 e 2 ) * .Now assuming that one of G, H is simply connected, it follows that ΩG ∧ ΩH is connected, so S(ΩG ∧ ΩH) is simply connected.Consequently Proposition 2.1.If one of G and H is simply connected, there is a homotopy equivalence:

Let
θ : S(ΩG ∧ ΩH) → T (e 1 e 2 ) be the projection and be the unique right inverse to θ which projects trivially onto T (1 + e 1 e 2 ).These maps determine a co-H space structure on T (e 1 e 2 ).
Proof.Since f and g are co-H maps, the squares commute up to homotopy.It follows that f and g induce maps that commute with e 1 and e 2 and hence with the equivalences of 2.1, θ and ψ.For part a, observe that the composition e 2 e 1 factors: where (1 ∧ ǫ)(ǫ 1 ∧ 1) is a right universe to (1 ∧ ν 2 )(ι ∧ 1).Thus we can apply 1.2 to see that SX • H ≃ X ∧ H with co-H structure given by the composite (1 ∧ ν 2 )(ι ∧ 1).This is precisely the co-H structure induced by ν 2 .Part b follows since S(G • H) is the telescope of e with co-H structure given by ν 1 ∧ ν 2 .Part c is a special case of part a: The last statement follows directly from 1.3.For the associativity assertion in Theorem 1, it will be convenient to describe an alternative definition of G • H.For this we assume that G is a retract of a space SX and H a retract of SY : We can then replace the telescope in the definition by the telescope of the composition: The co-H structures defined by these maps are equivalent to the structures defined by and we have a homotopy commutative ladder: Hence we have a commutative diagram where T is the telescope defined by 1 as co-H spaces, where the co-H structure on T is given by the equivalence At this point we will apply 2.4 to prove theorem 1 part e, the associativity formula.We will make repeated use of Lemma 2.6.There is a homotopy commutative square

S(G • H).
Proof.G ∧ H is a retract of S 2 X ∧ Y , so we may apply 1.2.
Proof.We suppose that G, H, and K are presented by retractions By 2.6, we have homotopy commutative diagrams: ≃ 8 8 q q q q q q q q q q Sψ / / S 2 X ∧Y ∧Z Suspending and applying 2.6 again we obtain a homotopy commutative diagram: from these diagrams it follows that S 2 (θ ′ ψ) is a homotopy equivalence and hence θ ′ ψ is as well.

3.
In this section we generalize the clutching construction [Gra88, Proposition 1] for fibrations over a suspension to fibrations over a co-H space.This allows for the decomposition results in theorems 2 and 3.
Proof.In the case G = SX, we have by [Gra88, Proposition 1] Consider the sequence of pull backs: Then we consider the composite: where the middle equivalence follows since the base is a suspension.Showing that the composite is a homotopy equivalence will take some work.
Since νǫ : SΩG → SΩG is an idempotent, we can decompose SΩG: We now observe that we can construct a quasifibration model for a fibration over a one point union.Suppose we have such a fibration with pull backs E A and E B and fiber F .Then we can construct the union of E A and E B with the subspace F identified.Then is a quasifibering by [DT58, 2.10].On the other hand Proof.Apply 3.1 to the path space fibration over G.
Proof of theorem 2. Construction: We now describe our generalization of the Whitehead product.Suppose G and H are co-H spaces and one of them is simply connected.The Whitehead product: is then defined as the composition: where ω is the inclusion of the fiber in the fibration sequence: Clearly ψ and ω are natural transformations, so W is as well.Before we prove the homotopy equivalence in theorem 2, we need to establish some results in theorem 3. We begin by constructing maps: For n > 0 we define ad n as the composition: Next we calculate the effect of ad n in loop space homology: To do this we need some notation.For each co-H space G, write σ −1 : H r (G) → H r−1 (ΩG) for the composition: Let {x i } be a basis for H * (G).Then H * (ΩG) is the tensor algebra on the classes {σ −1 (x i )}.Given two classes x ∈ H r (G), y ∈ H s (H) we will write for the class that corresponds to x ∧ − y ∈ H r+s (G ∧ H) under the isomorphism: Then the classes {x i • y j } form a basis for H * (G • H) where {x i } and {y i } respectively are bases for H * (G) and H * (H).
Proof.By lemma 2.6 where ξ is the standard homotopy equivalence SX ∧ Y ≃ X * Y .
Lemma 3.4.The composition: Proof.We first need to describe the homotopy equivalence Here we write points of the join as tx + (1 − t)y, 0 ≤ t ≤ 1, So X * Y is the quotient of X × I × Y given by the identifications (x, 0, y) ∼ (x ′ , 0, y) and (x, 1, y) ∼ (x, 1, y ′ ).Then ξ is given by the formula: Combining these we get with a 6-part formula: so the adjoint takes the pair (ω 1 , ω 2 ) to the product of loops ω −1 1 ω −1 2 ω 1 ω 2 .The effect of this on a primitive element is the graded commutator.Now the iterated circle product ad n (H)(G) has homology generated by classes of the form (. . . where the classes x and y i are thought of as classes in H * (G ∨ H).Proof.For part (a) apply 3.2 and theorem 3 part a.For part (b), we expand

Proof of theorem 3 part
using 2.5 and theorem 3 part a.Given a Theriault product P = G 1 • • • • • G s with some fixed association, let us write ℓ(P ) = s for the length of P .
Theorem 3.6.Suppose G and H are both simply connected co-H spaces and k 1.Then there is a locally finite collection of iterated Theriault products {P α (k)} of length ℓ α and iterated Whitehead product maps: and the factors of the righthand side are mapped to the lefthand side by the ω α (k).
Proof.For k = 1 we use the decomposition: where ΩG * ΩH is a boquet of iterated Theriault products of length at least 2 by 3.5(b).Now we proceed by induction on k.Among the finite list of products P α (k) of length k + 1, choose one which we label P .Then Ω( The second factor has one less product of length k + 1.If we repeat this process once for each P α (k) of length k + 1, we obtain: P α (k + 1) .Now add the P 1 . . .P m to the list of P α with ℓ(P α ) ≤ k to obtain all P α (k+1) with length ≤ k + 1.
Corollary 3.7.Suppose X is a finite dimensional co-associative co-H space and f: X → G∨H where G and H are simply connected co-H spaces.Then f is a sum of iterated Whitehead products.
Proof.Suppose dim X = k and f: X → G∨H is given.Decompose Ω(G∨H) as in theorem 3.6 and note that any product P α of length k is at least k connected.Consequently the restriction of Ωf : factors through the product Ω( ℓα≤k P α (k)) and the adjoint: is a sum of iterated Whitehead products.However f is the composition of this map with the co-H space structure map: which is a co-H map, so f is such a sum as well.
Proposition 3.8.If G and H are simply connected, then there is a homotopy equivalence The problem is to show that this map is a homotopy equivalence.We begin by observing that we can construct a right inverse ζ to Ωφ as the sum of the loops on the inclusions of G and H into C: ) is an epimorphism.We will complete the proof by showing that the rank of H k (ΩC) is less than or equal to the rank of H k (ΩG × ΩH).We will need several lemmas.
Lemma 3.9.Write ΩG * ΩH ≃ G • H ∨ SQ.Then the restriction: We first look at the homotopy commutative diagram Applying theorem 3a we see that the composition is a sum of maps γ i factoring through ad i (H)(G) → G ∨ H for i 2. By induction on i we see that is null homotopic for i 1.This follows since ad i factors It follows from 3.9 that the mapping cone of ω is homotopy equivalent to C ∨ S 2 Q.Recall that Ganea proved [Gan70]  Using such a factorization we can construct a homotopy equivalence: such that πΓ| S 2 Q is null homotopic.Replacing π with πΓ does not alter the homotopy type of the fiber of π, so we can form the following diagram of fibrations: The lefthand verticle fibration has a cross section since π 1 does, hence K is a co-H space and we have a splitting: Ω(ΩG * ΩH * Ω(G × H)) ≃ Ω(S 2 Q ⋊ ΩC) × ΩK.
Each space here is the loops on a co-H space, so the homologies are all tensor algebras.It follows that for each i 0 rank H i (ΩG * ΩH * Ω(G × H)) rank H i (S 2 Q ⋊ ΩC).
We now calculate the Poincaré series for each of these spaces.Suppose X (G) = 1 + gt and X (H) = 1 + ht where g and h are polynomial in t with positive integral coefficient.We then have the following consequences:

Theorem 3 .
Suppose G and H are simply connected co-H spaces.Then there are homotopy equivalences: (a) G ⋊ ΩH ≃ n 0 ad n (H)(G) where ι 1 corresponds to ad n on the appropriate factor (b) ΩG * ΩH ≃ i 0 j 1 ad j (H)(ad i (G)(G)) where ι 2 corresponds to ad j (ad i ) on to the appropriate factor (c) SΩG ≃ n 0 ad n (G)(G) where the composition SΩG → SΩG∨SΩG → G ∨ G corresponds to the appropriate iterated Whitehead product on each factor.It should be pointed out that the equivalence (c) generalizes the result of Theriault [The03, 1.1] where it is shown that a simply connected coassociative co-H space decomposes ΣΩG ≃ n 1 M n


(a): Now let G * = H * (G) and H * = H * (H).Let L(G * ⊕ H * ) be the free Lie algebra generated by G * and H * , and L(H * ) the free Lie algebra generated by H * .Then Neisendorfer has analyzed the kernel L(G * ∨ H * ) → L(H * ) ([Nei09, 8.7.4]).He has shown that this is the free Lie algebra n (H * )(G * )  The universal enveloping algebra is thus the tensor algebra generated by the elements ad n (H * )(G * ) for n 0. Consequently the fiber of the projection Ω(G ∨ H) → ΩH is Ω homotopy equivalent to Ω(G ⋊ ΩH) and the map n 0 ad n (H)(G) → G ∨ H which factors through G ⋊ ΩH establishes the homotopy equivalence in theorem 3. Corollary 3.5.(a) If G is simply connected, SΩG ≃ n 0 ad n (G)(G) (b) if both G and H are simply connected ΩG * ΩH ≃ i 0 j 1 ad j (H) ad i (G)(G) .
that given a fibration sequence F → E → B. One can construct a fibration sequenceF * ΩB → E ∪ CF π − → Bwhere π pinches the cone on F to a point.Apply this to the fibration sequence:ΩG * ΩH ω − → G ∨ H → G × H, to obtain: ΩG * ΩH * Ω(G * H) → C ∨ S 2 Q π − → G × H.It is possible that the map π| S 2 Q is nontrivial.However π| S 2 Q is the sum of the projections onto G and H, so it factors through C up to homotopy.
a homotopy equivalence.Now choose a right homotopy inverse for the map SX ∧ Y → T which projects trivially to T .It's composite with S ǫ 1 ∧ ǫ 2 will then project trivially to T (1 + e 1 e 2 ).Hence we have a homotopy commutative diagram: Proposition 2.4.Suppose G is represented as a retract of SX and H a retract of SY .Then G • H is homotopy equivalent to the telescope T of the composition.
α G • H / / SΩG ∧ ΩH / / G • Hand consequently the co-H structure on T is compatible under α with the co-H structure on G • H.We have proven ∧ν 2