Whitehead Products Research Articles

Goodwillie calculus and Whitehead products

Abstract We prove that iterated Whitehead products of length (n + 1) vanish in any value of an n-excisive functor in the sense of Goodwillie. We compare then different notions of homotopy nilpotency, from the Berstein–Ganea definition to the Biedermann–Dwyer one. The latter is strongly related to Goodwillie calculus and we analyze the vanishing of iterated Whitehead products in such objects.

On generalized Whitehead products

We define a symmetric monodical pairing G ◦ H among simply connected co-H spaces G and H with the property that S(G◦H) is equivalent to the smash product G∧H as co-H spaces. We further generalize the Whitehead product map to a map G ◦ H → G ∨ H whose mapping cone is the cartesian product. Whitehead products have played an important role in unstable homotopy. They were originally introduced [Whi41] as a bilinear pairing of homotopy groups: πm(X)⊗ πn(X) → πm+n−1(X), m, n > 1. This was generalized ([Ark62], [Coh57], [Hil59]) by constructing a map W: S(A ∧B) → SA ∨ SB. Precomposition with W defines a function on based homotopy classes: [SA,X]× [SB,X] → [S(A ∧B), X], 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 CO × CO → CO, (G,H) → G ◦H, and a natural transformation (1) W: G ◦H → G ∨H 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 coH 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. Received by the editors November 28, 2009 and, in revised form, June 8, 2010. 2010 Mathematics Subject Classification. Primary 55P99, 55Q15, 55Q20, 55Q25. 1In fact we can define G ◦ H for any two co-H spaces but require at least one of them to be either simply connected or a suspension in order to obtain the co-H space structure map on G◦H. Recently Grbic, Theriault and Wu have shown that the smash product of any two co-H spaces is the suspension of a co-H space, but their construction cannot satisfy Theorem 1(a) below [GTW]. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication 6143

Comultiplications on a wedge of two spheres

In this paper we study the set of comultiplications on a wedge of two spheres. We are primarily interested in the size of this set and properties of the comultiplications such as associativity and commutativity. Our methods involve Whitehead products in wedges of spheres and the Hopf-Hilton invariants. We apply our results to specific examples and determine the number of comultiplications, associative comultiplications and commutative comultiplications in these cases.

Properties of comultiplications on a wedge of spheres

In this paper we study the set of comultiplications on a wedge of a finite number of spheres. We are interested in group theoretic properties of these comultiplications such as associativity and commutativity and loop theoretic properties such as inversivity, power-associativity and the Moufang property. Our methods involve Whitehead products in wedges of spheres and the Hopf–Hilton invariants. We obtain extensive results for a restricted class of comultiplications, namely, the one-stage quadratic or cubic comultiplications.

Deformations of Whitehead Products, Symplectomorphism Groups, and Gromov-Witten Invariants

We provide a new way of understanding the multiplicative structure of the rational homotopy groups π∗(Xλ) ⊗ ℚ for a family of topological spaces, once we know enough about their additive structure. This allows us to interpret the condition of realizing as an Ak map a multiple of a map f : S1→G between two topological groups in terms of the existence of a rational Whitehead product of order k. Our main example will be when the Xλ are classifying spaces of symplectomorphism groups where ωλ is a symplectic deformation on the trivial ruled surface ⁠. Our method of detecting nontriviality is based on computations of equivariant Gromov–Witten invariants. One application gives a homotopy-theoretic counterpart to a geometric result obtained by Karshon. Another application concerns the ring structure of H∗(BSymp(S2×S2,ωλ)).

Whitehead products in function spaces: Quillen model formulae

We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f : X → Y , in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f . These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.

The rational homotopy Lie algebra of function spaces

In this paper we fully describe the rational homotopy Lie algebra of any component of a given (free or pointed) function space. Also, we characterize higher order Whitehead products on these spaces. From this, we deduce the existence of H -structures on a given component of a pointed mapping space \mathcal F_*(X,Y;f ) between rational spaces, assuming the cone length of X is smaller than the order of any non trivial generalized Whitehead product in π_*(Y) .

On the Hopf-James Homomorphism over B

We define the Hopf-James homomorphism and the Whitehead product in the category of fibrewise pointed spaces over B. Making use of them, we generalize Hardie-Jansen's result and obtain that a version of the generalized Hopf construction gives rise to an element which can be detected by a generalized Hopf invariant under some conditions.

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered.

External derivations of internal groupoids

If H is a G -crossed module, the set of derivations of G in H is a monoid under the Whitehead product of derivations. We interpret the Whitehead product using the correspondence between crossed modules and internal groupoids in the category of groups. Working in the general context of internal groupoids in a finitely complete category, we relate derivations to holomorphisms, translations, affine transformations, and to the embedding category of a groupoid.

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G *(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S 1-bundle over the n-dimensional torus is determined explicitly in the non-rational case.

Applications of combinatorial groups to Hopf invariant and the exponent problem

Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.

Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [Σ(V×W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan.

Higher homotopy operations

We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way.

The Whitehead products and powers in $W$-topology

We initiate a study of W'-topology. This is a modification of ordinary topology in which morphisms exist after smashing with a fixed space W. Several classical topics are considered in this setting, inter alia Whitehead products, Hopf invariants and the E-H-Δ sequence. The main emphasis is on detection of non-trivial elements in the W-homotopy groups of spheres.

Topology of symplectomorphism groups of rational ruled surfaces

LetMMbe eitherS2×S2S^2\times S^2or the one point blow-upCP2#CP¯2{\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2ofCP2{\mathbb {C}}P^2. In both casesMMcarries a family of symplectic formsωλ\omega _{\lambda }, whereλ>−1\lambda > -1determines the cohomology class[ωλ][\omega _\lambda ]. This paper calculates the rational (co)homology of the groupGλG_\lambdaof symplectomorphisms of(M,ωλ)(M,\omega _\lambda )as well as the rational homotopy type of its classifying spaceBGλBG_\lambda. It turns out that each groupGλG_\lambdacontains a finite collectionKk,k=0,…,ℓ=ℓ(λ)K_k, k = 0,\dots ,\ell = \ell (\lambda ), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute", i.e. all the higher Whitehead products that they generate vanish asλ→∞\lambda \to \infty. However, for each fixedλ\lambdathere is essentially one nonvanishing product that gives rise to a “jumping generator"wλw_\lambdainH∗(Gλ)H^*(G_\lambda )and to a single relation in the rational cohomology ringH∗(BGλ)H^*(BG_\lambda ). An analog of this generatorwλw_\lambdawas also seen by Kronheimer in his study of families of symplectic forms on44-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space ofωλ\omega _\lambda-compatible almost complex structures onMM.

Volumes, middle-dimensional systoles, and Whitehead products

Let X be a closed manifold of dimension 2m ≥ 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small vol- ume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.

Algebraic Invariants of Simple 4-Knots

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

Symmetricity of the Whitehead element

ABSTRACf. We study the symmetricity of the Whitehead element Wn E 1t2np_ 3(S2n-l) for an odd prime p. It is shown that Wo considered as a map S20p-3 -+ S20-1 factors through the p-fold covering map (]: S20p-3 -+ L 20 p-3 only when n is a power of p, and that wp' actually factors through (] if 0 :::;; i :::;; 4. This is some of an odd prime version of the results of Randall and Lin for the projectivity of the Whitehead product ['20-1, '20-1] E 1t40_3(S20-1).

T-spaces by the Gottlieb groups and duality

AbstractIt is shown that all the generalized Whitehead products vanish in X and all the components of XΣA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.