Smash Products for Secondary Homotopy Groups

Hans-Joachim Baues 0 Fernando Muro 0 F. Muro 0 0 Departament d'A` lgebra i Geometria, Universitat de Barcelona , Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain 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. The classical homotopy groups n X, n 0, of a pointed space X give rise to a graded abelian group X obtained by additivization in low dimensions. In particular n X = n X for n 2, 1 X = (1 X)ab is the abelianized fundamental group, and 0 X = Z[0 X] is the free abelian group on the pointed set of path components of X. The smash product on homotopy groups induces a natural homomorphism of graded abelian groups - X (X Y), The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract. which carries f g, with f : Sn X and g : Sm Y, to f g : Sn+m X Y. This shows that is a lax symmetric monoidal functor from pointed spaces to graded abelian groups. The smash product (1) can be used for example to define the Whitehead product on homotopy groups, compare Section 3.4. The purpose of this paper is to generalize these properties of primary homotopy groups on the level of secondary homotopy theory. Secondary homotopy operations like Toda brackets [22] or cup-one products [6, 18] are defined by pasting tracks, where tracks are homotopy classes of homotopies. Since secondary homotopy operations play a crucial role in homotopy theory it is of importance to develop the algebraic theory of tracks. We do this by introducing secondary homotopy groups of a pointed space X n, X = which have the structure of a quadratic pair module, see Sections 2.1 and 3.1. Here is a group homomorphism with cokernel n X for n 0 and kernel n+1 X for n 3. We define n, X for n 2 directly in terms of maps Sn X and tracks from such maps to the trivial map. For n 0 the functor n, is an additive version of the functor n, studied in [9]. We introduce and study the smash product morphism for additive secondary homotopy groups ,(X Y). Here one needs the symmetric monoidal structure of the category of quadratic pair modules qpm, which is based on the symmetric monoidal structure on the category of square groups constructed in [7]. The smash product morphism (2) is compatible with the associativity isomorphisms, but it is not directly compatible with the commutativity isomorphisms. In order to deal with commutativity we need the action of the symmetric track group Sym (n) on n, X in [10]. We show that in Eq. 2 is equivariant with respect to this action, and is commutative up to the action of a shuffle permutation. This leads to the definition of the symmetric monoidal category qpm0Sym with objects given by symmetric sequences of quadratic pair modules with extra structure. Then the morphism (2) induces a morphism in qpm0Sym for which the associativity and commutativity isomorphisms are compatible with the symmetric monoidal structure of qpm0Sym . Therefore , considered as a functor to the category qpm0Sym is, in fact, a lax symmetric monoidal functor. The smash product (2) is used to define the Whitehead product on secondary homotopy groups, compare Section 3.4. As an illustrating application of the results in this paper we prove a formula of BarrattJonesMahowald on unstable cup-one products, see Section 3.5. This formula was stated in [6], but a proof did not appear in the literature. A further application yields a formula for a triple Toda bracket which generalizes a well-known formula in [22], see Section 3.6. In a sequel of this paper we generalize the theory of secondary homotopy groups to symmetric spectra [8]. There we show that the smash product operation defined in this paper endows the secondary homotopy groups , R of a fibrant connective ring spectrum R with a graded algebra structure in the category of quadratic pair modules. The graded algebra , R determines all Toda brackets in R, which can be regarded as Massey products in , R. Moreover, , R determines the universal matrix Toda bracket in the category of finitely generated free R-modules. If R is in addition an E-ring spectrum then , R is a commutative algebra up to coherent homotopies in qpm0Sym which encodes not only Toda brackets, but also cup-one products in a purely algebraic way. The paper consists of three parts. The first part is concerned with the algebra needed for the statements of the main theorems. In Section 3 we present our main results and we give applications. Section 4 contains the construction of the smash product operation for additive secondary homotopy groups. There we prove all the properties which imply our main results. 2 Quadratic Pair Modules and Their Tensor Product In this part we describe the algebraic concepts needed for the structure of secondary homotopy groups. We introduce the category of quadratic pair modules and we show that this category is symmetric monoidal. The tensor product of quadratic pair modules is related to the exterior cup-products in the category Top. 2.1 Square Groups and Quadratic Pair Modules We first recall the notion of square group, see [13] and [7]. Definition 2.1.1 A square group X is a diagram X = where Xe is a group with an additively written group law, Xee is an abelian group, P is a homomorphism, H is a quadratic map, i.e. a function such that the cross effect (a|b )H = H(a + b ) H(b ) H(a) (1) (Px|b )H = 0, (a|Py) = 0, (2) P(a|b )H = [a, b ], (3) PH P(x) = P(x) + P(x). Here [a, b ] = a b + a + b is the commutator bracket. The cross effect induces a homomorphism (|)H : 2 Coker P Xee. Here 2 A = A A is the tensor square of an abelian group A. If Eq. 3 is an isomorphism we will say that the square group X is good. The function T = H P 1 : Xee Xee : Xe Xee : x (x|x)H H(x) + T H(x) is a homomorphism which satisfies T = . A morphism of square groups f : X Y is given by homomorphisms fe : Xe Ye, fee : Xee Yee, commuting with P and H. As an example of square group we can consider Znil = with P = 0 and H(n) = n2 = n(n21) . This is the unit object of the symmetric monoidal structure defined in the next section. We refer the reader to [7] where the quadratic algebra of square groups is developed. We need square groups for the definition of quadratic pair modules as follows. Definition 2.1.2 A quadratic pair module C is a morphism : C(1) C(0) between square groups such that ee = 1 : Cee Cee is the identity homomorphism. In particular C is completely determined by the diagram C(0) = C(1) = h0C = C0/(C1), h1C = Ker[ : C1 C0]. where = e, H1 = H and P0 = P. Morphisms of quadratic pair modules f : C D are therefore given by group homomorphisms f0 : C0 D0, f1 : C1 D1, fee : Cee Dee, commut (...truncated)