Note on Whitehead Products in Spheres.

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 [Whi...

Exotic spheres and the Whitehead space

I. Review: An obstruction theory. — Let Θn be the group of diffeomorphism classes of oriented smooth homotopy spheres of dimension n. Let Diff(Sn−1) be the simplicial group of orientation preserving diffeomorphisms Sn−1 → Sn−1. For n > 5, the natural homomorphism from π0Diff(Sn−1) to Θn is surjective by Smale’s h–cobordism theorem, and injective by Cerf’s pseudo–isotopy theorem. It is easily se...

Note on Cup-products

Explicit parallelizations on products of spheres

Kervaire’s proof does not provide an explicit parallelization on products of spheres. The only reference the author knows to provide explicit parallelizations is [Bru92], that considers the cases when one of the spheres is of dimension 1, 3, 5, 7, and uses some specific arguments of these low dimensions. In [Bru92] the general case is left as an open problem. The aim of this paper is to write a...

Quasilinear Actions on Products of Spheres

For some small values of f , we prove that if G is a group having a complex (real) representation with fixity f , then it acts freely and smoothly on a product of f +1 spheres with trivial action on homology.