### A chain rule in the calculus of homotopy functors.

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For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic...

We explore connections between our previous paper [J. Reine Angew. Math. 604 (2007)], where we constructed spectra that interpolate between bu and Hℤ, and earlier work of Kuhn and Priddy on the Whitehead conjecture and of Rognes on the stable rank filtration in algebraic K-theory. We construct a "chain complex of spectra" that is a bu analogue of an auxiliary complex used by Kuhn-Priddy; we conjecture that this chain complex is "exact"; and we give some supporting evidence. We tie this to work of...

Let V be an orthogonal representation of a compact Lie group G and let S(V),D(V) be the unit sphere and disc of V, respectively. If F: V → ℝ is a G-invariant C¹-map then the G-equivariant gradient C⁰-map ∇F: V → V is said to be admissible provided that ${\left(\nabla F\right)}^{-1}\left(0\right)\cap S\left(V\right)=\varnothing $. We classify the homotopy classes of admissible G-equivariant gradient maps ∇F: (D(V),S(V)) → (V,V∖0).

Let G be a finite group, ${\mathbb{O}}_{G}$ the category of canonical orbits of G and $A:{\mathbb{O}}_{G}\to \mathbb{A}$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $Ex{t}^{n-1}(A,A\otimes A)$. Then the case $G={\mathbb{Z}}_{{p}^{k}}$ leads to an example of infinitely...

A special case of G-equivariant degree is defined, where G = ℤ₂, and the action is determined by an involution $T:{\mathbb{R}}^{p}\oplus {\mathbb{R}}^{q}\to {\mathbb{R}}^{p}\oplus {\mathbb{R}}^{q}$ given by T(u,v) = (u,-v). The presented construction is self-contained. It is also shown that two T-equivariant gradient maps $f,g:(\mathbb{R}\u207f,{S}^{n-1})\to (\mathbb{R}\u207f,\mathbb{R}\u207f\setminus 0)$ are T-homotopic iff they are gradient T-homotopic. This is an equivariant generalization of the result due to Parusiński.