Numerical invariants of homotopies into spheres (Q760936)

A classical problem is: Given two points in the same component of a Riemannian manifold M, what is the length of the shortest path connecting them? The width of a homotopy \(X\times I\to M\) is the supremum of the lengths of paths traced by the points of X. Given a space X and two homotopic maps f,g: \(X\to M\), what is the width of the shortest homotopy between them? Let \(b_ q(M)\) be the infimum of all \(r\in {\mathbb{R}}\cup \{\infty \}\) such that for any q-dimensional normal space X and any two homotopic maps f,g: \(X\to M\) there is a homotopy of width \(\leq r\) between them. Let \(B_ q(M)\) be the infimum of all \(r\in {\mathbb{R}}\cup \{\infty \}\) such that for any q-dimensional normal space X, any homotopy \(H: X\times I\to M\) is homotopic keeping the endmaps fixed to a homotopy of width \(\leq r\). (It should be noted that these definitions differ slightly from those of the authors by allowing \(\infty\) as possible value.) The authors compute some of these numbers \(b_ q(M)\), \(B_ q(M)\) for spheres \(S^ n\), and for complex projective spaces \({\mathbb{C}}P^ n\). Of particular interest is the fact that \(b_{2n-2}(S^ n)\) detects the Hopf invariant one while \(B_{2n-2}(S^ n)\) does not.