Equivalence of Toda-Hopf invariants
Suppose that (X,Y) is an NDR pair with X and Y connected. Let A be the union of X and the cone over Y, and F be a fibration over A with fibre \(\Omega\) (\(\Sigma\) Y). The Gray invariant is the map G: \(F\to \Omega (X\wedge \Sigma Y)\) obtained by taking the natural map \(F\to \Omega (\Sigma F)\) and following it by the loop of the composite \(\Sigma\) \(F\to \Sigma (F/X)\to \Sigma X\wedge \Omega (\Sigma Y)\to^{eval}X\wedge \Sigma Y\). Recall that the James construction J(X) of a space X is the colimit of the subspaces \(J_ n(X)\), where \(J_ n(X)\) consists of the words of length \(\leq n\), topologized as a quotient of the n-fold product of X with itself. If \(X=\Sigma Y\), there is a Gray-Toda-Hopf invariant and Toda-Hopf invariant. Localized at p, the Toda-Hopf invariant T: \(\Omega\) (J\({}_{p-1}(S^{2m}))\to \Omega (S^{2pm-1})\) for p an odd prime is an H-map. The authors show that various existing constructions of Hopf invariant are equivalent to each other. Hence, there is one Toda- Hopf invariant with all of the properties.
scientific article; zbMATH DE number 4126312