A Borsuk-Ulam type theorem for a product of spheres

A parametrized Borsuk–Ulam theorem for a product of spheres with free ℤp–action and freeS1–action

“…These theorems extend the result proved by Koikara and Mukerjee in [7]. Further, in the particular case where G D ‫ޚ‬ p , we estimate the "size" of the ‫ޚ‬ p -coincidence set of a fibre-preserving map.…”

“…As in [7], we need to assume that the quotient bundle .X=G; x E; x ; B/, where G is ‫ޚ‬ p or S 1 , has the cohomology extension property and then the Leray-Hirsch theorem can be applied. There are two cases to consider, as follows.…”

“…The technique introduced by Dold to solve the parametrized problem by using characteristic polynomials was also used by Koikara and Mukerjee in [7] to show a parametrized version of the Borsuk-Ulam theorem for bundles whose fibre is a product of spheres, with the free ‫ޚ‬ 2 -action given by the product of the antipodal actions. The goal of this paper is to extend this result of Koikara and Mukerjee to all free Z p -actions, p > 2, and to all free S 1 -actions.…”

Some remarks on the parametrized Borsuk–Ulam theorem

“…The error lies in the proof of Corollaries 4.2 and 4.4, which is modelled on the somewhat misleading approach in [9] to the result stated there as Corollary 1.5. We trust that the argument used in [9,Corollary 1.5], and also in [12, Theorem 1.3], [10, Theorem 1.5] and [15,Theorem 1.3], will now be superseded by our proof of those theorems as applications (Examples 2.5) of Corollary 2.4.…”

A Borsuk-Ulam theorem for maps of fibre bundles with manifolds as fibres

“…Consequently we have obtained a Borsuk-Ulam theorem for maps of fibre bundles with manifolds as fibres, and for particular manifolds like product of spheres this procedure gives a better estimate of the Borsuk-Ulam set of the map of the bundle than that obtained using the method of characteristic polynomials (cf. [4]). …”