Maps of Stiefel manifolds and a Borsuk–Ulam theorem

Abstract

We are concerned with the following classical version of the Borsuk–Ulam theorem: Let f : S n → R k be a map and let A f = { x ∈ S n | fx = f (− x )}. Then, if k ≦ n , A f ≠φ. In fact, theorems due to Yang [17] give an estimation of the size of A f in terms of the cohomology index. This classical theorem concerns the antipodal action of the group G =ℤ 2 on S n . It has been generalized and extended in many ways (see a comprehensive expository article by Steinlein [16]). This author ([9, 10)] and Nakaoka [14] proved “continuous” or “parameterized” versions of the theorem. Analogous theorems for actions of the groups G = S 1 or S 3 have been proved in [11], and [12]; compare also [4, 5, 6].

Locations

• Proceedings of the Edinburgh Mathematical Society - View - PDF