Bourgin–Yang version of the Borsuk–Ulam theorem for ℤ_p^k–equivariant maps

Abstract

Let G D Z p k be a cyclic group of prime power order and let V and W be orthogonal representations of G withWe give an estimate for the dimension of the set f 1 f0g in terms of V and W . This extends the Bourgin-Yang version of the Borsuk-Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the G -coincidences set of a continuous map from S.V / into a real vector space W 0 .

General Bourgin–Yang theorems

“…Note that even though K * G is defined for locally compact G-spaces, a K * G -based length makes sense only for compact G-spaces: K * G is functorial only for proper maps, hence it is not possible to consider the map induced by p X : X → pt on K * G for a non-compact X. We mention this specifically because it happens to be a prevalent mistake in [13]. In fact, our work can be seen as an erratum for that paper: even though the arguments therein are flawed, the results are still true.…”

“…, g r } is a finite group (with a fixed ordering of elements), the following standard argument allows to reduce this problem to a Bourgin-Yang type situation (cf. [13], [21], [22]).…”

“…Furthermore, to the best of our knowledge, those which exist rely on having a sphere as the (co)domain of the equivariant map in question (e.g. Marzantowicz, de Mattos and dos Santos [13], [14]) or are specifically aimed at estimating the size of the so called coincidence sets (e.g. Munkholm [17], [18], Volovikov [20], [21]).…”

Bourgin–Yang versions of the Borsuk–Ulam theorem for p-toral groups

“…Recently, in [14] the authors considered the Bourgin-Yang problem for the case that G is a cyclic group of a prime power order, G = Z p k , k ≥ 1. Based on the result of [2] it was proved [14, Theorem 1.1] that if V, W are two orthogonal representations of the cyclic group Z p k and f : S(V ) → W an equivariant map then the covering dimension dim(Z f ) = dim(Z f /G) ≥ φ(V, W ), where φ is a function depending on dim V , dim W and the orders of the orbits of actions on S(V ) and S(W ) (cf.…”

“…Based on the result of [2] it was proved [14, Theorem 1.1] that if V, W are two orthogonal representations of the cyclic group Z p k and f : S(V ) → W an equivariant map then the covering dimension dim(Z f ) = dim(Z f /G) ≥ φ(V, W ), where φ is a function depending on dim V , dim W and the orders of the orbits of actions on S(V ) and S(W ) (cf. [14,Theorems 3.6 and 3.9]). In particular, if dim W < dim V /p k−1 , then φ(V, W ) ≥ 0, which means that there is no G-equivariant map from S(V ) into S(W ).…”