Zero sets of equivariant maps from products of spheres to Euclidean spaces

Some remarks on the parametrized Borsuk–Ulam theorem

“…Two cases in which these conditions are satisfied are described in Propositions 3.15 and 3.17, which pursue ideas introduced in [15] and [12].…”

“…(ii). ([9, Corollary 1.5], [12,Theorem 1.3]). Suppose that S(λ|F ) → F is homeomorphic to the projection S n1 × · · · × S nr → (S n1 × · · · × S nr )/{±1} from a product of r spheres to the quotient by the involution that acts antipodally on each factor.…”

“…(ii). ([9, Corollary 1.5], [12,Theorem 1.3]). Suppose that S(λ|F ) → F is homeomorphic to the projection…”

“…To be accurate, the papers [15,9,12,10] work in a rather more general setting than that considered here: the base B is not required to be a compact ENR and the involution on the vector bundle ξ is not necessarily antipodal. We shall not deal with more general base spaces here, but we explain in Section 5 how to incorporate more general involutions into the theory.…”

“…The first is quite elementary. But it leads to easy proofs of a number of results in the recent literature ([9, Corollary 1.5], [12,Theorem 1.3], [10,Theorem 1.5], [15,Theorem 1.3]). In the statement below, e(λ) = w 1 (λ) ∈ H 1 (E; F 2 ) is the F 2 -Euler class of λ. Theorem 1.1.…”