Borsuk-Ulam type spaces (Q2802999)

The classical Borsuk-Ulam theorem asserts that for any continuous mapping from the \(n\)-dimensional sphere \(\mathbb{S}^n\) into the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), \(f: \mathbb{S}^n \to \mathbb{R}^n\), there is a point \(x\in \mathbb{S}^n \) such that \(f(x)=f(-x)\). The Borsuk-Ulam theorem is equivalent to several other geometric results about spheres such as the Lusternik-Schnirelmann-Borsuk theorem, Tucker's lemma, the non-existence of an antipode-preserving-map \(f: \mathbb{S}^n \to \mathbb{S}^{n-1}\), etc. In a more general setting, given a topological space \(X\) with a free involution \(A: X\to X\), the pair \((X,A)\) is called a \(\text{BUT}_n\) (Borsuk-Ulam type space) if for any continuous mapping \(f: X \to \mathbb{\mathbb R}^n\) there is \(x\in X\) such that \(f(A(x))=f(x)\). And \((X,A)\) is said to be a \(\text{BUT}\) space if it is a \(\text{BUT}_n\) for \(n=\text{dim} \, X\). In [\textit{P. Bacon}, Can. J. Math. 18, 492--502 (1966; Zbl 0142.20804)], for \(X\) a normal topological space, the equivalence between \((X,A)\) being a \(\text{BUT}_n\), an \(\text{LS}_n\)(Lusternik-Schnirelmann space), a \(\text{T}_n\)(Tucker space), etc., was shown. In the present paper, the authors give interesting necessary and sufficient conditions for a space to be \(\text{BUT}\), some of them involving the use of Yang's cohomological index.