Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T:X→X{T:X\to X} and S:Y→Y{S:Y\to Y}, respectively. Suppose that there exists a sequence (Xi,Ti)⁢⟶hi⁢(Xi+1,Ti+1) for ⁢1≤i≤k,(X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i , Xi{X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution Ti{T_{i}}, such that Xk+1=X{X_{k+1}=X}, and hi:Xi→Xi+1{h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1≤i≤k{1\leq i\leq k}. To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n -acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n -acyclic spaces), we prove that there is no equivariant map f:(X,T)→(Y,S){f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

中文翻译：

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滤波空间的Borsuk–Ulam定理

令X和Y沿路径连通，并分别配备自由对合T：X→X {T：X \ to X}和S：Y→Y {S：Y \ to Y}的超紧Hausdorff空间。假设存在一个序列（Xi，Ti）⁢⟶hi⁢（Xi + 1，Ti + 1），⁢1≤i≤k，（X_ {i}，T_ {i}）\ overset {h_ {i} } {\ longrightarrow}（X_ {i + 1}，T_ {i + 1}）\ quad \ text {for}％1 \ leq i \ leq k，其中，对于每个i，Xi {X_ {i}}是配备有自由对合Ti {T_ {i}}的路径连接的超紧Hausdorff空间，使得Xk + 1 = X {X_ {k + 1} = X}，并且hi：Xi→Xi + 1 {h_ {i }：X_ {i} \到X_ {i + 1}}是等价映射，对于所有1≤i≤k{1 \ leq i \ leq k}。为了获得Borsuk–Ulam型定理，在文献中出现的几种结果中，陈述中涉及的空间X被假定为同调n-非循环空间。在本文中，