We compute the Nielsen–Borsuk–Ulam number for any selfmap of \(n-\)torus, \(\mathbb {T}^n\), as well as any free involution \(\tau \) in \(\mathbb {T}^n\), with \(n \leqslant 3\). Finally, we conclude that the tori, \(\mathbb {T}^1\), \(\mathbb {T}^2\) and \(\mathbb {T}^3\), are Wecken spaces in Nielsen–Borsuk–Ulam theory. Such a number is a lower bound for the minimal number of pair of points such that \(f(x)=f(\tau (x))\) in a given homotopy class of maps.

中文翻译：

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tori 之间映射的 Nielsen–Borsuk–Ulam 数

我们为\(n-\)环面的任何自映射\(\mathbb {T}^n\)以及\ (\mathbb中的任何自由对合\(\tau \)计算 Nielsen–Borsuk–Ulam 数{T}^n\)，其中\(n \leqslant 3\)。最后，我们得出结论，圆环\(\mathbb {T}^1\)、\(\mathbb {T}^2\)和\(\mathbb {T}^3\)是 Nielsen 中的 Wecken 空间– Borsuk-Ulam 理论。这样的数字是给定同伦映射类中满足 \(f(x)=f(\tau (x))\)的最小点对数的下界。