We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\mathbf{Diff}$ or $\mathbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $\mathbb{R}^n$ decidable? As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $\mathbb{R}^n$ is undecidable when $n-m$ is even and $11m \geq 10n+1$.

翻译：我们研究关于浸入理论的一个算法问题：对于哪些$m$、$n$以及$CAT=\mathbf{Diff}$或$\mathbf{PL}$，判断一个$m$维$CAT$流形是否可浸入$\mathbb{R}^n$的问题是可判定的？作为推论，我们证明当$n-m$为偶数且$11m \geq 10n+1$时，一个带边$m$流形在$\mathbb{R}^n$中的光滑可嵌入性是不可判定的。