论文标题
$ r $ $ immersions of $ \ mathbb r^n $的圆盘结合的空间
The space of $r$-immersions of a union of discs in $\mathbb R^n$
论文作者
论文摘要
对于歧管$ m $和整数$ r> 1 $，$ r $ immersions的$ m $ in $ \ mathbb r^n $的空间被定义为$ \ mathbb r^n $中$ m $的空间，以至于$ \ mathbb r^n $ copty in Mathbb r^n $ in Mathbb r^n $ coptiage copty的$ \ mathbb r^n $含量很少。当$ m $是$ k $ $ m $ $二维的二张光盘时，我们会考虑$ r $ immersions的空间，并证明它等同于$ r $ r $ -configuration $ k $ point in $ \ \ \ \ \ \ \ m m i \ mathbb r^n $和$ k^n $ and $ k^{$ text $ linb的产品r^m $ to $ \ mathbb r^n $。为了将Michael Weiss的歧管演算应用于$ r $ $ immersions的研究，需要此结果。嵌入空间的类似陈述是``著名''，但是在文献中很难找到详细的证据，并且现有的证据似乎使用了同位素扩展定理，即使只是为了方便起见。同位素扩展不适合$ r $ immmersions，因此我们阐明了避免使用它的证据的详细信息，并适用于$ r $ immmersions的空间。
For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that the preimage of every point in $\mathbb R^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb R^n$ and the $k^{\text{th}}$ power of the space of injective linear maps from $\mathbb R^m$ to $\mathbb R^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions.
