论文标题
理性映射空间的自忽视数量
Self-Closeness Numbers of Rational Mapping Spaces
论文作者
论文摘要
对于一个封闭的连接歧管$ m $的尺寸$ 2N $，Møller和Raussen证明了映射空间的组件从$ m $到$ s^{2n} $具有两种不同的理性同拷贝类型。但是，由于该组件的代数模型证明了这一结果，因此目前尚不清楚其他同型不变性是否区分了其合理同质副本类型。连接的CW复合物的自智度数量是整数最小的$ k $，因此，其自动图以$*\ le k $为$π_*$诱导同构的任何一个同构成都是同质的，并且到目前为止，映射空间的组成部分都没有结果。对于有限的$π_1$的理性庞加莱综合体$ x $ 2n $，我们完全确定了通过使用棕色szczarba型号，从$ x $到$ s^{2n} $的合理组件的自智能数量。作为推论，我们表明自倾性编号确实区分了组件的合理同副本类型。由于封闭的互联歧管是一个理性的庞加莱综合体，因此我们的结果部分概括了Møller和Raussen的综合体。
For a closed connected oriented manifold $M$ of dimension $2n$, it was proved by Møller and Raussen that the components of the mapping space from $M$ to $S^{2n}$ have exactly two different rational homotopy types. However, since this result was proved by the algebraic models for the components, it is unclear whether other homotopy invariants distinguish their rational homotopy types or not. The self-closeness number of a connected CW complex is the least integer $k$ such that any of its self-map inducing an isomorphism in $π_*$ for $*\le k$ is a homotopy equivalence, and there is no result on the components of mapping spaces so far. For a rational Poincaré complex $X$ of dimension $2n$ with finite $π_1$, we completely determine the self-closeness numbers of the rationalized components of the mapping space from $X$ to $S^{2n}$ by using their Brown-Szczarba models. As a corollary, we show that the self-closeness number does distinguish the rational homotopy types of the components. Since a closed connected oriented manifold is a rational Poincaré complex, our result partially generalizes that of Møller and Raussen.
