至多n值映射
At most n-valued maps
AI总结：
本文研究至多n值映射的四种模型及其包含关系，定义一般类C并给出配置空间观点，完整计算了C_n(S^1)的基本群和同调群。
摘要：
本文研究“至多n值映射”的各种模型。即多值映射$f:X\multimap Y$，其中对每个$x$，$f(x)$的基数至多为$n$。我们考虑文献中出现的4类这样的映射：$\mathcal U$，即恰好n值映射的集合，或此类映射的并集；$\mathcal F$，即由Crabb定义的n重映射的集合；$\mathcal S$，即对称积映射的集合；以及$\mathcal W$，即权重在$\mathbb N$中的加权映射的集合。我们的主要结果大致是这些类满足以下包含关系：\\[ \mathcal U \subsetneq \mathcal F \subsetneq \mathcal S = \mathcal W \\] 此外，我们定义所有至多n值映射的一般类$\mathcal C$，并证明$\mathcal C$中存在不属于上述任何其他类的映射。我们还为类$\mathcal C$描述了一个配置空间的观点，定义配置空间$C_n(Y)$，使得任何至多n值映射$f:X\multimap Y$自然对应于单值映射$f:X\to C_n(Y)$。我们给出了$C_n(S^1)$的基本群和同调群的完整计算。
英文摘要：
This paper concerns various models of ``at-most-$n$-valued maps''. That is, multivalued maps $f:X\multimap Y$ for which $f(x)$ has cardinality at most $n$ for each $x$. We consider 4 classes of such maps which have appeared in the literature: $\mathcal U$, the set of exactly $n$-valued maps, or unions of such; $\mathcal F$, the set of $n$-fold maps defined by Crabb; $\mathcal S$, the set of symmetric product maps; and $\mathcal W$, the set of weighted maps with weights in $\mathbb N$. Our main result is roughly that these classes satisfy the following containments: \[ \mathcal U \subsetneq \mathcal F \subsetneq \mathcal S = \mathcal W \] Furthermore we define the general class $\mathcal C$ of all at-most-$n$-valued maps, and show that there are maps in $\mathcal C$ which are outside of any of the other classes above. We also describe a configuration-space point of view for the class $\mathcal C$, defining a configuration space $C_n(Y)$ such that any at-most-$n$-valued map $f:X\multimap Y$ corresponds naturally to a single-valued map $f:X\to C_n(Y)$. We give a full calculation of the fundamental group and homology groups of $C_n(S^1)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
