At most n-valued maps

Abstract

This paper concerns various models of ``at-most-$n$-valued maps''. That is, multivalued maps $f:X⊸Y$ 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 $f:X⊸Y$ 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 $C_n(S^1)$.