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At most n-valued maps

Daciberg Lima Goncalves, Robert Skiba, P. Christopher Staecker

TL;DR

This work develops a unified configuration-space framework for at-most-$n$-valued maps by introducing the unordered configuration space $C_n(Y)$ and showing that an at-most-$n$-valued map $f:X\multimap Y$ corresponds to a single-valued map $F:X\to C_n(Y)$. It then analyzes four literature models—$\oldsymbol{\mathcal U}$, Crabb’s $\boldsymbol{\mathcal F}$, $\boldsymbol{\mathcal S}$, and $\boldsymbol{\mathcal W}$—establishing the containment chain $\Psi(\mathcal U) \subsetneq \Psi(\mathcal F) \subsetneq \Psi(\mathcal S)=\Psi(\mathcal W) \subsetneq \mathcal C$, with $\Psi(\mathcal S)$ and $\Psi(\mathcal W)$ shown to coincide. The paper also proves a splitting result for $\mathbb{Q}$-weighted maps in a broad connectivity/dimension regime and computes the topology of the at-most-$n$ configuration space on the circle, revealing a sphere-like homotopy type dependent on the parity of $n$. These results clarify the relationships among models, provide practical tools via the configuration-space viewpoint, and yield explicit topological invariants for $C_n(S^1)$ that reflect the underlying combinatorial structure of at-most-$n$-valued maps.

Abstract

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 $C_n(S^1)$.

At most n-valued maps

TL;DR

This work develops a unified configuration-space framework for at-most--valued maps by introducing the unordered configuration space and showing that an at-most--valued map corresponds to a single-valued map . It then analyzes four literature models—, Crabb’s , , and —establishing the containment chain , with and shown to coincide. The paper also proves a splitting result for -weighted maps in a broad connectivity/dimension regime and computes the topology of the at-most- configuration space on the circle, revealing a sphere-like homotopy type dependent on the parity of . These results clarify the relationships among models, provide practical tools via the configuration-space viewpoint, and yield explicit topological invariants for that reflect the underlying combinatorial structure of at-most--valued maps.

Abstract

This paper concerns various models of ``at-most--valued maps''. That is, multivalued maps for which has cardinality at most for each . We consider 4 classes of such maps which have appeared in the literature: , the set of exactly -valued maps, or unions of such; , the set of -fold maps defined by Crabb; , the set of symmetric product maps; and , the set of weighted maps with weights in . Our main result is roughly that these classes satisfy the following containments: Furthermore we define the general class of all at-most--valued maps, and show that there are maps in which are outside of any of the other classes above. We also describe a configuration-space point of view for the class , defining a configuration space such that any at-most--valued map corresponds naturally to a single-valued map . We give a full calculation of the fundamental group and homology groups of .

Paper Structure

This paper contains 14 sections, 40 theorems, 139 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A configuration space for at-most-$n$-valued maps
  3. Models of at-most-$n$-valued maps in the literature
  4. The class $\mathcal{U}$: unions of equicardinal maps
  5. The class $\mathcal{F}$: Crabb's $n$-fold maps
  6. The class $\mathcal{W}$: $\mathbb{N}$-weighted maps
  7. The class $\mathcal{S}$: Symmetric product maps
  8. $\Psi(\mathcal{U}) \subsetneq \Psi(\mathcal{F})$
  9. $\Psi(\mathcal{F}) \subsetneq \Psi(\mathcal{S})$
  10. $\Psi(\mathcal{S}) = \Psi(\mathcal{W})$
  11. $\Psi(\mathcal{W})\subsetneq \mathcal{C}$
  12. The class of $\{1,n\}$-valued maps
  13. A splitting result for $\mathbb{Q}$-weighted maps
  14. Topology of the at-most-$n$-valued configuration space of the circle

Key Result

Theorem 2.3

Let $Y$ be Hausdorff, and let $f:X\multimap Y$ be at-most-$n$-valued and lower semicontinuous. Then $f$ is also upper semicontinuous.

Figures (5)

  • Figure 1: The graph of a 2-valued function $f:\mathbb{R} \multimap \mathbb{R}$ which is upper semicontinuous but not lower semicontinuous.
  • Figure 2: An $\mathbb{N}$-weighted map which is not a union of equicardinal maps. Numbers on arcs of the graph give the weights. The blue arc are points of weight 2, the red arcs are points of weight 1.
  • Figure 3: An at-most-3-valued map which is not an $\mathbb{N}$-weighted map. The $x$-coordinates of the two branching points are $\frac{1}{3}$ and $\frac{2}{3}$. Arcs of the graph are labeled according to the argument given in Example \ref{['forkmap']}.
  • Figure 4: A union of equicardinal maps on the interval $[0,1]$ which is not a $\{1,n\}$-valued map.
  • Figure 5: Representatives for the two faces of $C_n(S^1)$.

Theorems & Definitions (83)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • ...and 73 more