The Injective category number on continuous maps

TL;DR

The paper defines the injective category number $IC(f)$ for continuous maps and establishes its basic and structural properties, including pullback stability and composition bounds. It provides a cohomological lower bound for surjective maps between manifolds of the same dimension and expresses $IC(f)$ in terms of finite multiple points, offering exact results in the finite-multiplicity case. For quotient maps arising from free $G$-spaces, it relates $IC(\mathfrak{q}^X)$ to the $2$-nd index $\mathrm{ind}_2(X,G)$ and proves sharp bounds when $G=\mathbb{Z}_2$, such as $IC(\mathfrak{q}^{S^n})=n+2$, connecting to Borsuk-Ulam-type results. Overall, the work links classical Borsuk-Ulam theory with contemporary topological methods to bound and compute injective category numbers, providing a framework for understanding when a locally injective map is globally injective.

Abstract

We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or \aspas{categorical structure} underlying the question: under what conditions is $f$ injective? More precisely, $\text{IC}(f)$ is the smallest positive integer $\ell$ such that $X$ can be covered by $\ell$ open subsets $U_1,\ldots,U_\ell$, with each restriction map $f_{\mid U}:U\to Y$ being injective. For instance, we examine the behaviour of $\text{IC}(f)$ under pullbacks and compositions of maps. In addition, we provide a cohomological lower bound for $\text{IC}(f)$. When $f$ has a finite number of multiple points, we express $\text{IC}(f)$ in terms of these points of non-injectivity. In the case that $f$ is the quotient map $\mathfrak{q}^X:X\to X/G$, where $X$ is a metric free $G$-space, we provide a lower bound for the injective category of $\mathfrak{q}^X$ in terms of the $2$-th index, $\text{ind}_2(X,G)$. When $G=\mathbb{Z}_2$, this lower bound is shown to be sharp. These results link a classical problem in Borsuk-Ulam theory to contemporary research developments in the study of injective category numbers.

Figures (3)

• Figure 1: Figure eight.
• Figure 2: The open subsets $U_1,U_2\subset S^1$.
• Figure 3: Immersion of the circle in the plane with a single multiple point of type $5$th.

Theorems & Definitions (37)

• Example 2.1
• Example 2.2
• Definition 2.3: Injective Category Number
• Remark 2.4
• Example 2.5
• Example 2.6
• Theorem 2.7: Under Pullback
• proof
• Proposition 2.8
• Remark 2.9