Equivariant category of wedges
Cesar A Ipanaque Zapata, Denise de Mattos
TL;DR
The paper studies equivariant Lusternik–Schnirelmann category $\text{cat}_G(X)$ and the equivariant/invariant topological complexities $\text{TC}_G(X)$ and $\text{TC}^G(X)$ for wedges and related quotients in the setting of a compact group $G$ acting on a space. Using an extension of the Cornea–Lupton–Oprea–Tanré framework, the authors introduce $y$-connectivity and based $G$-categorical covers to establish a sharp wedge formula $\text{cat}_G(X\vee Y)=\max\{\text{cat}_G(X),\text{cat}_G(Y)\}$ under suitable hypotheses. They derive corollaries including that $\bigvee_{i=1}^m X_i$ is $G$-contractible iff each $X_i$ is, compute $\text{cat}_G(X/A)$ when the inclusion $A\hookrightarrow X$ can be contracted to a fixed point, and obtain lower and, in many cases, exact upper bounds for $\text{TC}_G(X\vee Y)$ and $\text{TC}^G(X\vee Y)$. The results extend to products/actions, smash products, and quotients, and include concrete examples and conjectures about when the wedge equalities for TC hold in full generality.
Abstract
We prove the formula \begin{equation*} \text{cat}_G(X\vee Y)=\max\{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*} for the equivariant category of the wedge $X\vee Y$. As a direct application, we have that the wedge $\bigvee_{i=1}^m X_i$ is $G$-contractible if and only if each $X_i$ is $G$-contractible, for each $i=1,\ldots,m$. One further application is to compute the equivariant category of the quotient $X/A$, for a $G$-space $X$ and an invariant subset $A$ such that the inclusion $A\hookrightarrow X$ is $G$-homotopic to a constant map $\overline{x_0}:A\to X$, for some $x_0\in X^G$. Additionally, we also discuss the equivariant and invariant topological complexities for wedges. For instance, as applications of our results, we obtain the following equalities: \begin{align*} \text{TC}_G(X\vee Y)&=\max\{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\}, \text{TC}^G(X\vee Y)&=\max\{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*} for $G$-connected $G$-CW-complexes $X$ and $Y$ under certain conditions.