Equivariant category and Topological complexity of wedges

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 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. Keywords: (Equivariant) Lusternik-Schnirelmann category, equivariant and invariant topological complexities, $G$-spaces, wedge product, smash product