LS category of the quotient of a manifold by its involution

Question

$\DeclareMathOperator\cat{cat}$Let $\cat(X)$ denote the Lusternik–Schnirelmann (LS) category of $X$. This homotopy invariant has been well-studied for CW complexes: it is $0$ if and only if $X$ is contractible. I'm interested in studying the LS category of low-dimensional manifolds $M$. Let $M$ be a closed, connected, orientable manifold of dimension $n\le 4$. I have been stuck at the following question.

Let $\iota:M \to M$ be an involution of $M$ (possibly having fixed points) and let $\widehat{M}:= M/\iota$ denote the quotient space of $M$ by the involution $\iota$. I want to know if there is a general technique one can use to determine (or at least bound from above) the value $\cat(\widehat{M})$? Of course, $\cat(\widehat{M}) \le \dim(M)=n$, but is there a better upper bound one can obtain?

I understand that one can estimate LS-category in the case of coverings and in the case of quotients by group actions. But here, the situation is apparently different. Is there any literature that deals with calculating the LS-category of the quotients of nice manifolds with their nice involutions? Any references will be appreciated.

Answer 1

The equivariant LS category $\operatorname{cat}_G(X)$ of a space $X$ with $G$-action is the minimal $k$ such that $X$ admits a cover by $G$-invariant open sets $U_0,\ldots , U_k$ such that each inclusion $U_i\hookrightarrow X$ is $G$-homotopic to a map with values in a single orbit. See for example

Marzantowicz, Wacław, A G-Lusternik-Schnirelmann category of space with an action of a compact Lie group, Topology 28, No. 4, 403-412 (1989). ZBL0679.55001

for the case when $G$ is compact Lie. According to Proposition 1.3(viii) of the above, the inequality $\operatorname{cat}_G(X)\geq \operatorname{cat}(X/G)$ always holds.

In your case in which $G=C_2$ acts (possibly non-freely) on a manifold $M$, this might potentially give a useful upper bound on $\operatorname{cat}(M/\iota)$, provided you can get upper bounds on $\operatorname{cat}_{C_2}(M)$. The equivariant LS category for finite group actions has been studied in

Colman, Hellen, Equivariant LS-category for finite group actions., Cornea, O. (ed.) et al., Lusternik-Schnirelmann category and related topics. Proceedings of the 2001 AMS-IMS-SIAM joint summer research conference on Lusternik-Schnirelmann category in the new millennium, South Hadley, MA, USA, July 29–August 2, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2800-2/pbk). Contemp. Math. 316, 35-40 (2002). ZBL1035.55003

for example, you might find something relevant to your situation there. There is also some literature (by Colman, Ángel and others) about LS category of orbifolds, which might contain something useful. Generally though, when the action is non-free it seems to be somewhat difficult to get general upper bounds.