Deformational symmetries of smooth functions on non-orientable surfaces - Kuznietsova, Iryna; Maksymenko, Sergiy (Topological Methods in Nonlinear Analysis, 2025)
Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{∞}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f° h$ for $f\in \mathcal{C}^{∞}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$.Denote by $\mathcal{F}(M)$ the subset of $\mathcal{C}^{∞}(M,\mathbb{R})$ consisting of functions $\func\colon M→\mathbb{R}$ taking constant values on connected components of $\partial{M}$, having no critical points on $\partial{M}$, and such that at each of its critical points $z$ the function $f$ is $\mathcal{C}^{∞}$ equivalent to some homogenenous polynomial without multiple factors.In particular, $\mathcal{F}(M)$ contains all Morse maps.Let also$\mathcal{O}(f) = \{ f° h \mid h\in\mathcal{D}(M) \}$ be the orbit of $f$.Previously, the algebraic structure of $π_1\mathcal{O}(f)$ was computedfor all $f\in\mathcal{F}(M)$, where $M$ is any orientable compact surface distinct from $2$-sphere.In the present paper we compute the group $π_0\mathcal{S}(f,\partial\mathbb{M})$, where $\mathbb{M}$ is a Möbius band, and $\mathcal{S}(f,\partial\mathbb{M}) = \{ h\in\mathcal{D}(\mathbb{M}) \mid f° h = f, \ h|_{\partial \mathbb{M}} = id_{\mathbb{M}}\}$ is the subgroup of the corresponding stabilizer of $f$ consisting of diffeomorphisms fixed on the boundary $\partial \mathbb{M}$.As a consequence we obtain an explicit algebraic description of $π_1\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein bottle and projective plane.