Smooth functions that split a Klein bottle into two M\"obius bands

TL;DR

This paper computes the homotopy types of orbits of smooth functions on the Klein bottle that split it into two Möbius bands, extending previous results for functions on surfaces and revealing the structure of these function spaces.

Contribution

It explicitly determines the homotopy type of the orbit of certain smooth functions on the Klein bottle that divide it into two Möbius bands, linking it to the product of orbits on the bands.

Findings

Homotopy type of the orbit is equivalent to the product of orbits on the Möbius bands.

Provides explicit computations for the homotopy types of these orbits.

Extends the understanding of function spaces on non-orientable surfaces.

Abstract

Given a compact surface $M$, consider the right action $\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$, $(f, h) \mapsto f\circ h$, of the group $\mathcal{D}(M)$ of $\mathcal{C}^{\infty}$ diffeomorphisms of $M$ on the space $\mathcal{C}^{\infty}(M)$ of $\mathcal{C}^{\infty}$ functions on $M$. For $f\in\mathcal{C}^{\infty}(M)$ denote by $\mathcal{O}(f)$ its orbit, and by $\mathcal{O}_f(f)$ the path component of $\mathcal{O}(f)$ containing $f$. The paper continues a series of computations by many authors of homotopy types of orbits $\mathcal{O}_f(f)$ of smooth functions on compact surfaces. We provide here the computations of $\mathcal{O}_f(f)$ for a special class of functions $f\in\mathcal{C}^{\infty}(K)$ on the Klein bottle $K$ having the following properties: (i) at each critical point $f$ is smoothly equivalent to some homogeneous polynomial (e.g. $f$ is Morse), and (ii) there is a regular connected component $\alpha$ of a level set of $f$ such that $K\setminus\alpha$ is a disjoint union of two open M\"obius bands, with closures $M_1$ and $M_2$. Let $f_i = f|_{M_i}$ be the restriction of $f$ to the M\"{o}bius band $M_i$, $i=1,2$, and $\mathcal{O}_{f_i}(f_i)$ be the path component of $f_i$ in its orbit with respect to the above action of $\mathcal{D}(M_i)$. The possible homotopy types of $\mathcal{O}_{f_i}(f_i)$ are explicitly computed earlier. We prove that $\mathcal{O}_f(f)$ is homotopy equivalent to $\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$.