The orbit component of a smooth function on a Klein bottle that splits into two Möbius bands is homotopy equivalent to the product of the orbit components of its restrictions to the two bands.
Homotopy dimension of orbits of Morse functions on surfaces
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abstract
Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the connected components of $O$ have the homotopy type of a finite-dimensional CW-complex. Actually, these connected components are homotopy equivalent to a certain covering space of the $n$-th configuration space of the interior of $M$. As a consequence we obtain that the fundamental group of $O$ is a subgroup of the $n$-th braid group of $M$.
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Smooth functions that split a Klein bottle into two M\"obius bands
The orbit component of a smooth function on a Klein bottle that splits into two Möbius bands is homotopy equivalent to the product of the orbit components of its restrictions to the two bands.