Homotopy properties of smooth functions on the Möbius band

Abstract

Let $B$ be a Möbius band and $f:B\to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h=f$. Under certain assumptions on $f$ we compute the group ${\pi }_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to \mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials ${\mathbb{R}}^2\to \mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to \mathbb{R}$ on non-orientable surfaces $N$.