Homotopy Classes of Self-Maps of Annuli, Generalised Twists and Spin Degree

Let X be a [generalised] annulus and consider the space of continuous self-maps of X, that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak A}({\bf X}) := \left\{ \phi \in {\bf C}({\bf X}, {\bf X}) : \phi(x) = x \mbox{ for $x \in \partial {\bf X}$}\right\},$$\end{document} equipped with the topology of uniform convergence. In this article we address the enumeration problem for the homotopy classes of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak A}({\bf X})}$$\end{document} and introduce a topological degree (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi \mapsto {\bf deg}[\phi]}$$\end{document}) fully capable of describing the homotopy class of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi}$$\end{document} . We devise various methods for computing this degree and discuss some implications of the latter to problems of nonlinear elasticity. In particular we present a novel homotopy classification for all twist solutions to a displacement boundary value problem and single out an erroneous common belief that some natural classes of twists furnish solutions to the equilibrium equations of three dimensional elasticity (see, for example, Ciarlet in Mathematical elasticity: Three dimensional elasticity, vol 1, Elsevier, Amsterdam, p. 249, 1988).