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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, $${\mathfrak A}({\bf X}) := \left\{ \phi \in {\bf C}({\bf X}, {\bf X}) : \phi(x) = x \mbox{ for $x \in \partial {\bf X}$}\right\},$$ equipped with the topology of uniform convergence. In this article we address the enumeration problem for the homotopy classes of $${{\mathfrak A}({\bf X})}$$ and introduce a topological degree ($${\phi \mapsto {\bf deg}[\phi]}$$) fully capable of describing the homotopy class of $${\phi}$$ . 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).
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Publication status
- Published
Journal
Archive for Rational Mechanics and AnalysisISSN
0003-9527Publisher
SpringerExternal DOI
Issue
1Volume
197Page range
239-270Pages
32.0Department affiliated with
- Mathematics Publications
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- No
Peer reviewed?
- Yes
Legacy Posted Date
2012-02-06Usage metrics
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