Abstract
In this chapter, we use the torus to illustrate the basic ideas behind the topological classification of map**s. We introduce the map** class group of the torus and investigate its properties. The torus is very special in that we can easily algebraically characterize all its map**s and give systematic examples.
Mmm…donuts.
—Homer Simpson
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Notes
- 1.
We really mean an equivalence class of loops here, but from now on we will often not distinguish the two.
- 2.
This is also called the dilation, stretch factor, expansion constant, or growth.
References
Farb B, Margalit D (2011) A primer on map** class groups. Princeton University Press, Princeton, NJ
Hatcher A (2001) Algebraic topology. Cambridge University Press, Cambridge
Munkres JR (2000) Topology, 2nd edn. Prentice-Hall, Upper Saddle River, NJ
Sturman R, Ottino JM, Wiggins S (2006) The mathematical foundations of mixing: the linked twist map as a paradigm in applications: micro to macro, fluids to solids. Cambridge University Press, Cambridge
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Thiffeault, JL. (2022). Topological Dynamics on the Torus. In: Braids and Dynamics. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-031-04790-9_2
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DOI: https://doi.org/10.1007/978-3-031-04790-9_2
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