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000124757 001__ 124757 000124757 005__ 20190316234301.0 000124757 0247_ $$2doi$$a10.5075/epfl-thesis-4132 000124757 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4132-1 000124757 02471 $$2nebis$$a5573587 000124757 037__ $$aTHESIS 000124757 041__ $$aeng 000124757 088__ $$a4132 000124757 245__ $$aThe link between the infinite mapping class group of the disk and the braid group on infinitely many strands 000124757 269__ $$a2008 000124757 260__ $$aLausanne$$bEPFL$$c2008 000124757 300__ $$a119 000124757 336__ $$aTheses 000124757 520__ $$aφ For all finite n ∈ N, there is a well-known isomorphism between the standard braid group Bn and the mapping class group π0Hn. This isomorphism has been exhaustively studied in literature, and generalized in many ways. For some basic topological reason, this strong link between finite braid groups and finite mapping class groups can-not be extended to the infinite case in a straightforward way, and, in particular, is not yet well studied in literature. In our work, we define the infinite braid group B∞ to be the group of braids with infinitely many strands, all of which can be possibly nontrivial, i.e., not straight. In particular, this definition does not correspond to the group of finitary infinite braids, which is just the union of all finite braid groups. Similar to the maps π0φn for finite n, we introduce a map that, in particular, turns out not to be an isomorphism. However, we prove its injectivity, and identify its image in B∞. The study of the link between mapping class groups and braid groups in the infinite case is motivated by the study of homeomorphisms in H∞ that give rise to a homoclinic tangle. In fact, the map π0φ∞ attributes to each isotopy class of such a homeomorphism an element of the infinite braid group B∞, and so, allows us to describe the isotopy classes of these homeomorphisms in terms of their image in B∞. Using the fact that the map π0φ∞ is injective, we prove a result that can be applied to the study of the topological structure of homoclinic tangles. 000124757 6531_ $$ainfinite braid group 000124757 6531_ $$ainfinite mapping class group 000124757 6531_ $$ainfinite permutation group 000124757 6531_ $$ahomoclinic tangles 000124757 6531_ $$agroupe de tresses infini 000124757 6531_ $$amapping class group infini 000124757 6531_ $$agroupe de permutations infinies 000124757 6531_ $$aenchevêtrements homoclines 000124757 700__ $$aBrunner, Jan 000124757 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396 000124757 8564_ $$s1245918$$uhttps://infoscience.epfl.ch/record/124757/files/EPFL_TH4132.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text 000124757 909C0 $$0252139$$pUPHESS$$xU10968 000124757 909CO $$ooai:infoscience.tind.io:124757$$pthesis$$pthesis-bn2018$$pDOI$$pSV$$qDOI2$$qGLOBAL_SET 000124757 919__ $$aGR-HE 000124757 918__ $$aSB$$cIGAT$$dEDMA 000124757 920__ $$b2008 000124757 973__ $$aEPFL$$sPUBLISHED 000124757 970__ $$a4132/THESES 000124757 980__ $$aTHESIS