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1. Seigneur, Valentin. Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball.

Degree: Docteur es, Mathématiques, 2018, Lyon

Étant donnée une fonction lisse ˜ f définie sur un voisinage de la sphère euclidienne de dimension n dans la boule, peut-on l’étendre en une fonction définie sur la boule bordée par la sphère, de manière à ce que l’extension n’ait aucun point critique ? Cette thèse propose d’étudier cette question, en supposant que la restriction de ˜ f à la sphère, notée f, est Morse. Ce problème a été introduit pour la première fois par Blank et Laudenbach en1970, et a aussi été posé par Arnol’d en 1981. Nous donnons une condition nécessaire d’extension sans points critiques qui s’appuie sur le complexe de Morse de la fonction f, et de la répartition des points critiques de f en deux ensembles : ceux dont la dérivée normale est négative et ceux dont la dérivée normale est positive. Cette condition nécessaire permet alors de donner un cadre algébrique à ce problème venant de la topologie différentielle et s’appuie principalement sur lesgrandes théories de la deuxième moitié du XXème siècle, à savoir celle des cobordismes de Thom,Smale, Milnor etc. Elle permet notamment de donner des conditions nécessaires et suffisantesdans certains cas plus restrictifs, et donne lieu à une condition nécessaire plus faible qui présentel’intérêt d’être calculable.Le point de départ des résultats est celui de Barannikov, qui le premier a traduit le problèmed’extension de fonction avec des conditions de dérivées normales en un problème de chemin defonctions générique qui ne présente pas de singularité globale.

Given a smooth function ˜ f defined on a neighborhood of the euclidian sphere of dimension n in the ball, is it possible to extend it to a function defined on the ball which has no critical points ? This thesis studies this question, assuming the f, the restriction of ˜ f to the sphere, is Morse.This problem was first introduced by Blank and Laudenbach in 1970. We give a necessary condition of extension without critical points that is based on Morsehomology and the repartition of the critical set of f into two sets : the set of points whosenormal derivative to the sphere interior to the ball is negative and the set of points whosenormal derivative is positive. This necessary condition is of algebraic nature and uses great theories of the second half of the XXth century, namely cobordism theory of Thom, Smale,Milnor etc. It also leads to a sufficient condition in some interesting cases, and to a weaker necessary condition for a general function ˜ f which is easily computable.The point-of-view is the one of Barannikov, who was the first to tackle this problem bymeans of considerations about path of functions

Advisors/Committee Members: Ghys, Étienne (thesis director).

Subjects/Keywords: Théorie de Morse; Chemins de fonctions; H-cobordisme; Algèbre linéaire à coefficients entiers; Morse theory; Paths of functions; H-Cobordism; Linear algebra with integral coefficients

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APA (6th Edition):

Seigneur, V. (2018). Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball. (Doctoral Dissertation). Lyon. Retrieved from http://www.theses.fr/2018LYSEN082

Chicago Manual of Style (16th Edition):

Seigneur, Valentin. “Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball.” 2018. Doctoral Dissertation, Lyon. Accessed December 16, 2019. http://www.theses.fr/2018LYSEN082.

MLA Handbook (7th Edition):

Seigneur, Valentin. “Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball.” 2018. Web. 16 Dec 2019.

Vancouver:

Seigneur V. Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball. [Internet] [Doctoral dissertation]. Lyon; 2018. [cited 2019 Dec 16]. Available from: http://www.theses.fr/2018LYSEN082.

Council of Science Editors:

Seigneur V. Extensions de fonctions d'un voisinage de la sphère à la boule : Extensions of functions from a neighborhood of the sphere to the ball. [Doctoral Dissertation]. Lyon; 2018. Available from: http://www.theses.fr/2018LYSEN082

2. Rolland, Jeffrey Joseph. Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds.

Degree: PhD, Mathematics, 2015, University of Wisconsin – Milwaukee

In this dissertation we outline a partial reverse to Quilen's plus construction in the high-dimensional manifold categor. We show that for any orientable manifold N with fundamental group Q and any fintely presented superperfect group S, there is a 1-sided s-cobordism (W, N, N-) with the fundamental group G of N- a semi-direct product of Q by S, that is, with G satisying 1 -> S -> G -> Q -> 1 and actually a semi-direct product. We then use a free product of Thompson's group V with itself to form a superperfect group S and start with an orientable manifold N with fundamental group Z, the integers, and form semi-direct products of (S x S .... x S) with Z and cobordism (W1, N, N-), (W2, N-, N – ), (W3, N – , N – ) and so on and glue these 1-sided s-cobordisms together to form an uncoutable family of 1-ended pseudo-collarable manifolds V all with non-pro-isomorphic fundamental group systems at infinity. Finally, we generalize a result of Guilbault and Tinsley to show that in M is a manifold with hypo-Abelian fundamental group with an element of infinite order, then there is an absolutely inward tame manifold V with boundary M which fails to be pseudo-collaarable. Advisors/Committee Members: Craig Guilbault.

Subjects/Keywords: 1-Sided H-Cobordism; Cobordism; Manifold; Plus Construction; Pseudo-Collar; Mathematics

…decomposition as a sequence of compact cobordisms (W, M, M− ), where W is a semi-h-cobordism… …is a one-sided h-cobordism (a plus cobordism if the homotopy equivalence is simple… …of a semi-h-cobordism, a play on the traditional use of M + for the right-hand boundary of… …if e = h). [A 1-sided e-cobordism (W, N, N− ) is so-named presumably… …cobordism (W, M, M + ) or (W, M + , M ) is a 1-sided h-cobordism if M + ↪ W is… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Rolland, J. J. (2015). Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds. (Doctoral Dissertation). University of Wisconsin – Milwaukee. Retrieved from https://dc.uwm.edu/etd/916

Chicago Manual of Style (16th Edition):

Rolland, Jeffrey Joseph. “Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds.” 2015. Doctoral Dissertation, University of Wisconsin – Milwaukee. Accessed December 16, 2019. https://dc.uwm.edu/etd/916.

MLA Handbook (7th Edition):

Rolland, Jeffrey Joseph. “Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds.” 2015. Web. 16 Dec 2019.

Vancouver:

Rolland JJ. Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds. [Internet] [Doctoral dissertation]. University of Wisconsin – Milwaukee; 2015. [cited 2019 Dec 16]. Available from: https://dc.uwm.edu/etd/916.

Council of Science Editors:

Rolland JJ. Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds. [Doctoral Dissertation]. University of Wisconsin – Milwaukee; 2015. Available from: https://dc.uwm.edu/etd/916

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