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Simple homotopy theory and Whitehead groups SemesterSoSe 2019 Lecturer Georgios Raptis Type of course (Veranstaltungsart) Seminar Contents Algebraic topology is generally concerned with the notion of homotopy equivalence. This Seminar will be about a combinatorial approach to the definition of homotopy equivalence for finite CW complexes. We will ask the following fundamental question: is there is a collection of elementary geometrically defined homotopy equivalences from which every homotopy equivalence is generated? The definition of simple homotopy equivalence is based on such elementary homotopy equivalences, which act as building blocks or series of moves. The question then becomes: is every homotopy equivalence between finite CW complexes simple? The answer turns out to be "no", but in a very interesting way. There is a single algebraic invariant, called the Whitehead torsion, which decides whether a homotopy equivalence is simple. The Whitehead torsion of a homotopy equivalence is an element of an abelian group, called the Whitehead group, which depends only on the fundamental group. In this Seminar, we will first introduce and study simple homotopy equivalences, then we will define Whitehead groups and discuss their properties, and finally, we will prove the relationship between the two and answer the questions stated above. This answer is a beautiful (and rare) instance where topology and algebra match up exactly. The Seminar should be of interest to those interested in algebraic topology. The topic can also serve as an introduction to algebraic K-theory from a topological viewpoint. Depending on the interest in the topic, there may be a continuation of the Seminar next semester about geometric applications of simple homotopy theory and connections with differential topology (e.g. the s-cobordism theorem). The seminar talks will be in English or German. Literature The main reference for this Seminar is the book: M. M. Cohen "A Course in Simple-Homotopy Theory" Recommended previous knowledge Basic algebraic topology (covering spaces, CW complexes, singular and cellular homology) and some general familiarity with commutative algebra. Time/Date Dienstag 10-12 Location M009 Course homepage Details about the Seminar (e.g. schedule, notes, etc.) will appear on GRIPS and here http://graptismath.net/teaching.html (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
Mit Repetitorium, jeweils in der Woche vor den Vorträgen. Termin verhandelbar. Modules BSem, MV, MSem, LA-GySem ECTS Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn vor WS 15/16 |