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Die folgenden Informationen sind noch nicht freigegeben und deshalb unverbindlich: Simple homotopy theory and manifold topology SemesterSoSe 2026 Lecturer Marco Volpe Type of course (Veranstaltungsart) Seminar German title Einfache Homotopietheorie und Topologie von Mannigfaltigkeiten Contents Broadly speaking, simple homotopy theory studies the combinatorial aspects of algebraic topology. More specifically, a map between finite CW complexes is called a simple homotopy equivalence if it can be written as a finite composition of elementary expansions and collapses. A natural question is whether every homotopy equivalence is of this form. In general, the answer is no, and the failure of a homotopy equivalence to be simple is measured by the Whitehead torsion. The torsion takes values in the Whitehead group, a K-theoretic object that depends only on the fundamental group. Simple homotopy theory has been successfully applied in manifold topology. For example, work of Turaev and others shows that two orientable closed 3-manifolds are homeomorphic if and only if they are simple homotopy equivalent. In high dimensions, simple homotopy theory provides a criterion for determining when an h-cobordism is trivial, known as the s-cobordism theorem. In this seminar, we will give an introduction to simple homotopy theory and discuss some of its applications to manifold topology. Time permitting, and depending on the background and interests of the participants, we may also discuss Waldhausen’s more modern approach to the subject. Literature Cohen: A course in simple homotopy theory. Lurie: Algebraic K-theory and manifold topology. More references and a seminar program will appear in the website of the seminar. Recommended previous knowledge Algebraic topology: CW-complexes, homotopy groups, universal covering, singular/cellular homology. Some familiarity with commutative algebra. Some familiarity with differential topology. Based on the prerequisites of the participants, we may have a couple of talks where we review some of the above. Time/Date Wednesday 16-18 Location M009 Additional question session Time/Date: Decided individually with each speaker Location: M228 Course homepage https://sites.google.com/view/marco-volpe/seminars/simple-homotopy-theory-and-manifold-topology (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BSem, MV, MSem, LA-GySem ECTS 4.5 |