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| MATH0465-1 | Algebraic Topology
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| Duration : | 30h Th, 10h Pr, 20h Mon. WS |
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| Credits/ECTS : |
| Master in Mathematical Sciences, in-depth approach, 1st year | | First semester | | 8 |
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| Master in Mathematical Sciences, didactic approach, 1st year | | First semester | | 8 |
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| Master in Mathematical Sciences, professional focus in management, 1st year | | First semester | | 8 |
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| Master in Mathematical Sciences, professional focus in computer science, 1st year | | First semester | | 8 |
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| Master in Mathematical Sciences, specialized approach, 1st year | | First semester | | 8 |
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| Master in Mathematical Sciences | | First semester | | 8 |
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| Holder(s) : | Jean‑Pierre Schneiders |
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| Language : | French language |
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| Course contents : | This course constitutes an introduction to algebraic topology and homological algebra.
One starts by studying the singular homology of topological spaces in general and by treating explicitly a few significant examples (sphere, torus, sphere with handles, projective spaces, ...). To illustrate the usefulness of the preceding theory, we conclude this first part by establishing Jordan's theorem in arbitrary dimension.
In the second part, the study of product spaces leads us to define and study tensor products of complexes and to introduce the "Tor" functors. We conclude this part with Künneth's theorem.
The third part of the course is devoted to the duality between cohomology and homology and motivates the introduction and study of the "Ext" functors.
The course ends with a brief study of the homology and cohomology of topological manifolds and of Poincaré duality. |
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| Course objective : | Give the students an idea of what is algebraic topology and show how its study leads naturally to homological algebra and derived functors. Give a simple starting point for the more abstract developments of the algebraic analysis course. |
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| Prerequisites : | A good knowledge of basic algebra and topology is essential. |
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| Organization : | The course follows the official schedule given to the students at the beginning of the academic year. |
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| Written notes : | Lecture notes will be available. |
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| Assessment : | To be fixed with the students. |
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| Contacts : | Jean-Pierre Schneiders
Département de Mathématique (Bât. B37, Bureau 1/60)
Grande Traverse 12 - 4000 Liège (Sart-Tilman)
Tél. : (04) 366.94.01 - E-Mail : jpschneiders@ulg.ac.be
Web page : http://www.analg.ulg.ac.be/jps/ |
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