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Hinweis Bitte informieren Sie sich auf den jeweiligen GRIPS-Seiten über den digitalen Ablauf der Lehrveranstaltungen.  Note For our digital courses all relevant information can be found on the appropriate GRIPS sites.Algebraic K-theory and the Wall finiteness obstruction SemesterSoSe 2021 Lecturer Christoph Winges Type of course (Veranstaltungsart) Vorlesung Contents This course covers some applications of algebraic K-theory (in particular the class group K_0) in geometric/algebraic topology. We will primarily cover the Wall finiteness obstruction which is a K-theoretic invariant designed to detect whether certain topological spaces are homotopy equivalent to finite CW-complexes. After discussing the fundamentals of the finiteness obstruction, we will develop some K-theoretic machinery to give a proof of West's theorem. As an application of West's theorem, we will see that every compact topological manifold has the homotopy type of a finite CW-complex. Literature C.T.C. Wall. Finiteness Conditions for CW-Complexes. Ann. Math., 2nd series, 81, no. 1 (1965), 56-69. C.T.C. Wall. Finiteness Conditions for CW-Complexes II. Proc. Roy. Soc. London, Ser. A, 295, no. 1441 (1966) 129-139. Friedhelm Waldhausen. Algebraic K-theory of spaces. Algebraic and geometric topology, edited by A. Ranicki et al., LNM 1126 (1985), 318-419. James E. West. Mapping Hilbert Cube Manifolds to ANR's: A Solution of a Conjecture of Borsuk. Ann. Math., 2nd series, 106, no. 1 (1977), 1-18. W. Dwyer, M. Weiss, B. Williams. A parametrized index theorem for the algebraic K-theory Euler class. Acta Math. 190 (2003), 1-104. Recommended previous knowledge basic algebraic topology (it should be possible to attend this lecture concurrently with Algebraic Topology II), some category theory; some acquaintance with simplicial homotopy theory will be helpful for the later parts of the lecture, but the necessary material can also be covered in the lecture as we get to that point Time/Date Fri 10-12 Location Zoom Registration
BV, MV, MGAGeo ECTS 4,5 |