Algebraic Topology (lecture, 4+2)
- Beginning:
- 29 October 1997
- End:
- 21 February 1998
- Contents:
- I. CW Complexes
- Attaching Cells
- CW Complexes
- First Properties of CW Complexes
- Construction of CW Complexes
- Cellular Approximation
- II. Singular Homology
- Free Abelian Groups
- The Singular Homology Functor
- First Properties of Homology
- Homological Algebra (Long Exact Sequences)
- Exactness Axiom
- Chain Homotopy
- Method of Acyclic Models
- Homotopy Axiom
- Excision
- The Mayer-Vietoris Sequence
- Applications
- Eilenberg-Steenrod Axioms
- Cellular Homology
- Betti Numbers and Euler Characteristic
- Tensor Product of Groups and Complexes
- The Eilenberg-Zilber Theorem
- Exactness of Tensor Products
- Algebraic Künneth Theorem
- Topological Künneth Theorem, Universal Coefficients
- III. Cohomology, Products and Duality
- The Fuctor Hom
- Singular Cohomology
- The Functor Ext
- The Universal Coefficient Theorem for Cohomology
- The Künneth Theorem in Cohomology
- Products (cross, cup, slant, cap)
- Relative Versions of the Products
- Direct Limits
- Orientation
- Poincaré Duality
- Applications
- Alexander Duality
- The Lefschetz Fixed Point Theorem
- De Rham Cohomology
- Prerequisites:
- Basic notions from general topology; notions from algebra like
group, ring and module; The knowledge of
Topologie I
will help, but is not necessary.
- Recommendable Literature:
- M.J. Greenberg:
Lectures on Algebraic Topology, 1967/77
- W.S. Massey:
A Basic Course in Algebraic Topology, GTM 127, 1991
- J.J. Rotman:
An Introduction to Algebraic Topology; GTM 119, 1988/92
- E.H. Spanier:
Algebraic Topology, 1966
- R. Stöcker, H. Zieschang:
Algebraische Topologie, Teubner, 1988
- Exampleclasses:
- Were given by
Christoph Lossen
- This lecture will be given in the english language, because it
belongs to the programme
Mathematics International