Topology II

Lecturer:     Klaus Mohnke
                           office: Adlershof, Haus 1,  room 306
                           phone: (030) 2093 1814
                           fax: (030) 2093 2727
                           email: mohnke at address of institute

 
Lectures:  mondays 1-3 p.m., RUD 25, 1.115, wednesdays 11-1 p.m, RUD 25, 1.013

Tutorial:  mondays 3-5 p.m., RUD 25, 2.006
 
Office hours: wednesdays  2-4 p.m., RUD 25, 1.306 (my office)


News:



Homework

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we will largely follow:
(1) Hatcher: Algebraic Topology, http://www.math.cornell.edu/~hatcher

but also make use of :
(2) James W. Vick: Homology Theory, Springer, Graduate Texts in Mathematics

additionally literature (not a complete list)
(3) Greenberg, Harper: Algebraic Topology: A first course, Addison-Wesley
(4) Stöcker, Zieschang: Algebraische Topologie, Teubner
(5) Spanier: Algebraic Topology, Springer
(6) Munkres: Elements of Algebraic Topology, Addison-Wesley
(7) Fomenko et.al.: Homotopic topology, Akad. Kiadó

Subjects covered in class and in the tutorial:
 

                     - (singular) chain complex,                   
                     - singular homology
                     - funtoriality
                     - chain homotopy
                     - relative singular homology, long exact sequences
                     - excision and quotients of good pairs
                     - Mayer-Vietoris-Sequence
                     - singular homology of   spheres, Fixed point theorem  of Brouwer
                     - splitting short exact sequences, retracts and deformation retracts
                     - CW-complexes und cellular homology  (starting November 21)
                     - singular homology of projektive spaces

                     - cochains with coefficients
                     - singular cohomology, interpretation of cocycles and coboundaries
                     - relative singular cohomology, interpretation of relative cocycles and coboundarie
                     - functoriality, cochain homotopy, long exact sequences, excision (analogies to singular homology)
                     - cellular cohomology,
                     - universal coefficient theorem, Ext, Examples
                   
                     - cup-product, simple examples (spheres
                     - cup-product on relative cohomologies, functoriality
                     - cross-product, tori, cubes relative their boundaries
                     - projective spaces
                     - graded commutativity of cup-product  16.1. Hatcher pp. 215-217
                     - Künneth-formulas for cohomology of products  23.1.  Hatcher pp.218-223
                   
                    -  orientation of (topological) manifolds, fundamental classes  30.1.   Hatcher pp. 233-239
                    -  cap-product, Poincaré-duality   2.2.   Hatcher pp. 239-244
                    - cohomology with compact support, duality for non-compact manifolds  6.2.  Hatcher pp. 244-249
                    - cup and cap, examples  6.2.  Hatcher pp. 249-252
                    - Alexander duality  ???



Klaus Mohnke
Fr, February 10,  2017, 15:40