Seminar on selected topics in algebraic and differential topology

Marc Kegel

Wintersemester 2020/2021



Wednesdays, 15:00-16:30, online via Zoom

Veranstaltungsnummer: 3314520

Office hours / further discussion: after the lectures/exercise sessions via Zoom

Zoom meeting info: click here for Zoom meeting info

Mailling list: If you would like to receive information about this course by email via a mailing list, please send me a short email so I can add you to the mailing list.



Content: This seminar is intended for students with advanced knowledge of algebraic topology within the lectures Topology I and II. In this seminar we want to discuss some additional topics in algebraic and differential topology which have not been covered in the lectures.

There are still one free slot available. If you are interested in giving a talk, please contact me by e-mail.

To receive study points for this seminar, active participation in the seminar is absolutely necessary. In addition, an informative presentation (no longer than 90 minutes) must be given and clearly written notes must be submitted no later than one week after the presentation.

Announcements:



Talks:
04.11.20 Preliminary meeting
notes (pdf)
11.11.20 no seminar
18.11.20 De Rham cohomology and de Rham's theorem [Le, Chapter 11]
notes (pdf)
David Suchodoll
25.11.20 Obstruction theory and homotopy characterization of cohomology [FF, Lecture 18]
notes (pdf)
Tatiana Levinson
2.12.20 Characteristic classes [FF, Lecture 19]
notes (pdf)
Léo Mousseau
9.12.20 Homology of fibrations and spectral sequences [FF, Chapter 3; Ch]
notes (pdf)
Theo Müller
16.12.20 Whitneys embedding theorem [GP; Hi]
notes (pdf)
Max Huneshagen
23.12.20 no seminar
6.1.21 no seminar
13.1.21 Freudenthal suspension theorem [FF, Lecture 10]
notes (pdf)
Joshua Phillip Egger
20.1.21 Morse theory [Ma, Chapter 1]
notes (pdf)
Christoph Krpoun
27.1.21 Orbifolds
notes (pdf)
Adrian Dawid
3.2.21 Open book decompositions and fibered knots [Et; Ro; Gi; Ra, Appendix]
notes (pdf)
Milica Ðukić
10.2.21 The theorem of Hopf and the Pontrjagin construction [Mi]
notes (pdf)
Alberto Richtsfeld
17.2.21 Smale's theorem
Marc Kegel


There are still some free slots available. A list of further possible topics (I am very open to further suggestions) is:
The linking form and classifications of lens spaces [F]
Seifert fibered spaces
Čech homology and cohomology [W]
The h-cobordism theorem and the higher dimensional Poincare conjecture
Tubular neighborhood theorem [BJ]
Isotopy extension theorem [BJ]
The disk theorem [BJ]
Further topics in knot theory and gemeotric topology of 3-manifolds
Exotic 7-spheres
Existence of PL structures on smooth manifolds
Existence of smooth structures on topological manifolds



Literature:
[BJ] T. Bröcker und K. Jänich, Einführung in die Differentialtopologie, Springer, 1973.
[Ch] T. Chow, You could have invented spectral sequences, Notices Amer. Math. Soc. 53, (1973), 15-19.
[Et] J. Etnyre, Lectures on open book decompositions and contact structures, in: Floer homology, gauge theory, and low-dimensional topology, 2006.
[F] S. Friedl, Lecture notes for Algebraic Topology, available online at his homepage.
[FF] A. Fomenko and D. Fuchs, Homotopical topology, Springer, 2016.
[Gi] E. Giroux, What is... an open book?, Notices Amer. Math. Soc. 52, (2005).
[GP] V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, 1974.
[Hi] M. Hirsch, Differential topology, Springer, 1976.
[Le] J. Lee, Introduction to Smooth Manifolds, Springer, 2003.
[Ma] Y. Matsumoto, An introduction to Morse theory, American Mathematical Society, 2002.
[Mi] J. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965.
[Ra] A. Ranicki, High-Dimensional Knot Theory - Algebraic Surgery in Codimension 2, Springer, 1998.
[Ro] D. Rolfsen, Knots and Links, Publish or Perish, 1976.
[W] C. Wendl, Lecture notes for Topologie I and II, available online at his homepage.



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