Algebraisk Topologi F2003 (10 ECTS)

Praktiske oplysninger fra SIS.

Kursets organisering

Planen (men se nedenfor) er at gennemgå Chapter 1 - 3 i Allen Hatcher: Algebraic Topology. Bogen er udkommet på Cambridge University Press, men den kan også downloades fra forfatterens hjemmeside. Bogen bliver løbende opdateret, se her for updates.

Forudsætningerne er 3GT og 2AL. Kendskab til homologisk algebra er en fordel, men bestemt ikke et krav. Jeg forestiller mig tre modeller for kurset. I Model A gennemgår vi hele bogen, uden at behandle alle detaljer. Det skulle give et overblik over og et generelt indtryk af algebraisk topologi. I Model B starter vi på side 1, gennemgår alt, og stopper når semesteret er slut. Det skulle give mulighed for virkelig at forstå en mindre del af bogen. I Model C bruger vi slet ikke Hatcher men i stedet Part II i Munkres (se hjemmesiden for 3GT). Tænk over hvilken model, du foretrækker.

Her kan du se hjemmesiden for en tidligere version af kurset.

Undervisning på engelsk.

A rough outline of the course:

The fundamental group: 4 weeks
Singular homology: 5 weeks
Singular cohomology and Poincare duality: 6 weeks

You will be asked to hand in your solutions (in Danish, English, German, or French) to the exercises approximately every 5 weeks. I will decide if your solutions are good enough to get credit for this course.

The fundamental group (4 weeks)

Date	Lecture	Topics	Exercises
04.02	1.1 - 1.8	Paths, homotopy of paths, loops, fundamental group, the fundamental group of the circle	1.1.7, 1.1.8 Exercises §1.1 (dvi) (pdf)
06.02	1.8 - 1.16	Applications: Fundamental thm of algebra (Munkres §56), Brouwer's fixed point thm (Munkres §55), Borsuk- Ulam thm (Munkres §57)	1.1.10, 1.1.11, 1.1.13
11.02	1.14, 1.17 - 1.20	The fundamental group of the n-sphere The fundamental group and free homotopies Free coproducts of groups Discussion of exercises	1.2.4, 1.2.5
13.02	1.21 - 1.29	van Kampen (cf. Munkres p. 426) Wedge sum of spaces (cf. Munkres p. 434)	1.2.7, 1.2.14
18.02	1.21 - 1.29	Adjunction spaces (Munkres Ex 35.8). Attachment of 2-cells (cf. Munkres p. 438) Products and quotient spaces (Munkres Ex 29.11) Hawaiian earring group (cf. Munkres p. 436, 500)	Make a model or drawing of the orientable surface of genus 2 as a CW-complex
20.02	1.30 - 1.38	Covering spaces: Unique HLP (cf. Munkres §54) Discussion of exercises	1.3.9 , 1.3.14 (See Example 1.48) Exercises §1.3 (dvi) (pdf)
25.02	1.33 - 1.35	A lifting criterion The universal covering space	1.3.20 See Example 1.42 for the universal covering space action of the Klein Bottle
27.02	1.36 - 1.44	Classification of covering spaces Deck transformations. Notes on Classification of covering spaces (dvi) (pdf)	1.3.18 Hint: Z*Z made abelian is Z x Z There is a picture of the universal abelian covering space of S1 v S1 in the book!

Suggested student seminars

Knot groups (Example 1.24, Exercise 1.2.22)
Classification of compact surfaces (Munkres Chapter 12)
Universal covering space actions for compact surfaces (Example 1.42)
Nielsen-Schreier theorem on subgroups of free groups (Theorem 1A.4)
K(G,1) spaces (Appendix 1B)

Literature

J. Matousek: Using the Borsuk-Ulam theorem. Universitext. Springer 2003.

Singular homology (5 weeks)

Date	Lecture	Topics	Exercises
04.03	2.1 - 2.5	Singular chains and homology	Exercises §2.1 (pdf) 2.1.14
06.03	2.5 - 2.12	Homotopy invariance Notes on homotopy invariance (dvi)(pdf)	2.1.11
11.03	2.13 - 2.17	Relative homology	Exercises 2.1.16
13.03	2.18 - 2.22	Excision	2.1.20
18.03	pp. 102 - 107	$\Delta$ -complexes I am in Oberwolfach	2.1.3, 2.1.4, 2.1.5
20.03	pp. 102 - 107	$\Delta$ -complexes Produce one tex-file with the answers to the 6 exercises!	2.1.7. 2.1.8, 2.1.9
25.03	2.22 - 2.26	Homology of good pairs The equivalence of simplicial and singular homology	2.1.29
27.03	2.28 - 2.33	The degree of a self-map of a sphere CW-complexes	Exercises §2.2 (pdf) 2.2.2
01.04	2.34 - 2.37, 2.42	Cellular homology of CW-complexes Homology of compact surfaces and projective spaces	2.2.11
03.04		The Euler characteristic The Mayer-Vietoris sequence Homology with coefficients Applications of homology	2.2.23, 2.2.41

Suggested student seminars

Simplicial sets
The fundamental group and the first homology group (2.A)
The Jordan curve theorem (2.B)

Singular cohomology (6 weeks)

Participate in the problem sessions Monday 14.15 - 16.00 in Auditorium 8

Date	Lecture	Topics	Exercises
08.04	1.24	Torus Knots (Student seminar by Jesper)
10.04	3.1 - 3.3	Cohomology UCT	Exercises §3.1 (pdf) 3.1.3
15.04	3.3 - 3.5	Singular, cellular, and simplicial cohomology groups	3.1.11 (Cellular maps of CW-complexes induce (co)chain homomorphisms of cellular (co)chain complexes)
17.04		No lecture
22.04		No lecture
24.04	3.6 - 3.11	Mayer-Vietoris Cup product	Submit your homology exercises! 3.2.1 (See Example 3.13; local degree 2.30 is helpful)
29.04	Munkres	The classification of surfaces (Student seminar by Mia)	Identify a concrete surface
01.05		No lecture
06.05		Some cohomology rings	3.2.2 (Lysternik-Schnirelmann category)
08.05	3.12	Cohomology of projective spaces	Exercises §3.2 (pdf) 3.2.3, 3.2.7
13.05	3.14, 3.16	Commutativity of the cup product A Kunneth formula	3.2.6
15.05	3.20	Real division algebras	3.2.14
20.05	3.26	Orientation of manifolds	Submit your cohomology exercises!
22.05		LECTURE STARTS AT 14:15 Last lecture: Poincare duality

Suggested student seminars

The UCT for homology

Tilbage til Jespers hjemmside.

Jesper Michael Møller

Last modified: Fri Oct 29 10:18:25 CEST 2004

Algebraisk Topologi F2003 (10 ECTS)

Kursets organisering

The fundamental group (4 weeks)

Suggested student seminars

Literature

Singular homology (5 weeks)

Suggested student seminars

Singular cohomology (6 weeks)

Suggested student seminars

Links