Announcements
To book a time for the oral exam during the period March 16 - April 2 follow this link and make one choice. If you want to take the oral outside this time span, send an email to the examiner.
Notes products are now available under the heading Supplements in the menue to the left.
The final assignment is now available under the heading Examination in the menue to the left. Hand it in in connection with the oral exam.
Coursebook
Allen
Hatcher:
Algebraic Topology.The Section heading in the
program referes to this book.
The book is available for
free from Allen
Hatcher's
web page. Note the links to corrections and additional
exercises.
Program
Lectures
Treated topics will be marked green .| Lec # |
Date |
Time |
Room |
Topics | Sections |
|---|---|---|---|---|---|
| 1 | Jan 20 | 10.00 am - Noon | MVL14 | Categories and functors. Pushouts. $\Delta$-setsand $\Delta$-complexes and their homology. | 2.1, 2.3. |
| 2 | Jan 23 | 3.00 - 5.00 pm | MVL14 | Chain complexes, Singular homology, 0th and 1st homology groups. Hurewicz Theorem. | 2.1, 2.A. |
| 3 | Jan 28 | 3.15 - 5.15 pm | MVH12 | Homotopy invariance of homology. The homology cross product. Eilenberg-Zilber maps. Relative homology groups and long exact sequences. | 2.1,3.B. |
| 4 | Jan 30 | 3.00 - 5.00 pm | MVL14 | Excision and the locallity principle. Mayer-Vietoris sequences. | 2.1, 2.2. |
| 5 | Feb 4 | 3.00 - 5.00 pm | MVL15 | Classical applications of homology. | 2.B. |
| 6 | Feb 6 | 3.00 - 5.00 pm | MVL14 | Replacing relative homology groups by reduced. Explicit generators in $H_n(S^n)$ and $H_n(D^n,S^{n-1}).$ Axioms for homology. | 2.1, 2.3. |
| 7 | Feb 11 | 3.00 - 5.00 pm | MVL15 | Degree of $f:S^n\to S^n,\,n>0.$ Local calculation of degree. CW-complexes. | 2.2, Appendix, 0. |
| 8 | Feb 13 | 3.00 - 5.00 pm | MVL14 | Homology of CW-complexes. Euler characteristic. | 2.2. |
| 9 | Feb 18 | 3.00 - 5.00 pm | MVL15 | Simplicial complexes and simplicial approximation. Lefschetz Fixed Point Theorem. | 2.C. |
| 10 | Feb 20 | 3.00 - 5.00 pm | MVL14 | The Universal Coefficient Theorem and the Künneth Formula. | 3.A, 3.B. |
| 11 | Feb 25 | 3.00 - 5.00 pm | MVL15 | Cohomology groups. The Universal Coefficient Theorem for cohomology. | 3.1. |
| 12 | Feb 27 | 3.00 - 5.00 pm | MVL14 | Cup and cross products in cohomology. The Alexander-Whitney map. Künneth Formulas for cohomology. | 3.2. |
| 13 | Mar 3 | 3.00 - 5.00 pm | MVL15 | Cup and cross products in cohomology. The Alexander-Whitney map. Künneth Formulas for cohomology. | 3.2. |
| 13 | Mar 3 | 3.00 - 5.00 pm | MVL15 | Orientation of manifolds in homology. Poincaré duality. | 3.3. |
| 14 | Mar 5 | 3.00 - 5.00 pm | MVL14 | Poincaré duality (continued). | 3.3. |
| 15 | Mar 10 | 3.00 - 5.00 pm | MVL15 | Comparing axiomatic homology and cohomology therories. The telescope construction. | 2.3, 3.1, 3.F. |
| 16 | Mar 12 | 3.00 - 5.00 pm | MVL14 | Clearing up omissions. |
Examination
The course will be examined by assignments during the course and by a final oral exam. The assignments will consist of a choice of excercises in the coursebook.
Assignments
| Assignment |
Exercises |
Due |
|---|---|---|
| 1 | 2.1: 6, 8, 12, 17, 20, 27, 31. | Thursday Feb 13. |
| 2 | 2.2: 1, 3, 4, 12, 13 for (b) use that homotopic attaching maps result in homotopy equivalent spaces, 20, 21, 25, 33 , 36. | Tuesday March 3. |
| 3 | 3.1: 3, 5, 11, 3.2: 4, 7, 11, 3.A: 2, 3. | In connection with oral exam. |
Instructions for oral examination
The oral exam is arranged as follows. The student is assigned,
by chance, one of the topics in the list below and gets one
hour to prepare a short exposition (about 30 minutes) for the
examiner. This will be followed by a discussion of the
subject.
Note: to take the oral exam you need to have passed the
assignments.
To book a time for the oral exam during the period March 16
- April 2 follow
this
link and make one choice.
If you want to take the oral outside this time span, send an
email to the examiner.
Account for
- the definition of (relative) singular homology and its homotpy invariance,
- short exact sequences of chain complexes leading to long exact sequences and sketch of excision,
- CW-complexes and cellular homology,
- Degree of self-maps of spheres
- The Leftschetz number and the Lefschetz theorem
- Mayer-Vietoris sequences
- the definition of cohomology and its basic properties,
- the universal coefficient theorem and Ext-groups,
- the (simple) Künneth formula.
Supplementary notes
- Categories and functors.
- Delta sets and colimits.
- Geometric realisation of $\Delta$-sets and adjunctions.
- The homologoy cross product and homotopy invariance..
- The Hurewicz Theorem.
- Simplicial approximation.
- CW-complexes.
- The Künneth Formula and Acyclic Models.
- Products.
Supplementary notes will appear here.