#
MU:MU25019 Algebraic Topology I - Course Information

## MU25019 Algebraic Topology I

**Mathematical Institute in Opava**

Winter 2020

The course is not taught in Winter 2020

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- doc. RNDr. Michal Marvan, CSc. (lecturer)

doc. RNDr. Michal Marvan, CSc. (seminar tutor) **Guaranteed by**- doc. RNDr. Michal Marvan, CSc.

Mathematical Institute in Opava **Prerequisites**(in Czech)- TYP_STUDIA ( N )
**Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Geometry and Global Analysis (programme MU, NMgr-M)

**Course objectives**- Algebraic topology studies topological spaces by algebraic means. Among its typical problems is the that of finding whether a topological space can be continuously mapped onto another. A positive answer may be obtained by constructing such a map but a negative answer is more difficult to get. In this two-term course on algebraic topology, algebraic methods to solve such problems will be explained. During the first term, foundations of the homotopy theory are taught.
**Syllabus**- Categories, functors, category Top, Gr a Ab; products and sums, pull-back and push-out.

Homotopy of continuous mappings, relative homotopy; homotopical equivalence of topological spaces, contractibility.

Category Top_h, functors in algebraic topology, elementary problems of algebraic topology, homotopy extension property, Borsuk pairs.

Paths and loops, fundamental group, simply-connected spaces.

Covering spaces, covering path theorem, covering homotopy theorem, fundamental group, covering mapping theorem

Methods of calculation of homotopy groups, G-spaces, fundamental group of the orbit space; Seifert-Van Kampen theorem.

Superior homotopic groups, exact sequence of the homotopic groups.

- Categories, functors, category Top, Gr a Ab; products and sums, pull-back and push-out.
**Literature**- C. Kosniowski.
*A First Course in Algebraic Topology*. info - W. Fulton.
*Algebraic topology : a first course*. New York, 1995. ISBN 0-387-94327-7. info

*required literature*- A. Hatcher.
*Algebraic topology*. Cambridge, 2001. ISBN 0-521-79540-0. info - B. Gray.
*Homotopy theory : an introduction to algebraic topology*. New York, 1975. ISBN 0-12-296050-5. info

*recommended literature*- R.M. Switzer.
*Algebraic topology -- homotopy and homology*. NewYork, 1975. ISBN 0-387-06758-2. info

*not specified*- C. Kosniowski.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- The course can also be completed outside the examination period.
**Teacher's information**- To pass the course, the first step is to earn credit for tutorials by earning 70% on a written test. Then the student is allowed to attempt the final exam, which comprises a written part (4 problems to solve) and an oral part. The oral part tests the theoretical knowledge and understanding of the subject

- Permalink: https://is.slu.cz/course/sumu/winter2020/MU25019