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MU:MU25020 Algebraic Topology II - Course Information

## MU25020 Algebraic Topology II

**Mathematical Institute in Opava**

Winter 2021

The course is not taught in Winter 2021

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- RNDr. Jiřina Jahnová, Ph.D. (lecturer)

RNDr. Jiřina Jahnová, Ph.D. (seminar tutor) **Guaranteed by**- RNDr. Jiřina Jahnová, Ph.D.

Mathematical Institute in Opava **Prerequisites**(in Czech)-
**MU25019**Algebraic Topology I && 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. Main theme of the course in the second term is singular homology and cohomology, cellular homology and cohomology and de Rham's theorem.
**Syllabus**- Complexes of abelian groups, homology, morphisms of complexes, algebraic homotopy of morphisms of complexes.

Simplicial homology. Singular simplices, singular chains, singular (co)homology, homotopy invariance of singular (co)homology.

Long exact (co)homology sequences, barycentric subdivison, excision, Mayer-Vietoris sequence.

The degree of a mapping, methods of its computation.

CW complexes, cellular homology, isomorphism between cellular and singular homology.

De Rham's theorem.

- Complexes of abelian groups, homology, morphisms of complexes, algebraic homotopy of morphisms of complexes.
**Literature**- A. Hatcher.
*Algebraic topology*. Cambridge, 2001. ISBN 0-521-79540-0. info - J. R. Munkres.
*Elements of algebraic topology*. Mento Park, 1984. info

*required literature*- H. Sato.
*Algebraic Topology: An Intuitive Approach*. 1999. info - S. Mac Lane.
*Homology*. Springer, Berlin, 1963. info

*recommended literature*- R. Bott, L. Tu.
*Differential forms in Algebraic Topology*. 1982. info

*not specified*- A. Hatcher.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- The course can also be completed outside the examination period.
**Teacher's information**- The final exam consists of the presentation of solutions to 2 homework problems assigned to the student in advance and of answering correctly 2 theoretical questions (one in homology theory, the other in cohomology theory). To obtain the course credits it is neccessary to actively participate in the seminar and solve homework problems assigned by the instructor.

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