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
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.
Literature
    required 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
    recommended 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
    not specified
  • R.M. Switzer. Algebraic topology -- homotopy and homology. NewYork, 1975. ISBN 0-387-06758-2. info
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

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