Algebraic topology 1000-OG-EN-TA

Lecture:
I. Introduction.
1. Elements of general topology.
2. Basic facts on groups, rings and modules.
3. Language of categories and functors, diagrams
II. Homotopy theory
1. Homotopy of paths and maps.
2. The fundamental group.
3. Coverings, lifting properties.
4. Higher homotopy groups
III. Homology
1. Singular homology groups, relative homologies.
2. Exact sequence of homology groups.
3. Mayer-Vietoris sequence.
4. Axiomatic characterization of homology.
IV. Some applications: Euler characteristic, Brouwer fixed-point theorem.
Exercises: Calculating homotopy and homology groups in examples; completing details of some proofs arranged as independent problems.

Całkowity nakład pracy studenta

Lecture : 30 h. Exercise classes: 30 h. Self-study: reading reference materials, completion of notes:30 h. Self-study: solving problems: 45 h. Preparation to the exam -15 h Total: 150 h. – 6 ECTS

Efekty uczenia się - wiedza

W1: Knows the concept of homotopy, fundamental group and higher homotopy groups. (K_W01, K_W03) W2: Knows the role of covering spaces in determining the fundamental group. . (K_W02, K_W03) W3: Understands the construction of singular homology groups, their basic properties, in particular exact sequence of homology groups. . (K_W02, K_W03) W4: Is aware of axiomatic approach to homology theory. (K_W03) W5: Understands the functorial nature of the constructions of homology (homotopy) groups. (K_W03) W5: Knows selected applications of methods of algebraic topology.and the functoriality of this construction.(K_W04)

Efekty uczenia się - umiejętności

U1: Knows how to calculate fundamental groups in selected examples. (K_U01, K_U06, K_U10) U2: Determines homology groups of selected spaces, in particular using long exact sequence of homologies. .(K_U01, K_U06, K_U10) U3: Applies algebraic topology concepts to prove selected facts from topology and geometry. .(K_U01, K_U06) U4: Uses the terminology of algebraic topology and related branches of mathematics.(K_U10)

Efekty uczenia się - kompetencje społeczne

K1: Is able to evaluate his/her own knowledge and skills and to develop it independently. (K_K03)

Metody dydaktyczne

Lecture accompanied by exercise classes

Metody dydaktyczne podające

- wykład informacyjny (konwencjonalny)

Metody dydaktyczne poszukujące

- ćwiczeniowa

Wymagania wstępne

Knowedge of basic concepts of general topology and algebra (groups).

Koordynatorzy przedmiotu

Stanisław Kasjan

Kryteria oceniania

Oral examination : W1-W5, U4, K1.
Assessment of exercise classes: U1-U3.

Literatura

Recommended reading:
1. M. J. Greenberg, Wykłady z topologii algebraicznej, PWN, 1980. (oryg. ang. Lectures on Algebraic Topology)
2. E. H. Spanier, Algebraic Topology, Springer, 1966.
Supplementary reading:
3. A. Dold, ,Lectures on algebraic topology. Springer, 1995.
4. S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton, 1952.
5. R. Engelking, K. Sieklucki, Topology. A Geometric Approach, Heldermann Verlag, 1992.
6. S. MacLane, Categories for the Working Mathematician, Springer, 1978.
7. K. Sieklucki, Algebraic topology, IM PAN, 1968.

Więcej informacji

Dodatkowe informacje (np. o kalendarzu rejestracji, prowadzących zajęcia, lokalizacji i terminach zajęć) mogą być dostępne w serwisie USOSweb:

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