Algebraic topology

Knowledge of basic concepts of general topology (topological space, continuous map, compactness, connectedness) and algebra (groups, Abelian groups).

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

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)

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)

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

Lecture accompanied by exercise classes

- wykład informacyjny (konwencjonalny)

- ćwiczeniowa

The basic concepts of algebraic topology: homotopy groups and homology groups and their fundamental properties are introduced. Some applications are given.

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.

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.

Oral examination : W1-W5, U4, K1.

Assessment of exercise classes: U1-U3.