Introduction to geometric topology

2022/2023
Programme:
Mathematics, First Cycle
Year:
2 year
Semester:
second
Kind:
optional
ECTS:
5
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed course Point-set topology.

Content (Syllabus outline)

Quotient topology, continuous maps on quotients, adjunction spaces. Group actions and orbit spaces. Projective spaces.

Brouwer's fixed point theorem, the Jordan curve theorem, Brouwer's invariance of domain theorem.

Topological manifolds constructions of manifolds. Polyhedral surfaces, Euler characteristic. Classification of closed surfaces.

Simplicial complexes and polyhedra.

Readings

J. Dugundji: Topology, Allyn and Bacon, Boston, 1978.
W. S. Massey: Algebraic Topology: An Introduction, Springer, New York-Heidelberg, 1989.
J. R. Munkres: Topology : A First Course, Prentice Hall, Englewood Cliffs, 1975.

Objectives and competences

Student gets familiar with basic concepts of topology of Euclidian spaces and geometric topology, such as Jordan and Brouwer theorems, simplicial complexes and polihedra and manifolds.

Intended learning outcomes

Knowledge and understanding: Understanding of notions such as quotient topology, basic questions of topology of Euclidian spaces and relations between local and global picture of geometric objects. Knowledge of basic concepts of geometric objects.
Application: In the fields of mathematics, where geometric objects do appear (complex and global analysis, dynamic systems, numerical mathematics, mechanics, graph theory), in computing (graphics, pattern recognition), in physics, chemistry and other natural sciences and engineering.
Reflection: Understanding of the theory from the applications.
Transferable skills: Formulation of the problem in an appropriate language, the abilitiy to solve and analyze the progress on the cases, the transition from local to global properties.

Learning and teaching methods

Lectures, exercises, homework, seminar work, consultations

Assessment

Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Dušan Repovš:
KARIMOV, Umed H., REPOVŠ, Dušan. On generalized 3-manifolds which are not homologically locally connected. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2013, vol. 160, iss. 3, str. 445-449. [COBISS-SI-ID 16558681]
CÁRDENAS, Manuel, LASHERAS, Francisco F., QUINTERO, Antonio, REPOVŠ, Dušan. On manifolds with nonhomogeneous factors. Central European Journal of Mathematics, ISSN 1895-1074, 2012, vol. 10, no. 3, str. 857-862. [COBISS-SI-ID 16241753]
BANAKH, Taras, REPOVŠ, Dušan. Direct limit topologies in the categories of topological groups and of uniform spaces. Tohoku mathematical journal, ISSN 0040-8735, 2012, vol. 64, no. 1, str. 1-24. [COBISS-SI-ID 16215897]
CENCELJ, Matija, REPOVŠ, Dušan. Topologija, (Zbirka Pitagora). 1. ponatis. Ljubljana: Pedagoška fakulteta, 2011. XVI, 169 str., ilustr. ISBN 978-86-7735-051-2. [COBISS-SI-ID 254230528]
Sašo Strle:
OWENS, Brendan, STRLE, Sašo. A characterization of the Z [sup] n [oplus] Z([delta]) lattice and definite nonunimodular intersection forms. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 4, str. 891-913. [COBISS-SI-ID 16408153]
GRIGSBY, J. Elisenda, RUBERMAN, Daniel, STRLE, Sašo. Knot concordance and Heegaard Floer homology invariants in branched covers. Geometry & topology, ISSN 1364-0380, 2008, vol. 12, iss. 4, str. 2249-2275. [COBISS-SI-ID 14892121]
OWENS, Brendan, STRLE, Sašo. A characterisation of the n<1>[oplus]<3> form and applications to rational homology spheres. Mathematical research letters, ISSN 1073-2780, 2006, vol. 13, iss. 2, str. 259-271. [COBISS-SI-ID 13873241]