Algebraic topology 1

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M3
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Homotopy, homotopy equivalence, homotopy category, homotopy extensions and liftings. Cell complexes, cellular maps.

Fundamental group, Seifert - van Kampen theorem. Covering spaces, classification, deck transformations. Group of a knot, free groups, K(G,1) spaces.

Homology groups, homotopy invariance, exact sequences, excision. Degree of a map, cellular homology, Mayer-Vietoris sequence. Winding and linking numbers, index of a vector field, Lefschetz fixed point theorem, perrsistent homology, bordism, Khovanov homology.

Readings

A. Hatcher: Algebraic Topology, Ch. 0-2.

W.Massey: A Basic Course in Algebraic Topology, Ch. I-X.

E. Spanier: Algebraic Topology, Ch. 1-4.

Dodatna:

A. Dold: Lectures on Algebraic Topology, Ch. 1-6.

P. May, A Concise Course in Algebraic Topology

J. Munkres: Elements of Algebraic Topology, Ch. 1-4.

R. Switzer: Algebraic Topology – Homotopy and Homology

Objectives and competences

Student learns basic concepts of algebraic topology: homotopy, cellular spaces, fundamental group, homology groups.

Intended learning outcomes

Knowledge and understanding:
Basic concepts and techniques for the computation of the fundamental group and homology groups. Understanding of the concepts of homotopy invariance and of approaches to geometric problems by algebraic methods.
Application:
Parts of mathematics with strong geometric content (complex and global analysis, geometric and differential topology, graph theory), computer science (computer graphics, pattern recognition, topological data analysis, robotics), theoretical physics.
Reflection:
Understanding of theoretical concepts through examples and applications.
Transferable skills:
Recognition of algebraic structures in geometry, appropriate formulation of problems.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Exercise-based exam / written exam
Theoretical knowledge exam / oral exam

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Janez Mrčun:
– MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0 [COBISS-SI-ID 12683097]
– MOERDIJK, Ieke, MRČUN, Janez. Lie groupoids, sheaves and cohomology. V: EuroSchool PQR2003 on Poisson geometry, deformation quantisation and group representations, Université Libre de Bruxelles, June 13-17, 2003. GUTT, Simone (ur.), RAWNSLEY, John Howard (ur.), STERNHEIMER, Daniel (ur.). Poisson geometry, deformation quantisation and group representations, (London Mathematical Society lecture note series, ISSN 0076-0552, 323). Cambridge [etc.]: Cambridge University Press, cop. 2005, str. 147-272 [COBISS-SI-ID 13657689]
– MRČUN, Janez. Topologija, (Izbrana poglavja iz matematike in računalništva, 44). Ljubljana: DMFA - založništvo, 2008. VI, 147 str., ilustr. ISBN 978-961-212-207-2 [COBISS-SI-ID 243021824]

Petar Pavešić:
– PAVEŠIĆ, Petar CONNER, Gregory R., HERFORT, Wolfgang, PAVEŠIĆ, Petar. Some anomalous examples of lifting spaces. Topology and its Applications, ISSN 0166-8641. [Print ed.], April 2018, vol. 239, str. 234-243.
– PAVEŠIĆ, Petar, A topologist's view of kinematic maps and manipulation complexity. V: GRANT, Mark (ur.). Topological complexity and related topics : Mini-Workshop Topological Complexity and Related Topics, February 28 - March 5, 2016, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany, (Contemporary mathematics, ISSN 0271-4132, 702). Providence: American Mathematical Society.
– PAVEŠIĆ, Petar. Splošna topologija, (Izbrana poglavja iz matematike in računalništva, 43). Ljubljana: DMFA - založništvo, 2008. VI, 89 str., ilustr. ISBN 978-961-212-205-8 [COBISS-SI-ID 240425984]

Dušan Repovš:
– 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]
– 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]
– 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]
– 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:
– STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511 [COBISS-SI-ID 13135193]
– 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]
– OWENS, Brendan, STRLE, Sašo. Rational homology spheres and the four-ball genus of knots. Advances in mathematics, ISSN 0001-8708, 2006, vol. 200, iss. 1, str. 196-216 [COBISS-SI-ID 13875033]