Algebraic topology 1

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.

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

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

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.

Lectures, exercises, homeworks, consultations

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

Petar Pavešić:
PAVEŠIĆ, Petar. The Hopf invariant one problem, (Podiplomski seminar iz matematike, 23). Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, 1995. 65 str. ISBN 961-212-050-1. [COBISS-SI-ID 53969664]
PAVEŠIĆ, Petar. Reducibility of self-homotopy equivalences. Proceedings. Section A, Mathematics, ISSN 0308-2105, 2007, vol. 137, iss 2, str. 389-413. [COBISS-SI-ID 14371929]
PAVEŠIĆ, Petar, PICCININI, Renzo A. Fibrations and their classification, (Research and exposition in mathematics, vol. 33). Lemgo: Heldermann, cop. 2013. XIII, 158 str., ilustr. ISBN 978-3-88538-233-1. [COBISS-SI-ID 16616793]
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]
Sašo Strle:
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]
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]
Dušan Repovš:
KARIMOV, Umed H., REPOVŠ, Dušan. On the homology of the Harmonic Archipelago. Central European Journal of Mathematics, ISSN 1895-1074, 2012, vol. 10, no. 3, str. 863-872. [COBISS-SI-ID 16242009]
KARIMOV, Umed H., REPOVŠ, Dušan. On noncontractible compacta with trivial homology and homotopy groups. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2010, vol. 138, no. 4, str. 1525-1531. [COBISS-SI-ID 15382873]
HEGENBARTH, Friedrich, REPOVŠ, Dušan. Applications of controlled surgery in dimension 4: examples. Journal of the Mathematical Society of Japan, ISSN 0025-5645, 2006, vol. 58, no. 4, str. 1151-1162. [COBISS-SI-ID 14120537]