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Algebraic topology 2

2018/2019
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
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)

Cohomology groups, universal coefficients theorem. Cohomology ring. Čech cohomology. Orientation on manifolds, duality (Poincare - Lefschetz, Alexander). Künneth theorem, Bockstein homomorphism, transfer, group cohomology.

Homotopy groups, exact sequence of a pair and of a fibration, Whitehead theorem, homotopy excision. Hurewicz theorem. Abstract homotopy theory (H- and coH-spaces, Puppe sequences, spectra).

Readings

A. Hatcher: Algebraic Topology, Ch. 3-4.

W.Massey: A Basic Course in Algebraic Topology, Ch. XiI-XV.

E. Spanier: Algebraic Topology, Ch. 5-7.

Dodatna:

A. Dold: Lectures on Algebraic Topology, Ch. 7-8.

P. May, A Concise Course in Algebraic Topology

J. Munkres: Elements of Algebraic Topology, Ch. 5-8.

R. Switzer: Algebraic Topology – Homotopy and Homology

Objectives and competences

Student learns basic concepts of algebraic topology: homotopy, cellular spaces, homotopy groups and cohomology groups.

Intended learning outcomes

Knowledge and understanding:
Basic concepts and techniques for the computation of homotopy and cohomology 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 toology, 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

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

Lecturer's references

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:
HEGENBARTH, Friedrich, MURANOV, Jurij Vladimirovič, REPOVŠ, Dušan. Browder-Livesay filtrations and the example of Cappell and Shaneson. Milan journal of mathematics, ISSN 1424-9286, 2013, vol. 81, iss. 1, str. 79-97. [COBISS-SI-ID 16619097]
KARIMOV, Umed H., REPOVŠ, Dušan. Examples of cohomology manifolds which are not homologically locally connected. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2008, vol. 155, iss. 11, str. 1169-1174. [COBISS-SI-ID 14678105]
REPOVŠ, Dušan, SKOPENKOV, Mikhail, SPAGGIARI, Fulvia. On the Pontryagin-Steenrod-Wu theorem. Israel journal of mathematics, ISSN 0021-2172, 2005, vol. 145, str. 341-348. [COBISS-SI-ID 13451353]