Topics in topology

2022/2023
Programme:
Doctoral Programme Mathematics and Physics
Orientation:
Mathematics
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
0
Lab
0
Content (Syllabus outline)

The content consists of a selection of standard graduate topics in topology, such as general homotopy theory, obstruction theory, the theory of fibre bundles. K-theory, the theory of Lie groupoids, the theory of spectral sequences, Morse theory, knot theory etc. The choice depends on students' research interests.

Readings

[1] G. Burde, H. Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co., Berlin, 2003.
[2] R. E. Gompf, A. I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, AMS, Providence, 1999.
[3] P. Hilton, G. Mislin, J. Roitberg, Localization of nilpotent groups and spaces , Elsevier, Amsterdam 1975.
[4] D. Husemoller, Fibre bundles, Springer, New York, 1994.[5] H. B. Lawson, M. L. Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, 1989.
[6] J. McCleary, A user’s guide to spectral sequences , Cambridge University Press, Cambridge, 2001.
[7] J. Milnor, Morse theory, Annals of Mathematics Studies 51, Princeton University Press, Princeton, 1963.
[8] C. P. Rourke, B. J. Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982.[9] E. Spanier, Algebraic topology, Springer, New York - Heidelberg - Berlin, 1966.
[10] G. W. Whitehead, Elements of homotopy theory, Springer, New York - Heidelberg - Berlin, 1978.

Objectives and competences

The main goal of the course is to provide students with some important topics in topology.

Intended learning outcomes

Knowledge and comprehension of presented concepts.
Ability to use acquired knowledge and skills.

Learning and teaching methods

Lectures, consultations, problem sessions

Assessment

Writen exam (homeworks), oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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]
KALIŠNIK, Jure, MRČUN, Janez. A Cartier-Gabriel-Kostant structure theorem for Hopf algebroids. Advances in mathematics, ISSN 0001-8708, 2013, vol. 232, iss. 1, str. 295-310. [COBISS-SI-ID 16432473]
JELENC, Blaž, MRČUN, Janez. Homotopy sequence of a topological groupoid with a basegroup and an obstruction to presentability of proper regular Lie groupoids. ArXiv.org, 13 str. [COBISS-SI-ID 16400729]
FRANC, Aleksandra, PAVEŠIĆ, Petar. Spaces with high topological complexity. Proceedings. Section A, Mathematics, ISSN 0308-2105, 2014, vol. 144, iss. 4, str. 761-773. [COBISS-SI-ID 17096025]
PAVEŠIĆ, Petar. Fibrations between mapping spaces. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2014, vol. 178, str. 276-287. [COBISS-SI-ID 17141337]
PAVEŠIĆ, Petar. Induced liftings, exchange rings and semi-perfect algebras. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2010, vol. 214, iss 11, str. 1901-1906. [COBISS-SI-ID 15627865]
REPOVŠ, Dušan. A two-parameter control for contractive-like multivalued mappings. V: 2010 International Conference on Topology and its Applications, June 26-30, 2010, Nafpaktos, Greece. 2010 International Conference on Topology and its Applications, (Topology and its applications, ISSN 0166-8641, Vol. 159, iss. 7). Amsterdam [etc.]: Elsevier, 2012, str. 1899-1905. [COBISS-SI-ID 16224857]
GARITY, Dennis, REPOVŠ, Dušan. Homogeneity groups of ends of open 3-manifolds. Pacific journal of mathematics, ISSN 0030-8730, 2014, vol. 269, no. 1, str. 99-112. [COBISS-SI-ID 17071961]
HEGENBARTH, Friedrich, REPOVŠ, Dušan. Controlled homotopy equivalences and structure sets of manifolds. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2014, vol. 142, no. 11, str. 3987-3999. [COBISS-SI-ID 17080665]
SMREKAR, Jaka. Turning a self-map into a self-fibration. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2014, vol. 167, str. 76-79. [COBISS-SI-ID 16943705]
SMREKAR, Jaka. Homotopy type of space of maps into a K(G,n). Homology, homotopy, and applications, ISSN 1532-0073, 2013, vol. 15, no. 1, str. 137-149. [COBISS-SI-ID 16643929]
SMREKAR, Jaka. Homotopy type of mapping spaces and existence of geometric exponents. Forum mathematicum, ISSN 0933-7741, 2010, vol. 22, no. 3, str. 433-456. [COBISS-SI-ID 15638105]
RUBERMAN, Daniel, STRLE, Sašo. Concordance properties of parallel links. Indiana University mathematics journal, ISSN 0022-2518, 2013, vol. 62, no. 3, str. 799-814. [COBISS-SI-ID 16946265]
OWENS, Brendan, STRLE, Sašo. Dehn surgeries and negative-definite four-manifolds. Selecta mathematica. New series, ISSN 1022-1824, 2012, vol. 18, iss. 4, str. 839-854. [COBISS-SI-ID 16808025]
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]