Algebraic topology 2

2022/2023
Programme:
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

exercise-basd exam, theoretical knowledge exam
written 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]