Topics in topology

The lecturer chooses one or more important topics such as:
Knots and links. Basic invariants. Knots and surfaces, the Seifert form. Connected sum and decomposition into prime knots. Polynomial invariants; Alexander’s and Jones’ polynomials. Khovanov homology. Braid groups. Lattice homology.
3- and 4-manifolds. Constructions: handle decomposition, surgery, Heegaard splitting and covering spaces. Casson’s and Rohlin’s invariants. Fundamental group and its representations. The basics of Seiberg-Witten theory and Heegaard-Floer homology.
The topology of smooth manifolds. Morse theory. Differential forms and de Rham cohomology. Hodge theory. Contact and symplectic structures on manifolds.
Discrete Morse theory. Simplicial complexes and maps: abstract simplicial complexes, geometric realization, subdivision, simplicial approximation. Discrete Morse functions, gradient vector fields, the Morse chain complex and Morse homology, discrete Morse inequalities. Computational algorithms and implementation.
Topological robotics. Configuration space of a robot, motion and navigation plans. The concept of topological complexity of motion planning and navigation; upper and lower bounds on topological complexity.
Persistent homology. Filtrations on spaces, persistence complex and its homology; stability theorems.
Topological methods in group theory. Finiteness properties of groups. Cohomology of infinite groups. Homotopy theory of groups. PL Morse theory.
Characteristic classes of vector bundles. Cohomology ring of a smooth manifold and vector bundles over smooth manifolds. Stiefel-Whitney characteristic classes. Orientability and Thom’s theorem. Complexification; Chern and Pontryagin characteristic classes.

1. H. Edelsbrunner, J. L. Harer: Computational topology : an introduction, Providence : American Mathematical Society, cop. 2010.
2. M. Farber: Invitation to topological robotics, Zürich : European Mathematical Society Publishing House, cop. 2008.
3. R. Geoghegan: Topological methods in group theory, New York : Springer, cop. 2008.
4. R. E. Gompf, A. I. Stipsicz: 4-manifolds and Kirby calculus, Providence : American Mathematical Society, cop. 1999.
5. M. W. Hirsch: Differential topology, New York ; Heidelberg ; Berlin : Springer, 1976.
6. D. Husemoller: Fibre bundles, 3rd ed., New York : Springer, cop. 1994.
7. L. H. Kauffman: On knots, Princeton : Princeton Univ. Press, 1987.
8. J. W. Milnor, J. D. Stasheff: Characteristic classes, Princeton : Princeton University Press ; Tokyo : Univ. of Tokyo Press, cop. 1974.
9. L. Nicolaescu, Lectures on the geometry of manifolds, World Scientific Publishing Co., 2021, dostopno na: https://www3.nd.edu/~lnicolae/Lectures.pdf
10. D. Rolfsen: Knots and links, Berkeley : Publish or perish, 1976.
11. N. Saveliev: Lectures on the topology of 3-manifolds, An introduction to the Casson invariant, De Gruyter, 2012.
12. N. A. Scoville: Discrete Morse theory, Providence : American Mathematical Society, cop. 2019.

The objectives and competences coincide with those of the study program.

Students get acquainted with one or more important or advanced topics in topology to the extent of being able to be introduced to research problems.

Lectures, discussion, exercises, homework assignments.

Theoretical knowldege exam (oral or written), exercise-based exame (written), homework assignments.

Petar Pavešić:
GOVC, Dejan, MARZANTOWICZ, Wacław, PAVEŠIĆ, Petar. Estimates of covering type and the number of vertices of minimal triangulations. Discrete & computational geometry. Jan. 2020, vol. 63, iss. 1, str. 31-48. [COBISS-SI-ID 18627417]
PAVEŠIĆ, Petar. Topological complexity of a map. Homology, homotopy, and applications. Jan. 2019, vol. 21, no. 2, str. 107-130. [COBISS-SI-ID 18590297]
PAVEŠIĆ, Petar. Triangulations with few vertices of manifolds with non-free fundamental group. Proceedings. Section A, Mathematics. Dec. 2019, vol. 149, iss. 6, str. 1453-1463. [COBISS-SI-ID 18671705]
Jaka Smrekar:
SMREKAR, Jaka. Rational Reidemeister trace of an outer automorphism of finite order. Journal of Pure and Applied Algebra. July 2023, vol. 227, iss. 7, art. 107350 (13 str.). ISSN 0022-4049. [COBISS-SI-ID 141555715]
SMREKAR, Jaka. Gottlieb’s theorem for a periodic equivalence. Journal of fixed point theory and its applications. Dec. 2022, vol. 24, iss. 4, art. 73 (15 str.). ISSN 1661-7738. [COBISS-SI-ID 125772547]
FORSTNERIČ, Franc, SMREKAR, Jaka, SUKHOV, Alexandre. On the Hodge conjecture for q-complete manifolds. Geometry & topology. 2016, vol. 20, no. 1, str. 353-388. ISSN 1465-3060. [COBISS-SI-ID 17622361]
Sašo Strle:
RUBERMAN, Daniel, SLAPAR, Marko, STRLE, Sašo. On the Thom conjecture in CP^3. International mathematics research notices, ISSN 1687-0247, 2021, vol. 2022, 1 spletni vir (1 datoteka pdf (14 str.)). [COBISS-SI-ID 89922563]
LEVINE, Adam Simon, RUBERMAN, Daniel, STRLE, Sašo, GESSEL, Ira M. Nonorientable surfaces in homology cobordisms. Geometry & topology, ISSN 1465-3060, 2015, vol. 19, no. 1, str. 439-494. [COBISS-SI-ID 17557337]
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]