In this invited one-hour plenary lecture at the international conference dedicated to new results in geometry and topology, recent work of this project research team was presented, including the most important obtained results.

B.04 Guest lecture

COBISS.SI-ID: 16311641This was a colloquium at the Brandeis University in Boston, USA. The key results of our project group in low-dimensional geometric topology were presented.

B.05 Guest lecturer at an institute/university

COBISS.SI-ID: 16317785So far knots have been classified up to a certain number of crossings only for a handful of spaces: the 3-dimensional Euclidean space, the projective space, and the solid torus, the latter being classifed only up to a so-called flip. In this thesis we append the infinite family of lens spaces to this modest list. As a side product, we refine the case of the solid torus by providing a complete classification of knots in it. In both cases we classify knots up to four crossings and up to five crossings with a few exceptions. We also establish which of the knots in the solid torus are amphichiral. We see that for each lens space, a subset of prime knots in the solid torus gives the classification in the lens space. Since there are very few applicable invariants of links in L(p, q), a necessary condition formaking a classification in these spaces is to develop invariants of links in L(p, q). The first invariant we introduce is the HOMFLYPT skeinmodule. The HOMFLYPT skein module has so far only been calculated only for S^3 and the solid torus. We show that the HOMFLYPT skein module of L(p, 1) is a free R-module and we present a basis of this module for each p ) 1. The second invariant is the Khovanov homology of the Kauffman bracket skein module of \mathbb{R}P^3. Khovanov homology, an invariant of links in {\mathbb R}^3, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki, and Sikora generalized this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in I-bundles over surfaces, except for the surface \mathbb{R}P^2, where the construction fails due to the strange behavior of links when projected to the non-orientable \mathbb{R}P^2. We categorify the missing case of the twisted I-bundle over \mathbb{R}P^2, \mathbb{R}P^2 \widetilde{\times} I \approx \mathbb{R}P^3 \setminus \{\ast\}, by redefining the differential in the Khovanov chain complex in a suitable manner. The classification, the calculations of the HOMFLYPT skein modules of the knots, and the calculations of the Kauffman bracket skein modules of the knots are done by a computer program that is available at https://github.com/bgabrovsek/lpq-classification.

D.09 Tutoring for postgraduate students

COBISS.SI-ID: 16639833Several questions/conjectures in CAT(0) geometry are inspired by analogous theorems that are known to hold for Riemannian manifolds of nonpositive sectional curvature. This thesis deals with the one which was settled by Bangert and Schröder in early nineties for real analytic manifolds, [V Bangert, v Schröder, Existence of flat tori in analytic manifolds of nonpositive curvature. Ann. Sci. École Norm. Sup. 24 (1992), no. 4 pp. 605-634]. It is called the flat closing problem and it predicts a copy of \mathbb{Z}^m in any discrete group which acts properly and cocompactly by isometries on a CAT(0) space X containing an isometric copy of \mathbb{R}^m. We summarize results from [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700 and P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2 (2009), no. 4, pp. 701-746] about the full isometry group of a proper, cocompact and geodesically complete CAT(0) space. Then we apply those results to prove the main theorem from [P.-E. Caprace, G. Zadnik, Regular elements in CAT(0) groups. Preprint at http://arXiv.org/abs/1112.4637 (2011)], a very partial answer to the flat closing conjecture: "If a proper CAT(0) space X is a product of m geodesically complete factors, then discrete \Gamma, which acts properly and cocompactly on X, contains a copy of \mathbb{Z}^m." Even though the theorem above is far from the full generality of the flat closing problem, its proof uses a deep machinery from the structure theory of the isometry group of the corresponding CAT(0) space. The proof relies in an essential way to the solution of Hilbert's fifth problem (Theorem Gleason, Montgomery-Zippin). This solution leads to a dichotomy for the isometry group of a nice non Euclidean CAT(0) space - either it is a Lie group or a totally disconnected locally compact group. Applying this dichotomy to the irreducible factors from the theorem, we deal with two separated approaches. The first case is covered by older results from Lie group theory while the second relies to the geometric properties of CAT(0) space with totally disconnected isometry group, see [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700].

D.09 Tutoring for postgraduate students

COBISS.SI-ID: 16941401We are interested in the configurations of surfaces inside 4-manifolds. Given a closed connected smooth 4-manifold X and a finite set of classes C \subset H_2(X), we would like to represent C by the simplest possible configuration of surfaces inside X. The regular neighborhood of such a configuration is a plumbing of disk bundles over the surfaces. For each geometric intersection of two surfaces, the corresponding disk bundles are plumbed once. The plumbing may be represented by its Kirby diagram, which also describes the construction of the boundary 3-manifold using surgery. We investigate the double plumbing N of two disk bundles over spheres with Euler numbers m and n. Using the Kirby diagram of N as a surgery diagram, we obtain the Heegaard diagram of the boundary 3-manifold Y = \partial N. We calculate the Heegaard-Floer homology \widehat{HF}(Y; \mathfrak{s}) in all the torsion Spin^c structures \mathfrak{s} \in Spin^c(Y). Using a cobordism from Y to the known 3-manifold L(m; 1)\# S_1 \times S^2, we calculate the absolute gradings of the homology \widehat{HF}(Y). We use the correction terms of the 3-manifold Y to find obstructions for realizing the double plumbing N inside given 4-manifolds X z b^+_2(X) = 2. By a similar method we study single plumbings of disk bundles over spheres inside selected closed 4-manifolds.

D.09 Tutoring for postgraduate students

COBISS.SI-ID: 17052761