arXiv subjects

Wojciech Lubawski

Publications and source records attributed to Wojciech Lubawski.

7 recordsLinked to original sources

arXiv2015

Homotopy representations of the unitary groups

Let $G$ be a compact connected Lie group and let $ξ,ν$ be complex vector bundles over the classifying space $BG$. The problem we consider is whether $ξ$ contains a subbundle which is isomorphic to $ν$. The necessary condition is that for every prime $p$ the restriction $ξ|_{BN_p^G}$, where $N_p^G$ is a maximal $p$-toral subgroup of $G$, contains a subbundle isomorphic to $ν|_{BN_p^G}$. We provide a criterion when this condition is sufficient, expressed in terms of $Λ^*$-functors of Jackowski, McClure \& Oliver and we prove that this criterion applies if $ν$ is a universal bundle over $BU(n)$. Our result allows to construct new examples of maps between classifying spaces of unitary groups. While proving the main result, we develop the obstruction theory for lifting maps from homotopy colimits along fibrations, which generalizes the result of Wojtkowiak.

math.AT↗

arXiv2014

Mirror links have dual odd and generalized Khovanov homology

We show that the generalized Khovanov homology, defined by the second author in the framework of chronological cobordisms, admits a grading by the group $\mathbb{Z}\times\mathbb{Z}_2$, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring $\mathbb{Z}_π:=\mathbb{Z}[π]/(π^2-1)$ (here, setting $π$ to $\pm 1$ results either in even or odd Khovanov homology). The generalized homology has $\Bbbk := \mathbb{Z}[X,Y,Z^{\pm 1}]/(X^2=Y^2=1)$ as coefficients, and the above implies that most of automorphisms of $\Bbbk$ fix the isomorphism class of the generalized homology regarded as $\Bbbk$-modules, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching $X$ with $Y$ induces a derived isomorphism between the generalized Khovanov homology of a link $L$ with its dual version, i.e. the homology of the mirror image $L^!$, and we compute an explicit formula for this map. When specialized to integers it descends to a duality isomorphism for odd Khovanov homology, which was conjectured by A. Shumakovitch.

math.GT↗

arXiv2014

A new approach to the equivariant topological complexity

We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the the motion planning algoritm than the one introduced and studied by Colman and Grant. We show many bounds for the symmetric topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the Farber's topological complexity of the orbit space. We define the Whitehead version of it.

math.AT↗

arXiv2013

Multidimensional necklaces and measurable colorings of R^n

A well known generalization of Alon's "splitting nacklace theorem" by Longueville and Zivaljevic states that every k-colored n-dimensional cube can be fairly split using only k cuts in each dimension. Here we prove that for every t there exist a finite coloring (with at least (t+4)^d - (t+3)^d + (t+2)^d - 2^d + d(t+2) +3 different colors) of R^n such that no n-dimensional cube can be fairly split using at most t cuts in each dimension. In particular there is a finite coloring of R^n such that no two disjoint n-dimensional cubes have the same measure of each color.

math.CO↗

arXiv2013

On-line list coloring of matroids

A coloring of a matroid is proper if elements of the same color form an independent set. A theorem of Seymour asserts that a k-colorable matroid is also colorable from any lists of size k. In this note we generalize this theorem to the on-line setting. We prove that a coloring of a matroid from lists of size k is possible even if appearances of colors in the lists are recovered color by color by an adversary, while our job is to assign a color immediately after it is recovered. We also prove a more general weighted version of our result with lists of varying sizes. In consequence we get a simple necessary and sufficient condition for matroid list colorability in general case. The main tool we use is the multiple basis exchange property, which we give a simple proof.

math.CO↗

arXiv2012

Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erdős on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljević. The question how many axes aligned hyperplanes are needed for a fair splitting of a d-dimensional k-colored cube remains open.

math.CO↗

arXiv2012

Zredukowane homologie Khovanova

From the very beginning the Khovanov homology appears to be one of the most important invariant of knots; for computational and theoretical reasons it would be useful to operate with reduced version of it - nevertheless the definition given by Khovanov appears to be not natural in a sense that it requires choices of circles in every resolution of knot diagram. We propose a definition that generalizes the reduced odd Khovanov homology defined by Rasmussen, Ozsvath and Szabo to the case of Putyra's chronological homology and therefore gives a simple and natural way to reduce the standard Khovanov homology. Surprisingly the construction works as well for virtual knots and links.

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