arXiv subjects

Krzysztof Ziemiański

Publications and source records attributed to Krzysztof Ziemiański.

23 records · Page 2Linked to original sources

arXiv2018

Stable components of directed spaces

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit the unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces.

math.AT↗

arXiv2017

Directed path spaces via discrete vector fields

Let $K$ be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space $\vec{P}(K)_v^w$ of directed paths between two given vertices $v,w$ of $K$. In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1--1 correspondence with the generators of the homology of $\vec{P}(K)_v^w$.

math.AT↗

arXiv2016

Paths of the directed suspension

We prove that the loop space of the directed suspension of a directed space is homotopy equivalent to the James construction. In particular, it does not depend on the directed structure of a given directed space.

math.AT↗

arXiv2016

Spaces of directed paths on pre-cubical sets

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also $\square$--sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set $K$, a CW-complex $W(K)_v^w$ that is homotopy equivalent to the space of directed paths between given vertices $v$, $w$ of $K$. This construction is functorial with respect to $K$, and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of $W(K)_v^w$ are provided.

math.AT↗

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↗