arXiv subjects

Wacław Marzantowicz

Publications and source records attributed to Wacław Marzantowicz.

16 recordsLinked to original sources

arXiv2025

Fundamental groups of small simplicial complexes

The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a complete list of groups that arise as fundamental groups of simplicial complexes with at most $8$ vertices. In addition we give many examples of fundamental groups of complexes with $9$ vertices although the complete classification seems to be beyond reach at the moment. Our results lead to many applications, including progress on the Björner-Lutz conjecture regarding vertex-minimal triangulations of the Poincaré homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups.

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arXiv2025

Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces

The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the Möbius inversion formula. The set ${\mathcal AP}(f) = \{ n \,\colon\, a_n(f) \neq 0 \}$ is called the set of algebraic periods of $f$. Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical systems and periodic point theory. In this work we provide a description of surface homeomorphisms with bounded $(L(f^n))$ (quasi-unipotent maps) in terms of Dold coefficients. We also discuss the problem of minimization of the genus of a surface for which one can realize a given set of natural numbers as the set of algebraic periods. Finally, we compute and list all possible Dold coefficients and algebraic periods for a given orientable surface with small genus and give some geometrical applications of the obtained results.

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arXiv2025

Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism

A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the Möbius inversion formula. The set ${\mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods. We solve an open problem existing in literature by showing that for every finite subset ${\mathcal A}$ of natural numbers there exist an orientable surface $S_{\rm g}$, as well as a non-orientable surface $N_{\rm g}$, of genus ${\rm g}$, and a Morse$\unicode{x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{AP}(f)=\mathcal{A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{A}$. For the orientation-reversing Morse$\unicode{x2013}$Smale diffeomorphisms of $S_{\rm g}$, we identify strong restrictions on ${\mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{x2013}$Smale diffeomorphisms, which is exponential in ${\rm g}$.

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arXiv2024

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

We construct a correspondence between epimorphisms $φ\colon π_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such $φ$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups.

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arXiv2021

Estimate of number of simplices of triangulations of Lie groups

We present estimates of number of simplices of given dimension of classical compact Lie groups. As in the previous work \cite{GMP2} the approach is a combination of an estimate of number of vertices with a use of valuation of the covering type by cohomological argument of \cite{GMP} and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the case of exceptional Lie groups we made a complete calculation using the description of their cohomology rings given by the first and third author. For infinite increasing series of Lie groups of growing dimension $d$ the rate of growth of number of simplices of highest dimension is given which extends onto the case of simplices of (fixed) codimension $d-i$.

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arXiv2020

How many simplices are needed to triangulate a Grassmannian?

We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.

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arXiv2019

Estimates of covering type and the number of vertices of minimal triangulations

The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of $X$. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.

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arXiv2018

General Bourgin-Yang theorems

We describe a unified approach to estimating the dimension of $f^{-1}(A)$ for any $G$-equivariant map $f \colon X \to Y$ and any closed $G$-invariant subset $A\subseteq Y$ in terms of connectivity of $X$ and dimension of $Y$, where $G$ is either a cyclic group of order $p^k$, a $p$-torus ($p$ a prime), or a torus.

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arXiv2018

Geometrically distinct solutions given by symmetries of variational problems with the $O(N)$-symmetry

For variational problems with $O(N)$-symmetry the existence of several geometrically distinct solutions had been shown by use of group theoretic approach in previous articles. It was done by a crafty choice of a family $H_i \subset O(N)$ subgroups such that the fixed point subspaces $E^{H_i} \subset E$ of the action in a corresponding functional space are linearly independent, next restricting the problem to each $E^{H_i}$ and using the Palais symmetry principle. In this work we give a thorough explanation of this approach showing a correspondence between the equivalence classes of such subgroups, partial orthogonal flags in $\mathbb{R}^N$, and unordered partitions of the number $N$. By showing that spaces of functions invariant with respect to different classes of groups are linearly independent we prove that the amount of series of geometrically distinct solutions obtained in this way grows exponentially in $N$, in contrast to logarithmic, and linear growths of earlier papers.

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arXiv2018

Topology of unavoidable complexes

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $σ_1,\ldots, σ_r$ of $| K|$ such that $f(σ_1)\cap\ldots\cap f(σ_r)\neq\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.6, 3.9, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1\ast\ldots\ast K_s$ of $r$-unavoidable complexes.

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arXiv2017

Equivariant maps between representation spheres

Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.

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arXiv2016

Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$.

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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.

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arXiv2014

On Representation of the Reeb Graph as a Sub-Complex of Manifold

The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $Γ(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $π_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.

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arXiv2013

On the interspike-intervals of periodically-driven integrate-and-fire models

We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input functions, we generalize some results of other authors. In particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces. Using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide also a complete description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution. Our results allow to explain some facts concerning this distribution observed numerically by other authors. These theoretical findings are illustrated by carefully chosen computational examples.

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arXiv2012

Displacement sequence of an orientation preserving circle homeomorphism

We give a complete description of the behaviour of the sequence of displacements $η_n(z)=Φ^n(x) - Φ^{n-1}(x) \ \rmod \ 1$, $z=\exp(2π\rmi x)$, along a trajectory $\{φ^{n}(z)\}$, where $φ$ is an orientation preserving circle homeomorphism and $Φ:\mathbb{R} \to \mathbb{R}$ its lift. If the rotation number $\varrho(φ)=\frac{p}{q}$ is rational then $η_n(z)$ is asymptotically periodic with semi-period $q$. This convergence to a periodic sequence is uniform in $z$ if we admit that some points are iterated backward instead of taking only forward iterations for all $z$. If $\varrho(φ) \notin \mathbb{Q}$ then the values of $η_n(z)$ are dense in a set which depends on the map $γ$ (semi-)conjugating $φ$ with the rotation by $\varrho(φ)$ and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if $φ$ is $C^1$-diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism $φ$. Finally, we prove that even for the irrational rotation number $\varrho$ the displacement sequence exhibits some regularity properties.

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