Title and Author(s) 
Page 

Editor’s Preface
Wacław Marzantowicz and Aleksy Tralle

3


Morse homotopy and topological conformal field theory
Viktor Fromm
ABSTRACT.
By studying spaces of flow graphs in a closed oriented manifold,
we equip the Morse complex with the operations of an open topological
conformal field theory. This complements previous constructions due to
R. Cohen et al., K. Costello, K. Fukaya and M. Kontsevich and is also the
Morse theoretic counterpart to a conjectural construction of operations on
the chain complex of the Lagrangian Floer homology of the zero section of
a cotangent bundle, obtained by studying uncompactified moduli spaces of
higher genus pseudoholomorphic curves.


7


Measurable patterns, necklaces and sets indiscernible by measure
Siniša Vresćica and Rade Živaljevic
ABSTRACT.
In some recent papers the classical `splitting necklace theorem'
is linked in an interesting way with a geometric `pattern
avoidance problem', see Alon et al. (Proc. Amer. Math. Soc.,
2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and
Laso\'{n} (arXiv:1304.5390v1 [math.CO]). Following these authors
we explore the topological constraints on the existence of a
(relaxed) measurable coloring of $\mathbb{R}^d$ such that any two
distinct, nondegenerate cubes (parallelepipeds) are measure
discernible. For example, motivated by a conjecture of Laso\'{n},
we show that for every collection $\mu_1,\ldots,\mu_{2d1}$ of $2d1$
continuous, signed locally finite measures on $\mathbb{R}^d$,
there exist two nontrivial axisaligned $d$dimensional cuboids
(rectangular parallelepipeds) $C_1$ and $C_2$ such that
$\mu_i(C_1)=\mu_i(C_2)$ for each $i\in\{1,\ldots,2d1\}$. We also
show by examples that the bound $2d1$ cannot be improved in
general. These results are steps in the direction of studying
general topological obstructions for the existence of
nonrepetitive colorings of measurable spaces.


39


Homologies are infinitely complex
Mark Grant and András Szűcs
ABSTRACT.
We show that for any $k>1$, stratified sets of finite complexity are insufficient to realize all homology classes of codimension $k$ in all smooth manifolds. We also prove a similar result concerning smooth generic maps whose doublepoint sets are cooriented.


55


Computational topology of equipartitions by hyperplanes
Rade Živaljevic
ABSTRACT.
We compute a primary cohomological obstruction to the existence of
an equipartition for $j$ mass distributions in $\mathbb{R}^d$ by
two hyperplanes in the case $2d3j = 1$. The central new result is
that such an equipartition always exists if $d=6\cdot 2^k +2$ and
$j=4\cdot 2^k+1$ which for $k=0$ reduces to the main result of the
paper P. ManiLevitska et al., Topology and combinatorics of
partitions of masses by hyperplanes, Adv. Math. 207 (2006),
266296. The theorem follows from a BorsukUlam type result
claiming the nonexistence of a $\mathbb{D}_8$equivariant map $f
\colon S^{d}\times S^d\rightarrow S(W^{\oplus j})$ for an associated
real $\mathbb{D}_8$module $W$. This is an example of a genuine
combinatorial geometric result which involves
$\mathbb{Z}/4$torsion in an essential way and cannot be obtained
by the application of either StiefelWhitney classes or
cohomological index theories with $\mathbb{Z}/2$ or $\mathbb{Z}$
coefficients. The method opens a possibility of developing an
``effective primary obstruction theory'' based on $G$manifold
complexes, with applications in geometric combinatorics, discrete
and computational geometry, and computational algebraic topology.


63


Extension of functors to fibrewise pointed spaces
Petar Pavesic
ABSTRACT.
We describe a new general method for the fibrewise extension of a given endofunctor on the category of pointed topological spaces
to the category of fibrewise pointed spaces. We derive some properties of the construction and show how it can be profitably used to
build the WhiteheadGanea framework for the fibrewise LusternikSchnirelmann category and the topological complexity.


91


Estimating the discrete LusternikSchnirelmann category
Brian Green, Nicholas A. Scoville and Mimi Tsuruga
ABSTRACT.
Let $K$ be a simplicial complex and suppose that $K$ collapses onto $L$. Define
$n$ to be $1$ less than the minimum number of collapsible sets it takes to cover
$L$. Then the discrete geometric LusternikSchnirelmann category of $K$ is the
smallest $n$ taken over all such $L$. In this paper, we give an algorithm which
yields an upper bound for the discrete geometric category. We show our
algorithm is correct and give several bounds for the discrete geometric category
of wellknown simplicial complexes. We show that the discrete geometric
category of the dunce cap is $2$, implying that the dunce cap is ``further" from
being collapsible than Bing's house whose discrete geometric category is $1$.


103


Hadwiger integration of random fields
Matthew L. Wright
ABSTRACT.
Hadwiger integrals employ the intrinsic volumes as measures for integration of realvalued functions.
We provide a formula for the expected values of Hadwiger integrals of Gaussianrelated random fields.
The expected Hadwiger integrals of random fields are both theoretically interesting and potentially useful in applications such as sensor networks, image processing, and cell dynamics.
Furthermore, combining the expected integrals with a functional version of Hadwiger's theorem, we obtain expected values of more general valuations on Gaussianrelated random fields.


117


Random coverings of one complexes and the Euler characteristic
Rafał Komendarczyk and J. Pullen
ABSTRACT.
This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one dimensional case. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the covering ought to be {\em good}, which holds if the diameter of the covering elements does not exceed
a certain size), but no a priori requirements on their distribution. An upper bound for the coverage probability is also obtained as a consequence of the concentration inequality. The techniques rely on a formulation of the coverage criteria in terms of the Euler characteristic of the nerve complex associated to the random covering.


129


The module category weight of compact exceptional Lie groups
Younggi Choi
ABSTRACT.
We give a lower bound for the LusternikSchnirelmann category of compact exceptional Lie groups by computing the module
category weight through analyzing several
EilenbergMoore type spectral sequences.


157


Totally normal cellular stratified spaces and applications to the configuration space of graphs
Mizuki Furuse, Takashi Mukouyama and Dai Tamaki
ABSTRACT.
The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with
posets. A generalization, called totally normal
cellular stratified spaces, was introduced in
\cite{Bas+}, \cite{Tama} by relaxing two conditions;
face posets are replaced by acyclic categories and cells with
incomplete boundaries are allowed.
The aim of this article is to demonstrate the usefulness of totally
normal cellular stratified spaces by constructing a combinatorial
model for the configuration space of graphs.
As an application, we obtain a simpler proof of Ghrist's theorem on the
homotopy dimension of the configuration space of graphs. We also make
sample calculations of the fundamental group of ordered and unordered
configuration spaces of two points for small graphs.


169


Cohomological decomposition of complex nilmanifolds
Adela Latorre and Luis Ugarte
ABSTRACT.
We study \emph{pureness} and \emph{fullness} of invariant complex structures on nilmanifolds.
We prove that in dimension six, apart from the complex torus, there exist only two nonisomorphic complex structures
satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the
product of two almost complex manifolds which are pure and full is not necessarily full.


215


On the space of equivariant local maps
Piotr Bartłomiejczyk
ABSTRACT.
We introduce the space of equivariant local maps
and present the full proof of the splitting theorem for the set of otopy
classes of such maps in the case of a representation of a compact Lie group.


233


Nudged elastic band in topological data analysis
Henry Adams, Atanas Atanasov and Gunnar Carlsson
ABSTRACT.
We use the nudged elastic band method from computational chemistry to analyze highdimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches.


247


An algorithmic approach to estimating the minimal number of periodic points for smooth selfmaps of simplyconnected manifolds
Grzegorz Graff and Paweł Pilarczyk
ABSTRACT.
For a given selfmap $f$ of $M$, a closed smooth connected and
simplyconnected manifold of dimension $m\geq 4$, we provide an algorithm
for estimating the values of the topological invariant $D^m_r[f]$,
which equals the minimal number of $r$periodic points
in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$
introduced by G. Graff and J. Jezierski
[J. Fixed Point Theory Appl. 13 (2013), 6384].
An opensource implementation of the algorithm
programmed in C++ is publicly available
at {\tt http://www.pawelpilarczyk.com/combtop/}.


273


On representation of the Reeb graph as a subcomplex of manifold
Marek Kaluba, Wacław Marzantowicz and Nelson Silva
ABSTRACT.
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 $1$\nobreakdashdimensional
complex $\Gamma(f)$ 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 $\pi_1(M)$ 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 $M_g$.


287
