arXiv subjects

P. Christopher Staecker

Publications and source records attributed to P. Christopher Staecker.

At least 19 recordsLinked to original sources

arXiv2026

Nielsen coincidence theory of $(n,m)$-valued pairs of maps

We consider pairs of maps $(f,g)$, where $f$ is an $n$-valued map and $g$ is an $m$-valued map, defined on connected finite polyhedra. A point $x$ such that $f(x)\cap g(x)\neq \emptyset$ is called a coincidence point of $f$ and $g$. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all $(n, m)$-valued pairs of maps homotopic to $(f,g)$. The construction of such an invariant $N(f:g)$ was proposed in [J. Fixed Point Theory Appl. 14, 309--324 (2013)]. Unfortunately, this approach has some flaws. In this paper, we present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of $f$ and $g$. In the case of $(n, m)$-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.

math.GN↗

arXiv2025

Homotopy equivalence of digital pictures in $\mathbb{Z}^2$

We investigate the properties of digital homotopy in the context of digital pictures $(X,\kappa,\bar \kappa)$, where $X\subsetneq \Z^n$ is a finite set, $\kappa$ is an adjacency relation on $X$, and $\bar \kappa$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\Z^2$. We define a numerical homotopy-type invariant for digital pictures in $\Z^2$ called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by ``filling in the gaps'' of any row or column. We show that a digital picture $(X,c_i,c_j)$ is homotopy equivalent to only finitely many other digital pictures $(Y,c_i,c_j)$. At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture.

math.AT↗

arXiv2025

The Face Group of a Simplicial Complex

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the ordinary (topological) fundamental group of the spatial realization. We define a counterpart to the edge group that likewise gives a combinatorial version of the second (higher) homotopy group. Working entirely combinatorially, we show our group is an abelian group and also respects products. We show that our combinatorially defined group is isomorphic to the ordinary (topological) second homotopy group of the spatial realization.

math.AT↗

arXiv2025

At most n-valued maps

This paper concerns various models of ``at-most-$n$-valued maps''. That is, multivalued maps $f:X\multimap Y$ for which $f(x)$ has cardinality at most $n$ for each $x$. We consider 4 classes of such maps which have appeared in the literature: $\mathcal U$, the set of exactly $n$-valued maps, or unions of such; $\mathcal F$, the set of $n$-fold maps defined by Crabb; $\mathcal S$, the set of symmetric product maps; and $\mathcal W$, the set of weighted maps with weights in $\mathbb N$. Our main result is roughly that these classes satisfy the following containments: \[ \mathcal U \subsetneq \mathcal F \subsetneq \mathcal S = \mathcal W \] Furthermore we define the general class $\mathcal C$ of all at-most-$n$-valued maps, and show that there are maps in $\mathcal C$ which are outside of any of the other classes above. We also describe a configuration-space point of view for the class $\mathcal C$, defining a configuration space $C_n(Y)$ such that any at-most-$n$-valued map $f:X\multimap Y$ corresponds naturally to a single-valued map $f:X\to C_n(Y)$. We give a full calculation of the fundamental group and homology groups of $C_n(S^1)$.

math.GN↗

arXiv2024

On digital H-spaces

In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies. We explore algebraic properties of digital H-spaces preserved under digital homotopy equivalence, and we give a general construction that produces examples of digital H-spaces which are not homotopy-equivalent to digital topological groups in both categories. Further, we show that this construction essentially classifies all NP$_2$-digital H-spaces. In a short appendix, we resolve a question that was left unresolved in [17], and complete the full classification of digital topological groups.

math.AT↗

arXiv2024

Ahead of the Count: An Algorithm for Probabilistic Prediction of Instant Runoff (IRV) Elections

How can we probabilistically predict the winner in a ranked-choice election without all ballots being counted? In this study, we introduce a novel algorithm designed to predict outcomes in Instant Runoff Voting (IRV) elections. The algorithm takes as input a set of discrete probability distributions describing vote totals for each candidate ranking and calculates the probability that each candidate will win the election. In fact, we calculate all possible sequences of eliminations that might occur in the IRV rounds and assign a probability to each. The discrete probability distributions can be arbitrary and, in applications, could be measured empirically from pre-election polling data or from partial vote tallies of an in-progress election. The algorithm is effective for elections with a small number of candidates (five or fewer), with fast execution on typical consumer computers. The run-time is short enough for our method to be used for real-time election night modeling where new predictions are made continuously as more and more vote information becomes available. We demonstrate the algorithm in abstract examples, and also using real data from the 2022 Alaska state elections to simulate election-night predictions and also predictions of election recounts.

cs.CY↗

arXiv2023

A Second Homotopy Group for Digital Images

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.

math.AT↗

arXiv2022

Digital topological groups

In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define $\NP_1$- and $\NP_2$-digital topological groups, and investigate their properties and algebraic structure. The $\NP_2$ category is very restrictive, and we give a complete classification of $\NP_2$-digital topological groups. We also give many examples of $\NP_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.

cs.CV↗

arXiv2021

The Reidemeister Trace of an $n$-valued map

In topological fixed point theory, the Reidemeister trace is an invariant associated to a selfmap of a polyhedron which combines information from the Lefschetz and Nielsen numbers. In this paper we define the Reidemeister trace in the context of $n$-valued selfmaps of compact polyhedra. We prove several properties of the Reidemeister trace which generalize properties from the single-valued theory, and prove an averaging formula.

math.AT↗

arXiv2021

The two digital homology theories

In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca & Ege. We show that the two simplicial homology theories are isomorphic, and that the two cubical theories are isomorphic, but that the simplicial theory is not isomorphic to the cubical theory. Thus we obtain, up to isomorphism, two different homology theories: simplicial, and cubical. We discuss briefly the relationship between the two theories. Notably, the two theories always agree in dimension zero, and in dimension 1 there is always a surjection from the cubical homology to the simplicial homology. We give examples to show that there need be no similar relationships in higher dimensions.

math.AT↗

arXiv2021

Digital homotopy relations and digital homology theories

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories. We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

math.AT↗

arXiv2021

Partitions of $n$-valued maps

An $n$-valued map is a set-valued continuous function $f$ such that $f(x)$ has cardinality $n$ for every $x$. Some $n$-valued maps will "split" into a union of $n$ single-valued maps. Characterizations of splittings has been a major theme in the topological theory of $n$-valued maps. In this paper we consider the more general notion of "partitions" of an $n$-valued map, in which a given map is decomposed into a union of other maps which may not be single-valued. We generalize several splitting characterizations which will describe partitions in terms of mixed configuration spaces and mixed braid groups, and connected components of the graph of $f$. We demonstrate the ideas with some examples on tori. We also discuss the fixed point theory of $n$-valued maps and their partitions, and make some connections to the theory of finite-valued maps due to Crabb.

math.GN↗

arXiv2020

Lefschetz numbers and fixed point theory in digital topology

In this paper, we present two types of Lefschetz numbers in the topology of digital images. Namely, the simplicial Lefschetz number $L(f)$ and the cubical Lefschetz number $\bar L(f)$. We show that $L(f)$ is a strong homotopy invariant and has an approximate fixed point theorem. On the other hand, we establish that $\bar L(f)$ is a homotopy invariant and has an $n$-approximate fixed point result. In essence, this means that the fixed point result for $L(f)$ is better than that for $\bar L(f)$ while the homotopy invariance of $\bar L(f)$ is better than that of $L(f)$. Unlike in classical topology, these Lefschetz numbers give lower bounds for the number of approximate fixed points. Finally, we construct some illustrative examples to demonstrate our results.

math.GN↗

arXiv2019

Lifting classes for the fixed point theory of $n$-valued maps

The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron $X$ is extended to $n$-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicoténcatl. The liftings of an $n$-valued map $f$ split into self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the lift-factors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$-valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$-valued map equals its Reidemeister number. If an $n$-valued map splits into $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.

math.AT↗

arXiv2019

Strong homotopy of digitally continuous functions

We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We explore basic properties of strong homotopy, and give some equivalent characterizations. In particular we show that strong homotopy is related to ``punctuated homotopy,'' in which the function changes by only one point in each homotopy time step. We also show that strongly homotopic maps always have the same induced homomorphisms in the digital homology theory. This is not generally true for digitally homotopic maps, though we do show that it is true for any homotopic selfmaps on the digital cycle $C_n$ with $n\ge 4$. We also define and consider strong homotopy equivalence of digital images. Using some computer assistance, we produce a catalog of all small digital images up to strong homotopy equivalence. We also briefly consider pointed strong homotopy equivalence, and give an example of a pointed contractible image which is not pointed strongly contractible.

math.GN↗

arXiv2019

Fixed point sets in digital topology, 1

In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum $F(X)$ of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of $F(C_n)$ where $C_n$ is the digital cycle of $n$ points. For other digital images, we show that, if $X$ has at least 4 points, then $F(X)$ always contains the numbers 0, 1, 2, 3, and the cardinality of $X$. We give several examples, including $C_n$, in which $F(X)$ does not equal $\{0,1,\dots,\#X\}$. We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., in some cases the fixed point set is always connected.

math.GN↗

arXiv2018

Remarks on Fixed Point Assertions in Digital Topology

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that in many cases, researchers using these tools have derived conclusions that are incorrect or trivial.

math.GN↗

arXiv2017

The asymptotic density of Wecken maps on surfaces with boundary

The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed points among maps homotopic to $f$. When these numbers are equal, the map is called Wecken. A recent paper by Brimley, Griisser, Miller, and the second author investigates the abundance of Wecken maps on surfaces with boundary, and shows that the set of Wecken maps has nonzero asymptotic density. We extend the previous results as follows: When the fundamental group is free with rank $n$, we give a lower bound on the density of the Wecken maps which depends on $n$. This lower bound improves on the bounds given in the previous paper, and approaches 1 as $n$ increases. Thus the proportion of Wecken maps approaches 1 for large $n$. In this sense (for large $n$) the known examples of non-Wecken maps represent exceptional, rather than typical, behavior for maps on surfaces with boundary.

math.AT↗