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P. Christopher Staecker

Publications and source records attributed to P. Christopher Staecker.

At least 37 records · Page 2Linked to original sources
arXiv2017

Axioms for the fixed point index of n-valued maps, and some applications

We give an axiomatic characterization of the fixed point index of an $n$-valued map. For $n$-valued maps on a polyhedron, the fixed point index is shown to be unique with respect to axioms of homotopy invariance, additivity, and a splitting property. This uniqueness is used to obtain easy proofs of an averaging formula and product formula for the index. In the setting of $n$-valued maps on a manifold, we show that the axioms can be weakened.

math.AT↗
arXiv2016

An averaging formula for the coincidence Reidemeister trace

In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.

math.AT↗
arXiv2016

Homotopy relations for digital images

We introduce three generalizations of homotopy equivalence in digital images, to allow us to express whether a finite and an infinite digital image are similar with respect to homotopy. We show that these three generalizations are not equivalent to ordinary homotopy equivalence, and give several examples. We show that, like homotopy equivalence, our three generalizations imply isomorphism of fundamental groups, and are preserved under wedges and Cartesian products.

math.GN↗
arXiv2016

Fundamental Groups and Euler Characteristics of Sphere-like Digital Images

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but different notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in a paper by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.

math.GN↗
arXiv2015

Connectivity Preserving Multivalued Functions in Digital Topology

We study connectivity preserving multivalued functions between digital images. This notion generalizes that of continuous multivalued functions studied mostly in the setting of the digital plane $Z^2$. We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morpholological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.

cs.CV↗
arXiv2015

A Borsuk-Ulam theorem for digital images

The Borsuk-Ulam theorem states that a continuous function $f:S^n \to \R^n$ has a point $x\in S^n$ with $f(x)=f(-x)$. We give an analogue of this theorem for digital images, which are modeled as discrete spaces of adjacent pixels equipped with $\Z^n$-valued functions. In particular, for a concrete two-dimensional rectangular digital image whose pixels all have an assigned "brightness" function, we prove that there must exist a pair of opposite boundary points whose brightnesses are approximately equal. This theorem applies generally to any integer-valued function on an abstract simple graph. We also discuss generalizations to digital images of dimension 3 and higher. We give some partial results for higher dimensional images, and show a counter example which demonstrates that the full results obtained in lower dimensions cannot hold generally.

math.GN↗
arXiv2015

Remarks on pointed digital homotopy

We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eventually constant maps are often easier to work with than the trivial extensions that are key to the development of the fundamental group in Boxer (1999) and many subsequent papers. We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in Boxer (2005), but there was an error in the proof; here, we correct the error.

math.CO↗
arXiv2015

Some enumerations of binary digital images

The topology of digital images has been studied much in recent years, but no attempt has been made to exhaustively catalog the structure of binary images of small numbers of points. We produce enumerations of several classes of digital images up to isomorphism and decide which among them are homotopy equivalent to one another. Noting some patterns in the results, we make some conjectures about digital images which are irreducible but not rigid.

math.CO↗
arXiv2014

Homotopy equivalence of finite digital images

For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops a numerical digital homotopy invariant and begins to catalog all possible connected digital images on a small number of points, up to homotopy equivalence.

math.GN↗
arXiv2014

A Hadwiger Theorem for Simplicial Maps

We define the notion of valuation on simplicial maps between geometric realizations of simplicial complexes in $\mathbb{R}^n$. Valuations on simplicial maps are analogous to valuations on sets. In particular, we define the Lefschetz volumes, which are analogous to the intrinsic volumes of subsets of $\mathbb{R}^n$. Our definition not only provides a generalization of the Lefschetz number, but also yields a Hadwiger-style classification theorem for all such valuations.

math.AT↗
arXiv2013

Axioms for the Lefschetz number as a lattice valuation

We give new axioms for the Lefschetz number based on Hadwiger's characterization of the Euler characteristic as the unique lattice valuation on polyhedra which takes value 1 on simplices. In the setting of maps on abstract simplicial complexes, we show that the Lefschetz number is unique with respect to a valuation axiom and an axiom specifying the value on a simplex. These axioms lead naturally to the classical computation of the Lefschetz number as a trace in homology. We then extend this approach to continuous maps of polyhedra, assuming an extra homotopy invariance axiom. We also show that this homotopy axiom can be weakened.

math.GN↗
arXiv2012

Generalizing the rotation interval to vertex maps on graphs

Graph maps that are homotopic to the identity and that permute the vertices are studied. Given a periodic point for such a map, a {\em rotation element} is defined in terms of the fundamental group. A number of results are proved about the rotation elements associated to periodic points in a given edge of the graph. Most of the results show that the existence of two periodic points with certain rotation elements will imply an infinite family of other periodic points with related rotation elements. These results for periodic points can be considered as generalizations of the rotation interval for degree one maps of the circle.

math.DS↗
arXiv2011

The Wecken property for random maps on surfaces with boundary

A selfmap is Wecken when the minimal number of fixed points among all maps in its homotopy class is equal to the Nielsen number, a homotopy invariant lower bound on the number of fixed points. All selfmaps are Wecken for manifolds of dimension not equal to 2, but some non-Wecken maps exist on surfaces. We attempt to measure how common the Wecken property is on surfaces with boundary by estimating the proportion of maps which are Wecken, measured by asymptotic density. Intuitively, this is the probability that a randomly chosen homotopy class of maps consists of Wecken maps. We show that this density is nonzero for surfaces with boundary. When the fundamental group of our space is free of rank n, we give nonzero lower bounds for the density of Wecken maps in terms of n, and compute the (nonzero) limit of these bounds as n goes to infinity.

math.AT↗
arXiv2011

A Nielsen theory for coincidences of iterates

As the title suggests, this paper gives a Nielsen theory of coincidences of iterates of two self maps f, g of a closed manifold. The ideas is, as much as possible, to generalize Nielsen type periodic point theory, but there are many obstacles. Many times we get similar results to the "classical ones" in Nielsen periodic point theory, but with stronger hypotheses.

math.AT↗
arXiv2011

Dynamics of random selfmaps of surfaces with boundary

We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps. In this sense, we show that "almost all" such selfmaps have periodic points of every period, and that in fact the number of periodic points of period n grows exponentially in n. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps. Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.

math.AT↗
arXiv2011

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.

math.GN↗
arXiv2011

Axioms for the coincidence index of maps between manifolds of the same dimension

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in $\Z\oplus \Z_2$. We also show in each setting that the group of values for the index (either $\Z$ or $\Z\oplus \Z_2$) is determined by the axioms. Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which charaterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds.

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