37 research outputs found
Fixed poin sets in digital topology, 1
No full text[EN] 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(Cn) where Cn 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 Cn, in which F(X) does not equal
{0, 1, . . . , #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., for some digital images the fixed point set is always
connected.Boxer, L.; Staecker, PC. (2020). Fixed poin sets in digital topology, 1. Applied General Topology. 21(1):87-110. https://doi.org/10.4995/agt.2020.12091OJS87110211C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Continuous maps on digital simple closed curves, Applied Mathematics 1 (2010), 377-386. https://doi.org/10.4236/am.2010.15050L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, Fixed points and freezing sets in digital topology, Proceedings, 2019 Interdisciplinary Colloquium in Topology and its Applications, in Vigo, Spain; 55-61.L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474J. Haarmann, M. P. Murphy, C. S. Peters and P. C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302. https://doi.org/10.1007/s10851-015-0578-8B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 18 (1983). https://doi.org/10.1090/conm/014E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics (1987), 227-234.A. Rosenfeld, "Continuous" functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6P. C. Staecker, Some enumerations of binary digital images, arXiv:1502.06236, 2015
Digital homotopy relations and digital homology theories
No full text[EN] 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 -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.Staecker, PC. (2021). Digital homotopy relations and digital homology theories. Applied General Topology. 22(2):223-250. https://doi.org/10.4995/agt.2021.13154OJS223250222H. Arslan, I. Karaca and A. Öztel, Homology groups of n-dimensional digital images, in: Turkish National Mathematics Symposium XXI (2008), 1-13.L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10, no. 1 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Appl. Gen. Topol. 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, I. Karaca and A. Öztel, Topological invariants in digital images, J. Math. Sci. Adv. Appl. 11, no. 2 (2011), 109-140.L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Appl. Gen. Topol. 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474O. Ege and I. Karaca, Fundamental properties of digital simplicial homology groups, American Journal of Computer Technology and Application 1 (2013), 25-41.S.-E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171, no. 1-3 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.S. S. Jamil and D. Ali, Digital Hurewicz theorem and digital homology theory, arxiv eprint 1902.02274v3.T. Kaczynski, K. Mischaikow and M. Mrozek, Computing homology. Algebraic topological methods in computer science (Stanford, CA, 2001), Homology Homotopy Appl. 5, no. 2 (2003), 233-256. https://doi.org/10.4310/HHA.2003.v5.n2.a8I. Karaca and O. Ege, Cubical homology in digital images, International Journal of Information and Computer Science, 1 (2012), 178-187.D. W. Lee, Digital singular homology groups of digital images, Far East Journal of Mathematics 88 (2014), 39-63.G. Lupton, J. Oprea and N. Scoville, A fundamental group for digital images, preprint.W. S. Massey, A Basic Course in Algebraic Topology,Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4939-9063-4A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-
A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles
No full textWe give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups
Typical elements in free groups are in different doubly-twisted conjugacy classes
No full textAbstractWe give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random
Nielsen equalizer theory
No full textAbstractWe 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
On the uniqueness of the coincidence index on orientable differentiable manifolds
No full textAbstractThe fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous mappings on oriented differentiable manifolds, the most common setting for Nielsen coincidence theory
Remnant inequalities and doubly-twisted conjugacy in free groups
No full textAbstractWe give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms φ and ψ such that certain remnant words from φ are longer than the images of generators under ψ.Our first result is a remnant inequality condition which implies that two words u and v are not doubly-twisted conjugate. Further we show that if ψ is given and φ, u, and v are chosen at random, then the asymptotic probability that u and v are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if φ, u, and v are chosen at random, then u and v are not in the same singly-twisted conjugacy class with asymptotic probability 1.Our second result generalizes Kim’s “bounded solution length”. We give an algorithm for deciding doubly-twisted conjugacy relations in the case where φ and ψ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if φ has remnant words of length at least 2.As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of the sets of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism
Maps on graphs can be deformed to be coincidence-free
No full textWe give constructions which can remove coincidence points of mappings on bouquets of circles by changing the maps by homotopies. When the codomain is a bouquet of at least 2 circles, we show that any pair of maps can be changed by homotopies to be coincidence free. This allows us to demonstrate that there can be no function on bouquets of circles which satisfies the natural properties expected of a coincidence index: additivity, homotopy invariance, and agreement with the fixed point index for selfmaps. Consequently, the Nielsen coincidence number and coincidence Reidemeister trace are not well-defined and the results of our previous paper ``A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles are invalid
Axioms for the fixed point index of n-valued maps, and some applications
No full textWe 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
