Celebratio Mathematica — Kuperberg — Complete Bibliography

[1] K. Kuperberg and W. Kuperberg: “On weakly zero-dimensional mappings,” Colloq. Math. 22 : 2 (1971), pp. 245–248. MR 281172 Zbl 0215.23904 article

[2] K. Kuperberg: “An isomorphism theorem of the Hurewicz-type in Borsuk’s theory of shape,” Fund. Math. 77 : 1 (1972), pp. 21–32. MR 324692 Zbl 0247.55008 article

[3] K. M. Kuperberg: The shape theory analogues of some classical isomophism theorems for homology and homotopy groups. Ph.D. thesis, Rice University, 1974. Advised by K. Borsuk and W. H. Jaco. MR 2624229 phdthesis

William H. Jaco	Related
Karol Borsuk	Related

[4] K. Kuperberg: “A note on the Hurewicz isomorphism theorem in Borsuk’s theory of shape,” Fund. Math. 90 : 2 (1975–1976), pp. 173–175. MR 394648 Zbl 0316.55012 article

In shape theory, the role of the homotopy groups \( \pi_n \) is played by the so-called fundamental groups \( \underline{\pi}{}_n \), introduced by K. Borsuk, and the homology groups which are useful there, are of the Vietoris–Čech type. The classical Hurewicz isomorphism theorem gives a connection between the homotopy groups \( \pi_n \) and the singular homology groups \( H_n \) with integral coefficients. An example of a compactum \( X \) is constructed, showing that there is no exact analogue of the Hurewicz theorem in shape theory. The example is simple: \( X \) is the double suspension of the 3-adic solenoid. The compactum \( X \) is arcwise connected and it has the following properties:

\( \underline{\pi}{}_q(X) \approx 0 \), for \( q=1 \), 2, 3, and
\( \underline{\pi}{}_4(X) \) and \( \check{H}_4(X) \) are not isomorphic.

[5] K. Kuperberg and W. Kuperberg: “An example concerning the cancellability of cycles,” Topology Proc. 4 : 1 (1979), pp. 133–137. Edited by Ross Geoghegan. MR 583696 Zbl 0447.55009 article

Włodzimierz Kuperberg	Related
Ross Geoghegan	Related

[6] K. Kuperberg, W. Kuperberg, and W. R. R. Transue: “On the 2-homogeneity of Cartesian products,” Fund. Math. 110 : 2 (1980), pp. 131–134. Dedicated to the memory and Ralph Bennett. MR 600586 Zbl 0475.54025 article

Włodzimierz Kuperberg	Related
William Raoul Reagle Transue	Related
Ralph Blount Bennett	Related

[7] K. Kuperberg: “A locally connected microhomogeneous nonhomogeneous continuum,” Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 : 11–12 (1980), pp. 627–630. MR 628653 Zbl 0467.54022 article

[8] K. Kuperberg and C. Reed: “A rest point free dynamical system on \( \mathbf{R}^3 \) with uniformly bounded trajectories,” Fund. Math. 114 : 3 (1981), pp. 229–234. MR 644408 Zbl 0508.58036 article

In this paper, we show that if \( \epsilon > 0 \), then there exists a \( C^{\infty} \) transformation \( G \) from \( R^3 \) into \( R^3 \) such that the unique solution \( \Phi \) to the differential equation \( y^{\prime} = G(y) \) is a dynamical system (a continuous transformation from \( R\times R^3 \) into \( R^3 \) such that \begin{align*} & \Phi(0,p) = p,\\ & \Phi(t_1,\Phi(t_2,p)) = \Phi(t_1+t_2,p), \text{ and}\\ & \tfrac{\partial}{\partial t}\Phi(0,p) = G(p) \end{align*} with the following two properties:

for each point \( p \) in \( R^3 \) and each number \( t \), \( \Phi(t,p) \) is in the \( \epsilon \)-neighborhood for \( p \); and
for each integer \( n\neq 0 \), \( \Phi(n,p) \neq p \).

Notice that the Scottish Book problem number 110 of Ulam follows a corollary where \( f(p) = \Phi(1,p) \) and the manifold is \( R^3 \).

[9] K. Kuperberg: “Homogeneity and twisted products,” Topology Proc. 13 : 2 (1988), pp. 237–248. MR 1053699 Zbl 0714.54005 article

A topological space \( X \) is said to be homogeneous if for every two points \( p \) and \( q \) in \( X \) there exists a homeomorphism \( \phi: X\to X \) such that \( \phi(p) = q \). A Cartesian product of homogeneous spaces is homogeneous. However, if at least one of the Cartesian factors is homeomorphic to the Menger curve \( M \), then the Cartesian product does not have some of the stronger homogeneity-type properties, see [Kennedy Phelps 1980], [Kuperberg et al. 1980] and [Patkowska 1984]. Even more interesting are continua which are not Cartesian products but whose every point has a neighborhood homeomorphic to a Cartesian product with one or more factors homeomorphic to \( M \), see [Kuperberg 1990].

In this paper, twisted products are obtained by making certain identifications on \( M\times M \), \( M\times I \), or \( M\times S^1 \). The construction yields continua whose every point has a homogeneous neighborhood but the space might not be homogenous, see [Kuperberg 1980]. It is shown here that twisted products of two Menger curves are not (with one obvious exception) homeomorphic to the Cartesian product \( M\times M \), but many twisted products of \( M \) and \( I \) are homeomorphic to \( M\times S^1 \).

[10] K. Kuperberg: “A homogeneous nonbihomogeneous continuum,” Topology Proc. 13 : 2 (1988), pp. 399–401. MR 1053707 Zbl 0719.54042 article

[11] K. Kuperberg: “Fixed points of orientation reversing homeomorphisms of the plane,” Topology Proc. 14 : 1 (1989), pp. 195–199. Research announcement for article published in Proc. Am. Math. Soc. 112:1 (1991). MR 1081127 Zbl 0725.54033 article

[12] K. M. Kuperberg and C. S. Reed: “A dynamical system on \( \mathbf{R}^3 \) with uniformly bounded trajectories and no compact trajectories,” Proc. Am. Math. Soc. 106 : 4 (August 1989), pp. 1095–1097. MR 965244 Zbl 0676.58047 article

[13] K. Kuperberg: “A nonparallel cylinder packing with positive density,” Mathematika 37 : 2 (1990), pp. 324–331. MR 1099780 Zbl 0721.52012 article

[14] K. Kuperberg: “On the bihomogeneity problem of Knaster,” Trans. Am. Math. Soc. 321 : 1 (1990), pp. 129–143. MR 989579 Zbl 0707.54025 article

[15] K. Kuperberg: “Fixed points of orientation reversing homeomorphisms of the plane,” Proc. Am. Math. Soc. 112 : 1 (1991), pp. 223–229. A research announcement was published in Topology Proc. 14:1 (1989). MR 1064906 Zbl 0722.55001 article

[16] K. Kuperberg: “A lower bound for the number of fixed points of orientation reversing homeomorphisms,” pp. 367–371 in The geometry of Hamiltonian systems (Berkeley, CA, 5–16 June 1989). Edited by T. Ratiu. Mathematical Sciences Research Institute Publications 22. Springer (New York), 1991. MR 1123283 Zbl 0737.55002 incollection

[17] K. M. Kuperberg, W. Kuperberg, P. Minc, and C. S. Reed: “Examples related to Ulam’s fixed point problem,” Topol. Methods Nonlinear Anal. 1 : 1 (1993), pp. 173–181. MR 1215264 Zbl 0787.54041 article

Włodzimierz Kuperberg	Related
Piotr Minc	Related
Coke Stevenson Reed	Related

[18] K. Kuperberg: “A smooth counterexample to the Seifert conjecture,” Ann. Math. (2) 140 : 3 (November 1994), pp. 723–732. MR 1307902 Zbl 0856.57024 article

[19] K. Kuperberg and W. Kuperberg: “Translates of a starlike plane region with a common point,” pp. 205–216 in Intuitive geometry (Szeged, Hungary, 2–7 September 1991)). Edited by K. Böröczky and G. Fejes Tóth. Colloquia Mathematica Societatis Janos Bolyai 63. North-Holland (Amsterdam), 1994. MR 1383627 Zbl 0823.52005 incollection

Włodzimierz Kuperberg	Related
Gábor Fejes Tóth	Related
Károly J. Böröczky	Related

[20] É. Ghys: “Construction de champs de vecteurs sans orbite périodique (d’après Krystyna Kuperberg)” [Construction of vector fields without periodic orbits (after Krystyna Kuperberg)], pp. 283–307 in Séminaire Bourbaki: Volume 1993/94. Astérisque 227. Société Mathématique de France, (Paris), 1995. Exposé no. 785. MR 1321651 Zbl 0846.57019 incollection

[21] Continua (Cincinnati, OH, 12–15 January 1994). Edited by H. Cook, W. T. Ingram, K. T. Kuperberg, A. Lelek, and P. Minc. Lecture Notes in Pure and Applied Mathematics 170. Dekker (New York), 1995. This interpolates the “Houston Problem Book,” which includes a set of 200 problems accumulated over several years at the University of Houston. MR 1326830 Zbl 0813.00008 book

Howard Cook	Related
William Thomas Ingram, III	Related
Andrew Lelek	Related
Piotr Minc	Related

[22] K. M. Kuperberg, W. Kuperberg, and W. R. R. Transue: “Homology separation and 2-homogeneity,” pp. 287–295 in Continua (Cincinnati, OH, 12–15 January 1994). Edited by H. Cook, W. T. Ingram, K. T. Kuperberg, A. Lelek, and P. Minc. Lecture Notes in Pure and Applied Mathematics 170. Dekker (New York), 1995. MR 1326851 Zbl 0826.54026 incollection

The homology separation axiom is introduced to investigate homogeneity properties of Cartesian products of continua. We prove that under certain conditions of a homological nature the products are factorwise rigid. Also, if the product \( X\times Y \) of an \( n \)-dimensional representable continuum \( X \) and a non-degenerate continuum \( Y \) is 2-homogeneous, then \( X \) is locally \( n \)-acyclic. In particular, the product \( B\times Y \) of a \( \mu^n \)-manifold \( B \) (based on the \( n \)-dimensional Menger universal compactum \( \mu^n \)) with any non-degenerate continuum \( Y \) is not 2-homogeneous. These theorems generalize a previous result of the same authors and some results of J. Kennedy Phelps.

Włodzimierz Kuperberg	Related
Howard Cook	Related
William Raoul Reagle Transue	Related
William Thomas Ingram, III	Related
Andrew Lelek	Related
Piotr Minc	Related

[23] A. Bezdek, K. Kuperberg, and W. Kuperberg: “Mutually contiguous translates of a plane disk,” Duke Math. J. 78 : 1 (April 1995), pp. 19–31. MR 1328750 Zbl 0829.52008 article

Włodzimierz Kuperberg	Related
Andras Bezdek	Related

[24] W. Hurewicz: Collected works of Witold Hurewicz. Edited by K. Kuperberg. American Mathematical Society (Providence, RI), 1995. With contributions by Ryszard Engelking, Roman Pol, Edward Fadell, Solomon Lefschetz and Samuel Eilenberg. MR 1362795 Zbl 0831.01017 book

Witold Hurewicz	Related
Solomon Lefschetz	Related
Edward Richard Fadell	Related
Samuel Eilenberg	Related
Ryszard Engelking	Related
Roman Pol	Related

[25] G. Kuperberg and K. Kuperberg: “Generalized counterexamples to the Seifert conjecture,” Ann. Math. (2) 143 : 3 (May 1996), pp. 547–576. A corrected version was published in Ann. Math. 144:2 (1996). MR 1394969 article

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere \( S^3 \) has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on \( S^3 \) with no singular points has a periodic trajectory.

[26] G. Kuperberg and K. Kuperberg: “Generalized counterexamples to the Seifert conjecture,” Ann. Math. (2) 144 : 2 (September 1996), pp. 239–268. This is a corrected version of the article originally published in Ann. Math. 143:3 (1996). MR 1418899 Zbl 0856.57026 ArXiv math/9802040 article

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere \( S^3 \) has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on \( S^3 \) with no singular points has a periodic trajectory.

[27] K. Kuperberg: “Bihomogeneity and Menger manifolds,” pp. 175–184 in Proceedings of the international conference on set-theoretic topology and its applications, part 2 (Matsuyama, Japan, 12–16 December 1994), published as Topology Appl. 84 : 1–3. Issue edited by T. Nogura. Elsevier (Amsterdam), April 1998. MR 1611226 Zbl 0995.54029 ArXiv math/9804085 incollection

[28] K. Kuperberg: “Counterexamples to the Seifert conjecture,” pp. 831–840 in Proceedings of the International Congress of Mathematicians, volume 2: Invited lectures (Berlin, 18–27 August 1998), published as Doc. Math. Extra Volume. Issue edited by G. Fischer and U. Rehmann. Deutsche Mathematiker-Vereinigung (Berlin), 1998. Dedicated to the author’s son Greg. MR 1648130 Zbl 0924.58086 incollection

Ulf Rehmann	Related
Gregory John Kuperberg	Related
Gerd Fischer	Related

[29] M. Barge and K. Kuperberg: “Periodic points from periodic prime ends,” pp. 13–21 in Proceedings of the 1998 topology and dynamics conference (Fairfax, VA, 12–14 March 1998), vol. 23. Edited by G. Gruenhage. Topology Proc. Spring. Auburn University, 1998. MR 1743798 Zbl 0976.54047 incollection

This paper generalizes a theorem of Barge and Gillette asserting that if an orientation preserving plane homeomorphism \( F \) has a fixed prime end associated with an invariant continuum \( \Delta \) which separates the plane into exactly two domains, then \( F \) has a fixed point in \( \Delta \). The generalization goes in two directions. The Barge–Gillette theorem is proved for a continuum with more than two complementary domains if for all but one complementary domains \( U \), \( F \) has a fixed prime end in \( U \). The other generalization addresses the existence of periodic points with least period \( q \) provided certain conditions concerning \( F^q \) and periodic prime ends with the same least period are met.

Gary Gruenhage	Related
Marcy Mason Barge	Related

[30] K. Kuperberg: “A knotted minimal tree,” Commun. Contemp. Math. 1 : 1 (1999), pp. 71–86. MR 1670920 Zbl 0958.57002 ArXiv math/9806080 article

[31] K. Kuperberg, W. Kuperberg, J. Matoušek, and P. Valtr: “Almost-tiling the plane by ellipses,” Discrete Comput. Geom. 22 : 3 (1999), pp. 367–375. MR 1706602 Zbl 0952.52016 ArXiv math/9804040 article

Jíří Matoušek	Related
Włodzimierz Kuperberg	Related
Pavel Valtr	Related

[32] K. Kuperberg: “Aperiodic dynamical systems,” Notices Am. Math. Soc. 46 : 9 (October 1999), pp. 1035–1040. MR 1710669 Zbl 0936.37039 article

[33] Geometry and topology in dynamics (Winston-Salem, NC, 9–10 October 1998 and San Antonio, TX, 13–16 January 1999). Edited by M. Barge and K. Kuperberg. Contemporary Mathematics 246. American Mathematical Society (Providence, RI), 1999. MR 1732367 Zbl 0930.00046 book

[34] I. Bárány, K. Kuperberg, and T. Zamfirescu: “Total curvature and spiralling shortest paths,” pp. 167–176 in U.S.-Hungarian workshops on discrete geometry and convexity (Budapest, 12–16 July 1999 and Auburn, AL, 21–26 March 2000), published as Discrete Comput. Geom. 30 : 2. Issue edited by G. Fejes Tóth and W. Kuperberg. Springer (New York), August 2003. MR 2007957 Zbl 1043.52004 ArXiv math/0301338 incollection

This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in \( \mathbb{R}^3 \) cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an example is constructed to show that the total curvature of a shortest path on the boundary of a convex polyhedron in \( \mathbb{R}^3 \) can exceed \( 2\pi \). Another example shows that the spiralling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large.

Gábor Fejes Tóth	Related
Tudor I. Zamfirescu	Related
Włodzimierz Kuperberg	Related
Imre Bárány	Related

[35] Workshop: “Geometric theory of dynamical systems” (Krakow, Poland, 19–22 June 2002), published as Univ. Iagel. Acta Math. 41. Issue edited by K. Kuperberg, A. Pelczar, and R. Srzednicki. Jagiellonian University (Krakow, Poland), 2003. MR 2084749 Zbl 1359.00032 book

Roman Srzednicki	Related
Andrzej Pelczar	Related

[36] G. Kuperberg, K. Kuperberg, and W. Kuperberg: “Lattice packings with gap defects are not completely saturated,” Beitr. Algebra Geom. 45 : 1 (2004), pp. 267–273. MR 2070648 Zbl 1054.52010 ArXiv math/0303366 article

Włodzimierz Kuperberg	Related
Gregory John Kuperberg	Related

[37] K. Kuperberg: “2-wild trajectories,” Discrete Contin. Dyn. Syst. supplement (2005), pp. 518–523. MR 2192710 article

[38] A. Clark, J. Kennedy, and K. Kuperberg: “Preface,” pp. 154 in Proceedings of the US-Polish international workshop on geometric methods in dynamical systems (Newark, DE, 9–12 June 2004), published as Topology Appl. 154 : 13. Issue edited by A. Clark, J. Kennedy, and K. Kuperberg. Elsevier (Amsterdam), July 2007. MR 2332865 incollection

Judy Anita Kennedy	Related
Alexander Douglass Clark	Related

[39] Proceedings of the US-Polish international workshop on geometric methods in dynamical systems (Newark, DE, 9–12 June 2004), published as Topology Appl. 154 : 13. Issue edited by A. Clark, J. Kennedy, and K. Kuperberg. Elsevier (Amsterdam), July 2007. Zbl 1121.57001 book

Judy Anita Kennedy	Related
Alexander Douglass Clark	Related

[40] K. Kuperberg and K. Gammon: “A short proof of nonhomogeneity of the pseudo-circle,” Proc. Am. Math. Soc. 137 : 3 (2009), pp. 1149–1152. MR 2457457 Zbl 1158.54014 ArXiv 0803.1139 article

[41] A. Trybulec, A. Kornilowicz, A. Naumowicz, and K. Kuperberg: “Formal mathematics for mathematicians,” pp. 119–121 in Formal mathematics for mathematicians (New Orleans, LA, 6–9 January 2011), published as J. Automat. Reason. 50 : 2. Issue edited by A. Trybulec, A. Kornilowicz, A. Naumowicz, and K. Kuperberg. Springer (Berlin), 2013. Foreword to a special issue. MR 3016795 incollection

Andrzej Trybulec	Related
Adam Naumowicz	Related
Artur Korniłowicz	Related

[42] K. Kuperberg: “Periodicity generated by adding machines.” Edited by A. K. M. Libardi, M. Golasiński, V. V. Sharko, and S. Spiejz. Zb. Pr. Inst. Mat. NAN Ukr. 10 : 6 (2013), pp. 140–147. Zbl 1313.37007 article

Marek Golasiński	Related
Alice Kimie Miwa Libardi	Related
Volodymyr Vasylyovych Sharko	Related
Stanisław Spież	Related

[43] K. T. Kuperberg: “Andrzej Trybulec–in memoriam,” J. Automat. Reason. 55 : 3 (2015), pp. 187–190. MR 3402512 Zbl 1337.01014 article

[44] K. Kuperberg: “Two Vietoris-type isomorphism theorems in Borsuk’s theory of shape, concerning the Vietoris–Cech homology and Borsuk’s fundamental groups,” Chapter 22, pp. 285–313 in Studies in topology (Charlotte, NC, 14–16 March 1974). Edited by N. M. Stavrakas and K. R. Allen. Academic Press, New York. MR 383398 Zbl 0323.55021 incollection

In the realm of Borsuk’s theory of shape, where fundamental sequences play the role of mappings (see [Borsuk 1968] for definitions), we prove two isomorphism theorems of the Vietoris-type: one for Borsuk’s fundamental groups and one for Vietoris homology groups. In order to forumlate such theorems in shape theory, one has to introduce first the notion of a fundamental sequence onto (corresponding to a mapping onto) and a condition corresponding to Vietoris’ assumption on the acyclicity of the inverse images of points. Considering the special case of fundamental sequences generated by mappings, we get in Section 2 the following theorem: If \( f \) is a mapping of a pointed compactum \( (X,x_0) \) onto a pointed compactum \( (Y,y_0) \) such that \( f^{-1}(y) \) is approximately \( q \)-connected for each \( q = 0 \), 1, \( \dots, n \) and for each \( y\in Y \), then \( f \) induces an isomorphism between the \( n \)-th fundamental groups \( \underline{\pi}{}_n(X,x_0) \) and \( \underline{\pi}{}_n(Y,y_0) \). As a similar corollary for homology, we get the classical Vietoris isomorphism theorem (see [Begle 1950] and [Vietoris 1927]). A partial generalization of Smale’s Vietoris theorem for homotopy (see [Smale 1957]) is given in Section 3 as a further application of our theorem for Borsuk’s fundamental groups.

Keith Roy Allen	Related
Nicholas Mark Stavrakas	Related