Celebratio Mathematica

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

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

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.

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 \( R^3 \).

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

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

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

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

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

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

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.

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

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

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

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

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Krystyna Kuperberg

Krystyna Maria Trybulec Kuperberg

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