Celebratio Mathematica

J. Kister, P. Stein, S. Ulam, W. Walden, and M. Wells: “Experiments in chess,” Jour. Assoc. Computing Machinery 4 (1957), pp. 174–177. article

Stanisław Marcin Ulam	Related
Mark Brimhall Wells	Related
Paul Stein	Related

J. Kister: “Small isotopies in Euclidean spaces and 3-manifolds,” Bull. Am. Math. Soc. 65 : 6 (1959), pp. 371–373. MR 107232 Zbl 0089.39502 article

J. M. Kister: Isotopies in manifolds. Ph.D. thesis, University of Wisconsin, Madison, 1959. Advised by R. H. Bing. MR 2612903 phdthesis

J. M. Kister: “Isotopies in 3-manifolds,” Trans. Am. Math. Soc. 97 : 2 (November 1960), pp. 213–224. MR 120628 Zbl 0096.37906 article

In the years immediately following the elegant proofs of Alexander [1923] and Kneser [1926] on the realization of homeomorphisms through deformations, relatively little work on isotopic deformations appeared. The reason lies, no doubt, in the inherent difficulty of constructing homeomorphisms. One needs to know a good deal about the structure of the space, preferably linear structure, before constructing or extending homeomorphisms. Also the space needs to have rather strong homogeneity conditions, like those of a manifold, before isotopy considerations become fruitful. These restrictions are severe, especially when, in several respects, the questions that arise become interesting only in three or more dimensions. With recent increases in the knowledge of 3-manifolds such questions have become approachable, and therefore it is, perhaps, not unnatural to find a number of papers devoted to the theme of isotopy appearing at this time (e.g. see [Sanderson 1959, 1960; Fisher 1960; Hamstrom 1958]).

The main result of this paper is the following:

Suppose \( M \) is a 3-manifold with boundary having triangulation \( \Sigma \) and \( \rho \) is the natural metric for \( \Sigma \). Then for any \( \epsilon > 0 \) there is a \( \delta > 0 \) so that if \( f \) and \( g \) are homeomorphisms of \( M \) onto itself and \[ \rho(f(x),g(x)) < \delta \] for all \( x \) in \( M \), there is an \( \epsilon \)-isotopy of \( M \) taking \( g \) onto \( f \).

J. M. Kister: “Examples of periodic maps on Euclidean spaces without fixed points,” Bull. Am. Math. Soc. 67 : 5 (1961), pp. 471–474. MR 130929 Zbl 0101.15602 article

Let \( T \) be a map of period \( r \) on a Euclidean space \( E^n \). Smith seems to have been the first to consider fixed points of \( T \). He showed that \( T \) has a fixed point if \( r \) is a prime in [Smith 1934], extended this result to \( r \) a power of a prime, and raised the question concerning the existence of a fixed point for \( r \) not a prime power in [Smith 1941]; also cf. Problem 33 in [Eilenberg 1949]. Conner and Floyd gave an example of a contractible manifold \( M_r \) for every \( r \) not a prime power, and a map \( T \) of period \( r \) on \( M_r \) without fixed points [Conner and Floyd 1959]. They conjectured that \( M_r \) was a Euclidean space. This note shows that a slight modification of their example is Euclidean, hence:

If \( r \) is an integer which is not a power of a prime, then there exists a triangulation \( \tau \) of \( E^{9r} \), a map \( T \) of period \( r \) on \( E^{9r} \) without fixed points, and \( T \) is simplicial relative to \( \tau \).

J. M. Kister and L. N. Mann: “Isotropy structure of compact Lie groups on complexes,” Mich. Math. J. 9 : 1 (1962), pp. 93–96. MR 132120 Zbl 0111.35702 article

In this paper we prove the following conjecture of Floyd [Borel et al. 1960, p. 95]:

If \( G \) is a compact Lie group operating on a finite complex \( K \), then there are only finitely many distinct conjugate classes of isotropy subgroups.

In the proof we use a decomposition of \( K \) into a finite number of invariant open manifolds, each of which has an orientable covering manifold whose integral cohomology (with compact supports) is finitely generated. Lifting the action of \( G \) to the covering manifolds, we apply a result of Mann [1962] to establish the theorem. The theorem is false when \( K \) is locally-finite complex having finitely generated integral cohomology. To see this, consider the 2-complex consisting of a line and a sequence of closed discs, with centers on the line and going off to infinity. Define an action of the circle group \( S^1 \) on this complex by defining \( \theta \) in \( S^1 \) (\( 0 < \theta < 2\pi \)) to act as the rotation \( j\theta \) on the \( j \)th disc. Then \( Z_j \) (the subroup of \( S^1 \) isomorphic to the integers, modulo \( j \)) leaves the \( j \)th disc point-wise fixed, and therefore \( Z_j \) is an isotropy subgroup for each \( j \). On the other hand, the one-point compactification of this complex is of the same homotopy type as the circle, and therefore the finitely generated integral cohomology condition is satisfied.

J. M. Kister: “Uniform continuity and compactness in topological groups,” Proc. Am. Math. Soc. 13 : 1 (1962), pp. 37–40. MR 133392 Zbl 0103.01604 article

A topological group \( G \) will be said to have property \( U \) if every continuous real-valued function \( f \) on \( G \) is uniformly continuous (i.e. for every \( \epsilon > 0 \) there is a neighborhood \( N \) of the identity in \( G \) such that if \( x \) and \( y \) are in \( G \) with \( xy^{-1} \) in \( N \) then \( |f(x) - f(y)| < \epsilon \)). It is well known that any compact topological group has property \( U \) and clearly any discrete group has property \( U \). It is natural to conside whether the converse holds: is every topological group having property \( U \) compact or discrete?

This note shows that the answer in general is in the negative, but that affirmative results in this direction hold even in uniform spaces, and if one assumes the group is locally compact then the converse is true.

J. M. Kister: “A theorem on infinite regular neighborhoods and an application to periodic maps on \( E^n \),” pp. 221–222 in Topology of 3-manifolds and related topics (Athens, GA, 14 August–8 September 1961). Edited by M. K. Fort. Prentice-Hall (Englewood Cliffs, NJ), 1962. MR 140105 Zbl 1246.57067 incollection

J. M. Kister: “Questions on isotopies in manifolds,” pp. 229–230 in Topology of 3-manifolds and related topics (Athens, GA, 14 August–8 September 1961). Edited by M. K. Fort. Prentice-Hall (Englewood Cliffs, NJ), 1962. MR 140106 Zbl 1246.57065 incollection

J. M. Kister and L. N. Mann: “Equivariant imbeddings of compact abelian Lie groups of transformations,” Math. Ann. 148 : 2 (April 1962), pp. 89–93. MR 141727 Zbl 0209.44104 article

J. M. Kister and D. R. McMillan, Jr.: “Locally euclidean factors of \( E^4 \) which cannot be imbedded in \( E^3 \),” Ann. Math. (2) 76 : 3 (November 1962), pp. 541–546. MR 144322 Zbl 0115.40703 article

J. M. Kister: “Differentiable periodic actions on \( E^8 \) without fixed points,” Am. J. Math. 85 : 2 (April 1963), pp. 316–319. MR 154278 Zbl 0119.18801 article

J. M. Kister: “Microbundles are fibre bundles,” Bull. Am. Math. Soc. 69 : 6 (1963), pp. 854–857. An expanded version of this was published in Ann. Math. 80:1 (1964). MR 156359 Zbl 0117.16701 article

R. H. Bing and J. M. Kister: “Taming complexes in hyperplanes,” Duke Math. J. 31 : 3 (1964), pp. 491–511. MR 0164329 Zbl 0124.16701 article

In this paper we investigate conditions that suffice for an imbedding of a complex in a Euclidean space to be tame. We also consider the more general question of when two imbeddings are equivalent. We show, for example, with a dimension restriction that if the image under an imbedding of a \( k \)-complex is contained in a hyperplane of codimension \( k \), then the imbedding is tame. More precisely we prove:

Let \( K \) be a finite \( k \)-dimensional complex and \( h \) an imbedding of \( K \) into an \( n \)-plane \( E^n \) in \( E^{n+k} \), where \( k+2\leq n \). Let \( \varepsilon \) be any positive number. Then there exists an isotopy \( G_t \) (\( t\in I \)) of \( E^{n+k} \) onto itself such that

\( g_0 \) is the identity,
\( g_1 h \) is piecewise linear,
\( g_t \) is the identity outside an \( \varepsilon \)-neighborhood of \( h(K) \) for each \( t \) in \( I \),
each point of \( E^{n+k} \) moves along a polygonal path under \( g_t \) (\( t\in I \)) having length less than \( \varepsilon \).

J. M. Kister: “Microbundles are fibre bundles,” Ann. of Math. (2) 80 (1964), pp. 190–199. MR 0180986 Zbl 0131.20602 article

J. M. Kister: “Inverses of Euclidean bundles,” Mich. Math. J. 14 : 3 (1967), pp. 349–352. To Raymond L. Wilder on his seventieth birthday. MR 212830 Zbl 0149.40903 article

For each Euclidean bundle or microbundle it is useful to find another bundle of the same type, called an inverse bundle, such that the Whitney sum of the two is a trivial bundle. Milnor in [4] ingeniously showed how to construct an inverse to a microbundle over a finite-dimensional, locally finite, simplicial complex. Here we give a short and elementary proof of the existence of inverses for Euclidean bundles over paracompact spaces having a finiteness condition. This contains Milnor’s result, since one may regard a microbundle as a Euclidean bundle [2]. Hirsch [1] has also developed a new proof of the existence of the inverse of a bundle over a polyhedron, in his work on the stable existence and stable isotopy of normal microbundles.

J. M. Kister: “Homotopy types of ANR’s,” Proc. Am. Math. Soc. 19 : 1 (February 1968), pp. 195. MR 219031 Zbl 0161.42602 article

J. Cheeger and J. M. Kister: “Counting topological manifolds,” Topology 9 : 2 (May 1970), pp. 149–151. MR 256399 Zbl 0199.58403 article

We consider the class \( \mathscr{C} \) of all compact topological manifolds, boundaries permitted. It is known that there are only a countable number of homotopy types in \( \mathscr{C} \), [Mather 1965] and [Kister 1968]. The subclass \( \mathscr{C}_{PL} \), of piecewise linear manifolds has only a countable number of topological distinct elements, since each could be regarded as a finite simplicial complex and a simple argument shows there are only a countable number of those, up to isomorphism. Recently, however, Kirby and Siebenmann [1969] have discovered some examples of topological manifolds admitting no \( PL \) structure, so that route for counting homeomorphism types in \( \mathscr{C} \) is rather unpromising (whether manifolds can be triangulated without a \( PL \) structure is still open).

We take a direct approach and with the aid of a very useful result of Edwards and Kirby [1971] examine overlapping coordinate neighborhoods to show that \( \mathscr{C} \) has only countably many elements up to homeomorphism. Our argument is similar to one in a differential setting in [Kirby and Siebenmann 1969].

M. Brown and J. M. Kister: “Invariance of complementary domains of a fixed point set,” Proc. Amer. Math. Soc. 91 : 3 (1984), pp. 503–504. MR 744656 Zbl 0547.57010

A. Blass and J. M. Kister: “Free subgroups of the homeomorphism group of the reals,” pp. 243–252 in Special volume in honor of R. H. Bing (1914–1986) (San Marcos, TX, November 1984), published as Topology Appl. 24 : 1–3. Issue edited by S. Singh and T. L. Thickstun. Elsevier (Amsterdam), December 1986. MR 872496 Zbl 0604.57015 incollection

R H Bing	Related	Volume
Sukhjit Singh	Related
Thomas Lusk Thickstun	Related
Andreas Raphael Blass	Related

James M. Kister

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