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: “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 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,” Ann. of Math. (2) 80 (1964), pp. 190–199. MR 0180986 Zbl 0131.20602 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].

year	title	people

			clear

James M. Kister

James Milton Kister

Filter the Bibliography List