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].

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James M. Kister

James Milton Kister

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