Celebratio Mathematica — Kister — Complete Bibliography

Complete Bibliography

Works connected to Jeff Cheeger

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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].