David W. Henderson

Infinite-dimensional manifolds are open subsets of Hilbert space

dimensional Hilbert space, H, then M can be embedded as an open subset of H. Each infinite-dimensional separable Frechet space (and therefore each infinitedimensional separable Banach space) is homeomorphic to H. (See [I].) We shall use “F-manifold ” to denote “ metric manifold modeled on a separable infinite-dimensional Frenchet space “. Thus we have COROLLARY 1. Each separable F-matzifold ccl/z be embedded as an open subset of H. Recent results of Eells and Elworthy [6] and Kuiper and Burghelea [lo] and Moulis [14] combine to show (see [5]) that every homotopy equivalence between C”-Hilbert manifolds is homotopic to a C” diffeomorphism. Since open subsets of H have an induced C” structure, we have

Corrections and extensions of two papers about infinite-dimensional manifolds

Abstract In this paper we correct and extend two papers which were written jointly by the author and R. Schori and J. West and which prove or announce several classification theorems for infinite-dimensional topological manifolds.

Experiencing Meanings in Geometry

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences, and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the products of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth - the very experience out of which Love rises. (Plotinus, The Enneads, II.9.16; 1991, p. 129)

A simplicial complex whose product with any ANR is a simplicial complex

Abstract If X is a finite-dimensional, complete, metric AR (or ANR), then X×R∞ is R∞ (or an open subset of R∞).

SELF-UNLINKED SIMPLE CLOSED CURVES

interiors of a collection of disjoint 3-cells each of diameter less than S. A set X is locally tame at p if p has a closed neighborhood in X which is a tame complex in M. If X is not locally tame at p then p is a wild point of X. A set is called nicely wild if the union of its wild points is a tame 0-dimensional set. For J an arc or scc we make the following definitions, the first of which is used in [1]. The penetration index P(J,x) of J at a point x E J is the smallest cardinal number n such that there are arbitrarily small 2-spheres enclosing x and containing no more than n points of J. The penetration index P(J) of J is the least upper bound of the cardinal numbers P(J, x), for all x E J. If J is nicely wild, then the nice penetration index NP(J) of J is the smallest integer n such that, for every E > 0, the set of wild points of J can be covered by the interiors of a collection of disjoint 3-cells each with diameter less than s and such that the boundary of each 3-cell intersects J in no more than n points. (The union of members of this collection is called a taming s-set of J of index n.) CONJECTURE. There is a nicely wild scc J such that NP(J) # P(J).

How to Use History to Clarify Common Confusions in Geometry

Most people judge the size of cities simply from their circumference. So that when one says that Megalopolis is fifty stades in contour and Sparta forty-eight, but that Sparta is twice as large as Megalopolis, what is said seems unbelievable to them. And when in order to puzzle them even more, one tells them that a city or camp with the circumference of forty stades may be twice as large as one of the circumference of which is one hundred stades, what is said seems to them absolutely astounding. The reason of this is that we have forgotten the lessons in geometry we learnt as children. -Polybius, The Histories, 9.26a, 2 century B.C. (from the English translation by W.R. Paton, Cambridge, MA: Harvard University Press, 1922-1927)