The Recognition Problem : What Is a Topological Manifold ?

Abstract

setting, difficult to come by. A good solution probably should not involve the notion of homogeneity (see Supplement 5) since, in applications, the spaces constructed which are to be checked are obviously manifolds at some points, so that recognizing homogeneity is precisely the difficulty. Finally, a satisfactory solution should allow one to solve problems of independent interest. Before spring of this year (1977) no conjectured characterization of topological manifolds seemed to have a clear cut advantage over any other. But the situation has changed rapidly so that we can make the following conjecture with some confidence. 1.3. CONJECTURE. A topological «-manifold may be characterized as a generalized «-manifold satisfying a minimal amount of general position. (Definitions follow.) Prerequisites for understanding the conjecture in particular and the paper in general include a knowledge of basic homology theory (as presented, for example, in [33]) and basic PL topology (as presented, for example, in [57]). A good introduction to the particular point of view that we shall pursue (concerning tameness and wildness) appears in [19]. Nevertheless, even without those prerequisites the reader will probably understand and enjoy some of the historical material in §§2 and 3 and the discussion of Antoine's necklace in §§5 and 6. 1.4. DEFINITION. A generalized «-manifold M is a Euclidean neighborhood retract (ENR) (=>= retract of an open subset of some Euclidean space E) with the local homology groups at each point of Euclidean «-space E: H+(M, M ~{x};Z) s H*(E9 E n (0};Z) (for each x E M). The probable appropriate general position condition for « > 5 is the disjoint disk property. 1.4'. DEFINITION. A space M satisfies the disjoint disk property if arbitrary maps ƒ, g: B-+ M from the 2-dimensional disk B into M can be approximated by maps/ , g': B -» M with ƒ'(B) n g\B) = 0 . The conjecture, as completed by Definitions 1.4 and 1.4', was proved during the spring of this year (1977) for a large class of generalized manifolds by J. W. Cannon [22] and Cannon, J. L. Bryant, and R. C. Lacher [23] (see Supplement 4). The fertile source of generalized manifolds supplied by cell-like upper semicontinuous decompositions of manifolds was then, for « > 5, completely mastered by R. D. Edwards [36] (see Supplement 4); his result confirmed the conjecture for all cell-like decompositions of «-manifolds, « > 5. An infinite dimensional analogue of the conjecture for Q manifolds was proved early in the year by H. Torunczyk [65] (see Supplement 3). An easy consequence of the work, and one of its great motivations, is the famous double suspension conjecture: 1.5. THEOREM. The double suspension S//" of any homology «-sphere is homeomorphic with the (« + 2)-sphere S*. (A homology «-sphere H is an n-manifold satisfying H^(H;Z) s H+(S;Z); the kth suspension of a space is the join of that space with the (k — 1)-sphere S~; see Definition 1.6 for the definition of a sphere.)

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@inproceedings{Cannon2007TheRP, title={The Recognition Problem : What Is a Topological Manifold ?}, author={James W. Cannon}, year={2007} }