hilbert cube manifold
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2018 ◽  
Vol 12 (04) ◽  
pp. 1073-1101
Author(s):  
Shijie Gu ◽  
Craig R. Guilbault

This paper is concerned with compactifications of high-dimensional manifolds. Siebenmann’s iconic 1965 dissertation [L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton Univ. (1965), MR 2615648] provided necessary and sufficient conditions for an open manifold [Formula: see text] ([Formula: see text]) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where [Formula: see text] is noncompact with compact boundary; however, when [Formula: see text] is noncompact, the situation is more complicated. The goal becomes a “completion” of [Formula: see text], i.e. a compact manifold [Formula: see text] containing a compactum [Formula: see text] such that [Formula: see text]. Siebenmann did some initial work on this topic, and O’Brien [G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl. 16 (1983) 303–324, MR 722123] extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here, we provide such a characterization. Our second main theorem involves [Formula: see text]-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann [T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171–208, MR 0425973] guarantee [Formula: see text]-compactifiability for a manifold [Formula: see text]. We cannot answer that question, but we do show that those conditions are satisfied if and only if [Formula: see text] is [Formula: see text]-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem — an application that partly explains our current interest in that topic, and also illustrates the utility of the [Formula: see text]-condition found in that theorem.



Author(s):  
Sergei M. Ageev ◽  
Duŝan Repovŝ

AbstractWe study Banach-Mazur compacta Q(n), that is, the sets of all isometry classes of n-dimensional Banach spaces topologized by the Banach-Mazur metric. Our main result is that Q(2) is homeomorphic to the compactification of a Hilbert cube manifold by a point, for we prove that Qg(2) = Q(2) / {Eucl.} is a Hilbert cube manifold. As a corollary it follows that Q(2) is not homogeneous.



1989 ◽  
Vol 80 (406) ◽  
pp. 0-0 ◽  
Author(s):  
H. Toruńczyk ◽  
J. West


1985 ◽  
Vol 120 (1) ◽  
pp. 153-178 ◽  
Author(s):  
Scott Metcalf


1984 ◽  
Vol 112 (2) ◽  
pp. 407-426
Author(s):  
Luis Montejano Peimbert


1981 ◽  
Vol 12 (1) ◽  
pp. 19-33
Author(s):  
T.A. Chapman




1976 ◽  
Vol 137 (0) ◽  
pp. 171-208 ◽  
Author(s):  
T. A. Chapman ◽  
L. C. Siebenmann


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