Graduate Thesis Or Dissertation
 

A thin codimension-one decomposition of the Hilbert Cube

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  • For cell-like upper semicontinuous(usc) decompositions G of finite dimensional manifolds M, the decomposition space M/G turns out to be an ANR provided M/G is finite dimensional ([Dav07], page 129 ). Furthermore, if M/G is finite dimensional and has the Disjoint Disks Property (DDP), then M/G is homeomorphic to M ([Dav07], page 181). For an infinite dimensional M modeled on I∞, we can construct cell-like usc decompositions G associated with defining sequences. But it is more complicated to check whether M/G is an ANR. We need an additional special property of the defining sequence. To check whether or not M/G is homeomorphic to M is even more difficult. We need M/G to be an ANR which has the DDP and which also satisfies the Disjoint Cech Carriers Property. We give a specific cell-like decomposition X of the Hilbert Cube Q with the following properties: The nonmanifold part N of X is complicated in the sense that it is homeomorphic to a Hilbert Cube of codimension 1 in Q. X is still a factor of Q because X × I² ≅ Q. If A is any closed subspace of N of codimension ≥ 1 in N, then the decomposition of Q over A is homeomorphic to Q. In particular, the nonmanifold nature of X is not detectable by examining closed subsets of codimension ≥ 1. This example is produced by combining mixing techniques for producing a nonmanifold space whose nonmanifold part is a Cantor set, with decompositions arising from a generalized Cantor function.
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