Graduate Thesis Or Dissertation
 

An intrinsic codimension two cellular decomposition of the Hilbert cube

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  • Cellular sets in the Hilbert cube are the intersection of nested sequences of normal cubes. One way of getting cellular maps on the Hilbert cube is by decomposing the Hilbert cube into cellular sets and using a quotient map. By using a cellular decomposition of the Hilbert cube, an example of a cellular map is given to show that the image of the Hilbert cube under a cellular map can have complex non manifold part, not be a Hilbert cube manifold, and still be a Hilbert cube manifold factor. The non degenerate decomposition elements are shown to satisfy the cellularity criteria. To measure how far the image is from being a Hilbert cube manifold, the idea of covering codimension in finite dimensions is generalized by using a homological codimension approach. In finite dimensional settings, the two codimensions are equivalent. The complexity of the non manifold part of the image space is measured in terms of intrinsic codimension, which uses the homological codimension of the image of the union of nondegenerate decomposition elements. The intrinsic codimension of the map in this example is found to be exactly two. Using a characterization of Hilbert cube manifolds, it is shown that the decomposition space is not the Hilbert cube, but is a factor of the Hilbert cube.
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