| dc.description |
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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