Hatcher 0.7

Hatcher, Algebraic Topology, Chapter 0

7. Fill in the details in the following construction from [Edwards 1999] of a compact space Y\subset\mathbb{R}^3 with the same properties as the space Y in Exercise 6, that is, Y is contractible but does not deformation retract to any point. To begin, let X be the union of an infinite sequence of cones on the Cantor set arranged end-to-end, as in Figure 1 below. Next, form the one-point compactification of X\times\mathbb{R}. This embeds in \mathbb{R}^3 as a closed disk with curved “fins” attached along circular arcs, and with the one-point compactification of X as a cross-sectional slice. The desired space Y is then obtained from this subspace of \mathbb{R}^3  by wrapping one  more cone on the Cantor set around the boundary of the disk.

Hatcher 0.7
Figure 1
The spaces mentioned above

Proof.

     \square

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