Algebraic Topology · Problem 7

The Fundamental Group — Allen Hatcher

Mathematics — Topology · contractible-spaces

Problem

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 the figure. 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.