Hatcher-Algebraic-Topology · Problem 1.3.7

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

Let

Y

be the quasi-circle shown in the figure, a closed subspace of

\mathbb{R}^2

consisting of a portion of the graph of

y = \sin(1/x)

, the segment

[-1,1]

in the

y

-axis, and an arc connecting these two pieces. Collapsing the segment of

Y

in the

y

-axis to a point gives a quotient map

f : Y \to S^1

. Show that

f

does not lift to the covering space

\mathbb{R} \to S^1

, even though

\pi_1(Y) = 0

. Thus local path-connectedness of

Y

is a necessary hypothesis in the lifting criterion.