Hatcher-Algebraic-Topology · Problem 1.3.25

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

Let

\varphi : \mathbb{R}^2 \to \mathbb{R}^2

be the linear transformation

\varphi(x,y) = (2x, y/2)

. This generates an action of

\mathbb{Z}

X = \mathbb{R}^2 - \{0\}

. Show this action is a covering space action and compute

\pi_1(X/\mathbb{Z})

. Show the orbit space

X/\mathbb{Z}

is non-Hausdorff, and describe how it is a union of four subspaces homeomorphic to

S^1 \times \mathbb{R}

, coming from the complementary components of the

x

-axis and the

y

-axis.