Hatcher-Algebraic-Topology · Problem 1.3.29

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

Let

Y

be path-connected, locally path-connected, and simply-connected, and let

G_1

and

G_2

be subgroups of

\mathrm{Homeo}(Y)

defining covering space actions on

Y

. Show that the orbit spaces

Y/G_1

and

Y/G_2

are homeomorphic iff

G_1

and

G_2

are conjugate subgroups of

\mathrm{Homeo}(Y)