Group of covering transformations
of a regular covering
The group of those homeomorphisms
of the space
onto itself such that
. (
and
are understood to be connected, locally path-connected, Hausdorff spaces.)
The group of covering transformations of the covering of the circle by the real line given by
is thus the group of translations
,
.
is a discrete group of transformations of
acting freely (that is,
for some
implies
), and
is naturally isomorphic to the quotient space
. The group
is isomorphic to the quotient group of the fundamental group
, where
, by the image of the group
, where
, under the homomorphism induced by the mapping
. In particular, if
is the universal covering, then
is isomorphic to the fundamental group of
.
References
[1] | S.-T. Hu, "Homotopy theory" , Acad. Press (1959) |
Comments
See also Covering; Universal covering.
References
[a1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. 2 |
Group of covering transformations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_of_covering_transformations&oldid=42469