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