# 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 |

**How to Cite This Entry:**

Group of covering transformations.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Group_of_covering_transformations&oldid=15034