Covering Space Actions

Covering Space Actions

Recall the definition of a group action:

A set equipped with an action of is called a -set.

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We can treat it as a group homomorphism . We usually assume the action is faithful (i.e., is injective). If not, we can replace with the quotient group .

Covering Space Action

A -action on a space is called a covering space action if for any point , there exists an open neighborhood of such that for any two distinct elements , the sets and are disjoint. When a group acts on a space , we can form the (quotient) orbit space . The points of are the orbits of points in under the action of .

Properties of Covering Space Actions

Theorem

If a group acts on a space via a covering space action, and with projection , then:

1. is a normal covering space.
2. If is also connected, then the group of deck transformations is isomorphic to the group , i.e., .
3. If is also path-connected and locally path-connected, then the group of deck transformations is isomorphic to the quotient of the fundamental groups: .

Proof (1) First, we show is a covering space. Let be a neighborhood of a point as in the definition of a covering space action. Then is an open set in . Its preimage is . The restriction of to each disjoint open set is a homeomorphism onto . These sets form a basis of evenly covered neighborhoods for the topology on , so is a covering map. To show the covering is normal, we must show that for any such that , there exists a deck transformation with . By definition of the quotient space, means that and are in the same orbit, so there exists some such that [cite: 29]. The action of this (i.e., the map ) is a deck transformation, and since , the condition is satisfied and the covering is normal. (2) If is connected, any deck transformation is uniquely determined by its action on a single point, say . Since is a deck transformation, we must have . This implies that and are in the same fiber, so there must be some such that . Since the action of is also a deck transformation that sends to , and since deck transformations are uniquely determined by their action on one point, we must have . Thus, every deck transformation corresponds to an element of . (3) This is a standard result from covering space theory which states that for a normal covering space with a path-connected base space, the group of deck transformations is isomorphic to .

e.g.

• The group of integers acts on the real line by addition: . The quotient space is the circle, . This shows .
• The group acts on the -sphere via the antipodal map (). The quotient space is the -dimensional real projective space, . So, for .
• The cyclic group acts on the -sphere , viewed as the unit sphere in . A generator of the group acts by multiplication by a -th root of unity, . The quotient space is called a lens space, denoted .
• Consider genus- surface for . The group acting on by rotating each component of genus . The quotient space is a surface with genus .