Covering space

Definition

A covering space is a topological space that maps onto another space in a way that each point in the target space has an open neighborhood evenly covered by the pre-image of the covering space. This concept is crucial because it helps us understand the structure of spaces through their coverings, revealing information about their fundamental groups and how they relate to loops and paths within them.

5 Must Know Facts For Your Next Test

1. Every covering space of a connected, locally path-connected space is path-connected.
2. The fundamental group of a covering space is related to the fundamental group of the base space; specifically, there is a correspondence between the deck transformations of the covering space and the normal subgroups of the fundamental group.
3. The lifting property allows paths and homotopies to be lifted uniquely under certain conditions, which provides a powerful tool for studying spaces via their covering spaces.
4. Covering spaces can be classified as regular or normal if the action of the deck transformation group on the fibers is transitive.
5. For any finite covering space, its number of sheets corresponds to the index of the subgroup of the fundamental group of the base space.

Review Questions

• How does the concept of covering spaces relate to the fundamental group, particularly in terms of loops and paths?
  • Covering spaces provide a framework for understanding how loops and paths in a base space can be represented in a more complex structure. Specifically, when considering loops in the base space, each loop corresponds to one or more lifts in the covering space. The fundamental group reflects these connections, as it classifies loops based on their equivalence under continuous deformations. This relationship highlights how covering spaces can help reveal deeper properties about the topological structure of spaces.
• Discuss how Van Kampen's theorem applies to covering spaces and how it helps compute fundamental groups.
  • Van Kampen's theorem plays a critical role in understanding how covering spaces can be used to compute fundamental groups by breaking down complex spaces into simpler pieces. When applying this theorem, one typically considers how to decompose a space into path-connected open sets whose intersections also have well-defined fundamental groups. This decomposition allows for an understanding of how these groups combine, and since covering spaces are often constructed from such decompositions, they offer insights into how these fundamental groups interact with one another.
• Evaluate the significance of covering spaces in algebraic topology and their implications for understanding higher-dimensional manifolds.
  • Covering spaces are significant in algebraic topology as they provide essential tools for studying properties of spaces, particularly when analyzing their fundamental groups and paths. The ability to lift paths and homotopies in covering spaces allows mathematicians to explore deeper relationships within spaces that might not be apparent at first glance. Additionally, these concepts extend beyond basic topological structures into higher-dimensional manifolds, where covering spaces can offer insights into complex behaviors such as fiber bundles and higher homotopy groups, enriching our overall understanding of topology.

Related terms

lift: To lift a path or loop means to find a corresponding path in the covering space that projects down to the original path in the base space.

path-connected:

A space is path-connected if any two points in the space can be connected by a continuous path, which is an important property for defining covering spaces.

fundamental group: The fundamental group is an algebraic structure that captures information about the loops in a topological space, particularly focusing on their equivalence classes under continuous deformation.