Projection map

Definition

A projection map is a continuous function from a topological space to another that 'projects' points in the domain onto a subset of the codomain, often relating to covering spaces. In the context of covering spaces, this map helps to illustrate how one space can be covered by another, revealing the structure and relationships between different topological spaces.

5 Must Know Facts For Your Next Test

1. The projection map plays a crucial role in defining how covering spaces relate to their base spaces by establishing a relationship between points in both spaces.
2. Every covering space has a natural projection map that takes points in the covering space and assigns them to their corresponding points in the base space.
3. The fibers of a projection map are discrete sets when dealing with covering spaces, reflecting how locally each point in the base space corresponds to several points in the covering space.
4. If a path in the base space is lifted to the covering space using the projection map, then it can be shown that this lift is unique up to homotopy if we fix the starting point.
5. The projection map is crucial for understanding fundamental groups, as it helps relate different topological spaces through their associated covering spaces.

Review Questions

• How does the projection map relate covering spaces to their base spaces?
  • The projection map acts as a bridge between a covering space and its base space by continuously mapping points from the covering space down to points in the base space. This relationship ensures that every point in the base space has an open neighborhood evenly covered by the covering space. Understanding this mapping helps visualize how different topological structures interact and provides insight into their properties.
• Discuss the significance of fibers in relation to projection maps and how they impact our understanding of covering spaces.
  • Fibers are essential when discussing projection maps because they represent all points in the covering space that map to a specific point in the base space. These fibers are discrete sets, which indicates that locally, around any point in the base space, there are several distinct points in the covering space. This understanding is key for grasping how paths can be lifted and how the structure of covering spaces reveals information about fundamental groups.
• Evaluate how projection maps contribute to the lifting property and its implications for path lifting in covering spaces.
  • Projection maps enhance our understanding of the lifting property by allowing us to lift paths from a base space to its covering space uniquely up to homotopy when starting from a specific point. This means that given a path in the base space, we can find a corresponding path in the covering space, preserving topological features. The uniqueness of such lifts underlies many important results in algebraic topology, including those related to homotopy groups and fundamental groups.

Related terms

Covering Space: A covering space is a topological space that 'covers' another space such that each point in the base space has an open neighborhood evenly covered by the covering space.

Lifting Property: The lifting property refers to the ability to lift paths or homotopies from a base space to its covering space via the projection map.

Fiber: In the context of a projection map, a fiber is the preimage of a point in the codomain, representing all points in the domain that map to that single point.