5.1 Definition and properties of covering spaces
4 min read•july 30, 2024
Covering spaces are a powerful tool in topology, allowing us to study complex spaces by breaking them down into simpler pieces. They're like a puzzle where each piece mirrors the whole, giving us insights into the structure of the original space.
In this topic, we'll explore the definition and key properties of covering spaces. We'll see how they relate to fundamental groups, how they inherit properties from base spaces, and dive into the fascinating world of local homeomorphisms and evenly covered neighborhoods.
Covering spaces and their properties
Definition and structure of covering spaces
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- consists of topological space E with continuous surjective map p: E → B to base space B
- Map p called covering map, E called total space, B called base space
- Each point in B has open neighborhood evenly covered by p
- Preimage in E forms disjoint union of open sets each homeomorphic to neighborhood
- Fiber of point b in B defined as preimage p^(-1)(b)
- Discrete subspace of E
- Covering spaces possess unique
- Continuous paths in B lift uniquely to paths in E given starting point
- Degree of covering map equals cardinality of fiber over any point in B
- Constant for path-connected spaces
- Covering spaces inherit topological properties from base spaces (compactness, path-connectedness)
Key properties and characteristics
- Local established between E and B by covering map p
- Base space B viewed as quotient space of E under equivalence relation induced by fibers of p
- of E isomorphic to subgroup of fundamental group of B
- Isomorphism induced by covering map
- Correspondence between subgroups of fundamental group of B and connected covering spaces of B
- Known as Galois correspondence for covering spaces
- Automorphism group of covering () isomorphic to quotient of fundamental group of B by image of fundamental group of E
- Regular (normal) covering spaces correspond to normal subgroups of fundamental group of B
- Universal cover of B (if exists) corresponds to trivial subgroup of fundamental group of B
- Has total space E
Relationship between spaces
Topological connections
- Covering map p: E → B establishes local homeomorphism between covering space E and base space B
- Base space B represents quotient space of E under equivalence relation induced by fibers of p
- Covering spaces inherit many topological properties from base spaces
- Examples include compactness, path-connectedness, and dimension
- Local structure of covering space mimics base space due to evenly covered neighborhoods
Algebraic relationships
- Fundamental group of E isomorphic to subgroup of fundamental group of B
- Isomorphism induced by covering map p
- Galois correspondence links subgroups of fundamental group of B to connected covering spaces
- Automorphism group of covering (deck transformations) isomorphic to quotient group
- Quotient of fundamental group of B by image of fundamental group of E
- Regular covering spaces correspond to normal subgroups of fundamental group of B
- Universal cover (if exists) corresponds to trivial subgroup of fundamental group of B
- Total space E of universal cover is simply connected
Local homeomorphism of covering spaces
Properties of local homeomorphisms
- Local homeomorphism defined as continuous map that is locally invertible and preserves local topological structure
- Covering map p: E → B functions as local homeomorphism
- For each point e in E, open neighborhood U of e exists where p|U is homeomorphism onto its image
- Local homeomorphism property ensures covering space locally resembles base space
- Preserves local topological features
- Allows transfer of local topological information between covering space and base space
- Essential for defining and understanding concept of evenly covered neighborhoods
Applications and implications
- Local homeomorphism property crucial for proving various theorems about covering spaces
- Examples include and unique path lifting property
- All covering maps are local homeomorphisms, but not all local homeomorphisms are covering maps
- Exponential map on complex plane serves as counterexample
- Local homeomorphism property enables study of global structure through local properties
- Facilitates construction of local sections of covering map
- Important for various proofs in covering space theory
Evenly covered neighborhoods in covering spaces
Definition and characteristics
- Evenly covered neighborhood U of point b in B defined as open set
- Preimage p^(-1)(U) forms disjoint union of open sets in E, each homeomorphic to U via p
- Existence of evenly covered neighborhoods for every point in B defines covering spaces
- Components of p^(-1)(U) called sheets over U
- Number of sheets constant for path-connected spaces
- Evenly covered neighborhoods provide local trivialization of covering
- Enables systematic study of global structure through local properties
Significance and applications
- Evenly covered neighborhoods crucial for constructing local sections of covering map
- Essential for various proofs in covering space theory
- Concept generalizes to fiber bundles
- Plays similar role in defining local trivializations
- Understanding evenly covered neighborhoods vital for proving path lifting and homotopy lifting properties
- Allows for detailed analysis of local structure of covering spaces
- Facilitates computation of important invariants (degree of covering map)
- Enables construction of new covering spaces from existing ones
- Example: product of covering spaces
Key Terms to Review (14)
Base of a topology: A base of a topology on a set is a collection of open sets such that every open set in the topology can be expressed as a union of elements from this collection. The base serves as a building block for the topology, allowing for the construction of all other open sets through unions of these base sets. Understanding the base is essential for grasping how topological spaces are formed and manipulated.
Circle covering space: A circle covering space is a topological space that maps onto a circle (denoted as $S^1$) such that every point on the circle has a neighborhood that is evenly covered by the covering space. This means that locally, the covering space looks like several copies of the circle, which are homeomorphic to the original circle, and each point in the circle corresponds to multiple points in the covering space. Circle covering spaces provide insight into properties like connectedness and fundamental groups.
Covering space: 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.
Deck Transformations: Deck transformations are homeomorphisms of a covering space that permute the fibers above points in the base space. They play a crucial role in understanding the structure of covering spaces, as they form a group known as the deck transformation group, which captures the symmetries of the covering space relative to the base space. This group helps to classify different types of covering spaces and reveals important properties about their topology.
Existence Theorem: An existence theorem is a fundamental concept in mathematics that asserts the existence of a certain object or structure under specific conditions. In the context of covering spaces and lifting properties, these theorems provide crucial insights into when a covering space can be constructed and how paths can be lifted from the base space to its covering space, ensuring that certain mappings and properties hold true.
Fundamental Group: The fundamental group is an algebraic structure that captures the notion of loops within a topological space and their equivalence under continuous deformations. It provides a way to classify spaces based on their shape, focusing on the idea of paths that can be continuously transformed into each other, and is vital in understanding properties like connectedness and hole structure.
Homeomorphism: A homeomorphism is a continuous function between two topological spaces that has a continuous inverse, establishing a one-to-one correspondence that preserves the topological structure. This means that two spaces are considered homeomorphic if they can be transformed into each other through stretching, bending, or twisting, without tearing or gluing. Homeomorphisms are fundamental in determining when two spaces can be regarded as essentially the same in a topological sense.
Homotopy Lifting Property: The homotopy lifting property is a fundamental concept in algebraic topology that describes how homotopies between maps can be lifted through covering spaces. This property allows us to extend paths and homotopies defined on a base space to their corresponding covering spaces, maintaining the structure of the original maps. It is crucial for understanding the relationship between different topological spaces and their coverings, particularly in exploring the nature of continuous functions and the behavior of loops and paths.
Locally path-connected: A space is locally path-connected if, for every point in the space and every neighborhood of that point, there exists a path-connected neighborhood contained within it. This concept is crucial in understanding the behavior of spaces under various types of continuous transformations, especially when considering homotopy and the structure of covering spaces. Locally path-connected spaces allow for the extension of paths and help ensure that small enough neighborhoods behave nicely in terms of path-connectedness.
Path Lifting Property: The path lifting property refers to a feature of covering spaces that allows a continuous path in the base space to be lifted to a continuous path in the covering space, starting from a specific point in the covering space. This property is essential for understanding how covering spaces relate to their base spaces, as it establishes a connection between paths and their corresponding lifts. It plays a crucial role in the study of homotopy and helps to demonstrate the uniqueness of lifts given specific conditions.
Projection map: 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.
Simply Connected: A space is simply connected if it is path-connected and every loop in the space can be continuously contracted to a point without leaving the space. This concept indicates that there are no 'holes' in the space that would prevent such contraction, making it essential for understanding properties like homotopy and fundamental groups.
Uniqueness Theorem: The uniqueness theorem in the context of covering spaces states that given a covering space of a topological space and a point in that space, there is exactly one way to lift paths from the base space to the covering space starting from the corresponding point in the covering space. This theorem is crucial as it ensures that the behavior of paths in the covering space is well-defined and consistent, leading to essential properties such as homotopy lifting and the ability to classify covering spaces up to isomorphism.
Universal covering space: A universal covering space is a special type of covering space that covers a given topological space in such a way that it can be used to study the fundamental group of that space. It is unique up to homeomorphism and has the property that every other covering space of the original space can be obtained as a quotient of the universal covering space. This concept plays a crucial role in understanding the properties and classifications of covering spaces.