Why are covering spaces important?

So one answer to your question of why they are more useful than more general covers is that they provide exactly the right setting for realizing spaces with fundamental groups that are subgroups of your original space.

What do you mean by covering spaces?

Given a topological space X, we’re interested in spaces which “cover” X in a nice way. Roughly speaking, a space Y is called a covering space of X if Y maps onto X in a locally homeomorphic way, so that the pre-image of every point in X has the same cardinality.

Why do we cover space when studying?

In algebraic topology, covering spaces help us understand the homotopy classes of maps between spaces better, get a handle on the fundamental group of spaces etc.

Why is algebraic topology important?

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Why is algebraic topology hard?

Algebraic topology, by it’s very nature,is not an easy subject because it’s really an uneven mixture of algebra and topology unlike any other subject you’ve seen before. However,how difficult it can be to me depends on how you present algebraic topology and the chosen level of abstraction.

What is covering map in mathematics?

In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function from a topological space to a topological space such that each point in has an open neighborhood evenly covered by (as shown in the image).

How do you read an algebraic topology?

Algebraic topology starts by putting a group structure on topological spaces using simplicial or cell complexes. Once you’ve placed a group structure on a topological spaces you can start to reason about what homomorphisms between two groups that represent homeomorphic topological spaces look like.

What is the importance of covering spaces?

Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group.

What is a covering space in geometry?

The definition implies that every covering map is a local homeomorphism . Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps.

What is a covering of a topological n-manifold?

Since coverings are local homeomorphisms, a covering of a topological n-manifold is an n-manifold. (One can prove that the covering space is second-countable from the fact that the fundamental group of a manifold is always countable.) However a space covered by an n-manifold may be a non-Hausdorff manifold.

What is the base space of a covering map?

The map is called the covering map, the space is often called the base space of the covering, and the space is called the total space of the covering. For any point in the base the inverse image of in is necessarily a discrete space called the fiber over .