How many 3-fold covering spaces does S1 V S1 have?

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In summary, the conversation discusses finding all 3-fold covering spaces of the one-point union of two circles, known as S1 V S1. One solution is W X {1, 2, 3}, which consists of three disjoint copies of W, the wedge sum. The speaker is unsure of how to construct the other covering spaces and wonders if there is a way to use the covering spaces of the circle to do so. They also mention having six diagrams so far and ask if anyone knows the total number of 3-fold covers or how to determine it.
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Homework Statement



Find all 3-fold covering spaces of S1 V S1 (the one-point union, or wedge sum, of two copies of the circle, S1).



Homework Equations



There is, as a hint, diagrams of the 3-fold covering spaces of the circle itself.



The Attempt at a Solution



Call the wedge sum W.

One 3-fold covering space is W X {1, 2, 3}. This is just the space consisting of 3 disjoint copies of W. But this was pretty easy.

I do not know how to construct the other covering spaces. Is there a way to use the covering spaces of the circle to construct covering spaces of W? Or at least a way to think about the covering space of the circle that could give me some insight? So far, I've only managed to stare at my sheet in a dazed fashion (other than the one covering space I did think of).
 
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I have six diagrams so far. Anybody know off-hand how many 3-fold covers of S1 V S1 there are or how one might determine how many there are?
 

FAQ: How many 3-fold covering spaces does S1 V S1 have?

What is a covering space of S1 V S1?

A covering space of S1 V S1 is a topological space that maps onto the torus (S1 V S1) in a continuous and surjective manner. It is a type of covering space that is formed by taking a space and replicating it multiple times over the original space, with each copy overlapping and covering the entire space.

What are the properties of a covering space of S1 V S1?

One of the main properties of a covering space of S1 V S1 is that it is a universal covering space, meaning that it is simply connected and covers all other covering spaces of S1 V S1. Additionally, it has a deck transformation group, which is a group of homeomorphisms that preserve the covering map.

How is a covering space of S1 V S1 different from a regular torus?

A covering space of S1 V S1 is different from a regular torus in that it is a topological space that maps onto the torus, rather than being the torus itself. This means that it may have different geometric properties and can be used to study different aspects of the torus.

Why are covering spaces of S1 V S1 important in mathematics?

Covering spaces of S1 V S1 are important in mathematics because they provide a way to study the properties of the torus in a more general and abstract manner. They also have applications in fields such as algebraic topology and differential geometry.

How are covering spaces of S1 V S1 used in real-world applications?

Covering spaces of S1 V S1 have various real-world applications, such as in the study of fluid dynamics and electromagnetic fields. They are also used in image processing and computer graphics to create seamless textures and patterns on curved surfaces.

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