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

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SUMMARY

The discussion focuses on identifying all 3-fold covering spaces of the wedge sum of two circles, denoted as S1 V S1. One confirmed covering space is W X {1, 2, 3}, which consists of three disjoint copies of the wedge sum W. Participants seek methods to derive additional covering spaces from known covering spaces of the circle S1, emphasizing the need for diagrams and insights into the structure of these spaces. The conversation highlights the complexity of determining the total number of 3-fold covers for S1 V S1.

PREREQUISITES
  • Understanding of covering spaces in topology
  • Familiarity with wedge sums, specifically S1 V S1
  • Knowledge of the properties of S1 and its covering spaces
  • Ability to interpret and create topological diagrams
NEXT STEPS
  • Research the properties of covering spaces of the circle S1
  • Explore the concept of wedge sums in algebraic topology
  • Study the classification of covering spaces and their corresponding diagrams
  • Investigate the relationship between covering spaces and fundamental groups
USEFUL FOR

Mathematicians, particularly those specializing in algebraic topology, students studying covering spaces, and anyone interested in the properties of wedge sums and their applications in topology.

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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?
 

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