Fundamental Group Coset to preimage bijection

In summary: The map I constructed earlier is continuous, g is its inverse, so it's continuous as well. I'm trying to show injectiveness, but it's not as obvious as I thought it'd be. I'm trying to use the uniqueness of the lifting property to show that if g(c_1)=g(c_2) then c_1=c_2, but I'm struggling to make that connection.In summary, the conversation involves proving a bijection between the collection of right cosets of the homomorphism induced by a covering map and the preimage of a point under the covering map. The strategy suggested is to focus on the most natural map between the two sets and use the fact that E is
  • #1
PsychonautQQ
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Homework Statement


Let p: E-->B be a covering map, let p(e_0)=b_0 and let E be path connected. Show that there is a bijection between the collection of right cosets of p*F(E,e_0) in F(B,b_0) (where p* is the homomorphism of fundamental groups induced by p and F(E,e_0),F(B,b_0) are the fundamental groups of E and B based at e_0 and b_0 respectively) and the preimage of b_0 under the covering map p.

Homework Equations

The Attempt at a Solution


I have already showed that the homomorphism induced by p injects F(E,e_0) into F(B,e_0). Now I'm trying to show why the cosets defined above have a bijective correspondence to the preimage of the covering map of b_0... E being path connected definitely has something to do with it. Does anyone have any good hints/tips/insights? I seem to be going in circles >.<
 
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  • #2
A good strategy in trying to prove a bijection is to focus on the most natural map one can think of, between the two sets. Usually it turns out to be what one wants.

What might be a nice natural map between the collection of right cosets and the pre-image ##H\triangleq p^{-1}(b_0)##?

Every element of a right coset is a loop in B. If we could associate it with a path in E, we might reasonably expect that path to start and end on a point in ##H##.

To try and head in that direction, we might start by defining A as the set of all paths in E that start at ##e_0## and end at a point in ##H##. Then partition that set based on the point at which the path ends. Then there is a bijection between the partition - call it R - and H.

Now we need to try to relate R to the right coset ##Q\triangleq \pi_1(B,B_0)\ /\ p^*\left(\pi_1(E,e_0)\right)##.

Pick a coset in Q and choose a representative element ##f##, which is a loop in B. Can we associate f with an element of R? We can't just take a path ##p^{-1}\circ f## in E because ##p^{-1}## is not a function. But since the domain of ##f## is ##[0,1]##, which is compact, maybe we can find an open cover of ##[0,1]## that, upon reducing to a finite open cover, gives us a series of homeomorphisms between a finite collection of open sets in B, whose union contains ##Im\ f##, and corresponding open sets in E. Since each sufficiently small open set ('evenly covered neighbourhood') in B will correspond to multiple homeomorphic images of it in E (the 'sheets' making up the 'fibre' above that nbd), we need a way of choosing one of those sheets. Since we want to relate ##f## to an element of R, which is a set of paths that start at ##e_0##, we can do that by requiring the open set from our cover that contains 0, to map to ##e_0##. Do you think you can go on from there, using the finite sub-cover of ##[0,1]##, to show that there is a unique path in E that corresponds to the loop ##f##?

If we can do that then we have found a function, call it ##\phi##, from the set of loops in B, to R, by setting ##\phi(f)## to be the partition component containing the unique path (from ##A##) corresponding to ##f##.

Next we'd like to show that if loops ##f_1,f_2## are in the same coset in Q then ##\phi(f_1)=\phi(f_2)##. If so then we can define a map ##\phi^*## from Q to R.

Then all we have to do is show that ##\phi^*## is surjective and injective.

PS This is a pretty first-principles approach. It is possible there are theorems that have already been given to you that could considerably short-cut this.
 
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  • #3
andrewkirk said:
Pick a coset in Q and choose a representative element ##f##, which is a loop in B. Can we associate f with an element of R? We can't just take a path ##p^{-1}\circ f## in E because ##p^{-1}## is not a function. But since the domain of ##f## is ##[0,1]##, which is compact, maybe we can find an open cover of ##[0,1]## that, upon reducing to a finite open cover, gives us a series of homeomorphisms between a finite collection of open sets in B, whose union contains ##Im\ f##, and corresponding open sets in E. Since each sufficiently small open set ('evenly covered neighbourhood') in B will correspond to multiple homeomorphic images of it in E (the 'sheets' making up the 'fibre' above that nbd), we need a way of choosing one of those sheets. Since we want to relate ##f## to an element of R, which is a set of paths that start at ##e_0##, we can do that by requiring the open set from our cover that contains 0, to map to ##e_0##. Do you think you can go on from there, using the finite sub-cover of ##[0,1]##, to show that there is a unique path in E that corresponds to the loop ##f##?

The unique path in E that corresponds to that loop, you mean the unique lifting of ##f## that must exist because ##E## is a covering space of ##B##?
 
  • #4
PsychonautQQ said:
The unique path in E that corresponds to that loop, you mean the unique lifting of ##f## that must exist because ##E## is a covering space of ##B##?
The terminology I am familiar with is similar, but slightly different. Under that terminology the path we are referring to is

'the unique lift ##\gamma## of ##f## such that ##\gamma(0)=e_0##'

There is a different 'lift' of ##f## for every point in the fibre over ##b_0##, but only one of those starts at ##e_0##.
 
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  • #5
Btw thanks, I've made a lot of progress on this proof. I found a mapping and proved that it's well defined, my professor verified what I did. Now I'm trying to show injectiveness, he says surjective will be easy.

I'm defining a map let's call it g such that g sends equivalence classes in the fundamental group of B to the end point of their unique lifting into E, thus it's a map that can send elements from the cosets of Q to preimages of b_0
 

Related to Fundamental Group Coset to preimage bijection

1. What is a fundamental group coset?

A fundamental group coset is a set of elements in a group that can be multiplied together to form any element in that group. It is essentially a subset of the group that represents all possible combinations of elements.

2. What is a preimage bijection?

A preimage bijection is a mapping or function that connects two sets, where each element in the first set has a unique corresponding element in the second set. This means that the function is both one-to-one and onto, and every element in the second set has a preimage in the first set.

3. How are fundamental group cosets and preimage bijections related?

In mathematics, fundamental group cosets and preimage bijections are related through the concept of group homomorphisms. A group homomorphism is a function that preserves the group structure, meaning that the operation between two elements in the first group will result in the same operation between their corresponding elements in the second group. This allows for a bijection between the cosets of the fundamental group and the preimages of the homomorphism.

4. What is the significance of the fundamental group coset to preimage bijection?

The fundamental group coset to preimage bijection is significant in algebraic topology, as it helps in understanding the structure and properties of topological spaces. It allows for the simplification and analysis of complex spaces by breaking them down into smaller, more manageable pieces.

5. How is the fundamental group coset to preimage bijection used in real-world applications?

The fundamental group coset to preimage bijection has various applications in mathematics, physics, and engineering. It is used to study the behavior of systems with symmetries, in the analysis of mathematical models, and in the design of algorithms for data compression and error correction. It is also essential in the development of new technologies, such as coding theory and cryptography.

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