Construct an explicit isomorphism

In summary, An explicit isomorphism is a mathematical concept that describes the relationship between two structures or objects that are essentially the same. It is constructed by identifying the elements of the two structures or objects and creating a one-to-one mapping between them. The purpose of constructing an explicit isomorphism is to demonstrate equivalence and gain a deeper understanding of the underlying similarities and relationships. It differs from an implicit isomorphism in that it is a direct and clear mapping. However, there may be limitations to constructing an explicit isomorphism, such as difficulty finding a mapping or not capturing all of the relationships and properties.
  • #1
bedi
81
0
$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle

I'm trying to construct an explicit isomorphism from ##E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}## to ##T = [0, 1] × R/ ∼## where ##(0, t) ∼ (1, −t)##. I verified that ##\Bbb{R}P^1## is homeomorphic to ##\Bbb{S}^1## which is homeomorphic to ##[0,1]/∼## where ##0∼1##. So this is the map I have in my mind: ##([x],v)\to (x,(1-x)v+xe^v)##. Does that work? It doesn't look very natural.
 
Physics news on Phys.org
  • #2
How about pulling back the bundle using the homeomorphism?
 

1. What is an explicit isomorphism?

An explicit isomorphism is a mathematical concept that describes the relationship between two structures or objects that are essentially the same. It is a way of showing that two things are equivalent or identical by mapping the elements of one onto the elements of the other.

2. How is an explicit isomorphism constructed?

An explicit isomorphism is constructed by identifying the elements of the two structures or objects and then creating a mapping between them. This mapping must be one-to-one and preserve the structure and relationships between the elements.

3. What is the purpose of constructing an explicit isomorphism?

The purpose of constructing an explicit isomorphism is to demonstrate that two structures or objects are equivalent or identical. It allows for a deeper understanding of the underlying similarities and relationships between the two, and can be useful in solving problems or proving theorems.

4. How is an explicit isomorphism different from an implicit isomorphism?

An explicit isomorphism is a clear and direct mapping between two structures or objects, while an implicit isomorphism is a more abstract and indirect way of showing equivalence. Explicit isomorphisms are often easier to construct and understand, but implicit isomorphisms can also be useful in certain contexts.

5. Are there any limitations to constructing an explicit isomorphism?

Yes, there are some limitations to constructing an explicit isomorphism. It may not always be possible to find a one-to-one mapping between two structures or objects, and even if a mapping exists, it may be difficult or impossible to construct. Additionally, an explicit isomorphism may not capture all of the underlying relationships and properties of the two structures or objects.

Similar threads

  • Topology and Analysis
2
Replies
43
Views
825
Replies
2
Views
3K
Replies
2
Views
1K
  • Topology and Analysis
Replies
8
Views
1K
  • Topology and Analysis
Replies
1
Views
724
  • Linear and Abstract Algebra
Replies
1
Views
933
Replies
4
Views
209
  • Calculus and Beyond Homework Help
Replies
24
Views
671
Replies
27
Views
3K
  • Math Proof Training and Practice
2
Replies
69
Views
3K
Back
Top