Understanding 1/2 of Van Kampen's Theorem

  • Thread starter sammycaps
  • Start date
  • Tags
    Theorem
In summary, the easy 1/2 of Van Kampen's theorem states that for sets X, A, and B with a basepoint x, where A and B are open in X and path-connected, the fundamental group of X is generated by the fundamental groups of A and B. This is achieved through the homomorphism induced by the inclusion of A and B in X, which may not be injective. Additionally, the theorem states that any pointed loop in X can be homotoped into a concatenation of loops in either A or B.
  • #1
sammycaps
91
0
So I'm having a little trouble with the part of Van Kampen's theorem my professor presented to us. He called this the easy 1/2 of Van Kampen's theorem.

Theorem (1/2 of Van Kampen's) - Let X,x=A,x U B,x (sets with basepoint x) where A and B are open in X and A[itex]\bigcap[/itex]B is path-connected. Then [itex]\pi[/itex]1(X) is generated by [itex]\pi[/itex]1(A) and [itex]\pi[/itex]1(B).

[itex]\pi[/itex]1(A) and [itex]\pi[/itex]1(B) are not necessarily subsets of [itex]\pi[/itex]1(X), at least in general. So if anyone can enlighten me on exactly what the Professor meant. I would think he just means the embedding of [itex]\pi[/itex]1(A) and [itex]\pi[/itex]1(B) in [itex]\pi[/itex]1(X) but I don't think, at least in general, the homomorphism induced by the inclusion is injective.

Thanks very much.
 
Last edited:
Physics news on Phys.org
  • #2
Your interpretation is correct on all points. When we talk of [itex]\pi_1(A)[/itex] and [itex]\pi_1(B)[/itex] "in [itex]\pi_1(X)[/itex]", we mean their image by the homomorphism induced by the canonical inclusion. And of course, this homomorphism is usually not injective (i.e. not an embedding). Consider for instance the homomophism induced by the inclusion of A:=R²\{0} into X:=R².
 
  • #3
quasar987 said:
Your interpretation is correct on all points. When we talk of [itex]\pi_1(A)[/itex] and [itex]\pi_1(B)[/itex] "in [itex]\pi_1(X)[/itex]", we mean their image by the homomorphism induced by the canonical inclusion. And of course, this homomorphism is usually not injective (i.e. not an embedding). Consider for instance the homomophism induced by the inclusion of A:=R²\{0} into X:=R².

Thanks very much! (I changed my original post because I felt I was being too talkative). So, even if A and B have nontrivial fundamental groups, their images under homomorphism induced by the inclusion may both be trivial, in which case they would "generate" the trivial group, right?
 
  • #4
Right.

And in general, the easy 1/2 of V-K's thm states that any pointed loop [itex]\gamma[/itex] in X is homotopic to some concatenation of loops [itex]\sigma_1 * \ldots * \sigma_n[/itex] with each [itex]\sigma_i[/itex] lying entirely either in A or in B.
 
  • #5
Great! Thanks very much.
 

What is Van Kampen's Theorem?

Van Kampen's Theorem is a fundamental theorem in algebraic topology that allows us to compute the fundamental group of a space by decomposing it into simpler spaces.

What is the significance of understanding 1/2 of Van Kampen's Theorem?

Understanding 1/2 of Van Kampen's Theorem allows us to compute the fundamental group of a space that can be decomposed into two simpler spaces. This is useful in many applications, such as in knot theory and the study of surfaces.

What is the first part of Van Kampen's Theorem?

The first part of Van Kampen's Theorem states that if we have a space that can be decomposed into two open sets that overlap in a path-connected and simply connected space, then the fundamental group of the space is the free product of the fundamental groups of the two open sets.

What is the second part of Van Kampen's Theorem?

The second part of Van Kampen's Theorem states that if we have a space that can be decomposed into open sets that overlap in a path-connected and simply connected space, then the fundamental group of the space is the quotient of the free product of the fundamental groups of the open sets by the normal subgroup generated by the elements representing the loops in the overlap space.

What are some applications of Van Kampen's Theorem?

Van Kampen's Theorem has many applications in topology, including in the study of homotopy groups, covering spaces, and the classification of surfaces. It is also used in other fields such as knot theory and algebraic geometry.

Similar threads

Understanding the Fundamental Group: Exploring Pi1(X.x) and Its Definition
    • Differential Geometry
Replies
19
Views
4K
I Clarification about submanifold definition in ##\mathbb R^2##
    • Differential Geometry
Replies
4
Views
2K
Van Kampen's theorem and fundamental groups
    • General Math
Replies
7
Views
7K
I Closure & Interior as Dual Notions .... Proving Willard Theorem 3.11 ...
    • Topology and Analysis
Replies
2
Views
1K
MHB Implicit function theorem for f(x,y) = x^2+y^2-1
    • Calculus
Replies
1
Views
954
I The far reaching ramifications of the work-energy theorem
    • Classical Physics
Replies
31
Views
2K
Intermediate Value Theorem Problem on a String
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
I Looking For Help Understanding Bell's Theorem, Hidden Variables & QM
    • Quantum Physics
3
Replies
80
Views
3K
A Proof of Seifert-Van Kampen Theorem
    • Topology and Analysis
Replies
2
Views
2K
MHB Lusin's Theorem - Infinite Measure and not Real-Valued Extensions
    • Topology and Analysis
Replies
2
Views
4K

Hot Threads

Recent Insights

Back
Top