Proving Theorem 51.3 in Munkres 2e Edition on Homotopy Paths

  • Thread starter Thread starter valtz
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on proving Theorem 51.3 from Munkres' "Topology" (2nd Edition), which addresses homotopy paths. The theorem states that for a path \( f \) in a space \( X \) and a sequence of numbers \( a_0, a_1, \ldots, a_n \) with \( 0 = a_0 < a_1 < \ldots < a_n \), the path \( f_i: I \rightarrow X \) is defined as the positive linear map of \( I \) onto the interval \([a_{i-1}, a_i]\) followed by the path \( f \). The conclusion drawn is that the homotopy class of \( f \) can be expressed as the product of the homotopy classes of the paths \( f_1, \ldots, f_n \).

PREREQUISITES
  • Understanding of homotopy theory
  • Familiarity with paths in topological spaces
  • Knowledge of Munkres' "Topology" (2nd Edition)
  • Basic skills in constructing linear maps
NEXT STEPS
  • Study the concept of homotopy equivalence in topology
  • Learn about the construction of homotopy paths in detail
  • Explore examples of paths in topological spaces using Munkres' framework
  • Investigate visual representations of homotopy paths and their proofs
USEFUL FOR

Mathematics students, particularly those studying topology, educators teaching homotopy theory, and researchers interested in path homotopy concepts.

valtz
Messages
7
Reaction score
0
i stuck when i want to prove theorem 51.3 in munkres 2en editions about homotopy paths

Let f be a path in X , and let a0 , ... , an be numbers such that 0= a0 < a1 < ... < an. Let fi : I → X be the path that equals the positive linear map of I onto [ai-1, ai] followed by f then

[f] = [f1] * ... * [fn]


any idea to start prove this theorems?
 
Physics news on Phys.org
draw a picture.
 

Similar threads

What is the Acyclic Carrier Theorem and how can it be applied using chain maps?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Doubt about theorem in Calculus on Manifolds
  • · Replies 9 ·
Replies
9
Views
3K
Question regarding Munkres' proof of the Tychonoff theorem
  • · Replies 13 ·
Replies
13
Views
6K
An detail in proving the homotopy invariance of homology
  • · Replies 3 ·
Replies
3
Views
4K
Proving theorem for polynomials
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Is My Understanding of Homotopy and the Fundamental Group Correct?
  • · Replies 7 ·
Replies
7
Views
4K
Graduate Proof of Seifert-Van Kampen Theorem
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Application of the Stone Weierstrass Theorem
  • · Replies 9 ·
Replies
9
Views
2K
Linear algebra, field morphisms and linear independence
  • · Replies 1 ·
Replies
1
Views
1K