Section 51: Problem 1 Solution »

Section 51: Homotopy of Paths

Two continuous funcitons and from to are called homotopic if there is a continuous map ( ) such that and for all . itself is called a homotopy between and .
  • Notation: .
  • is an equivalence relation.
  • If is constant, is called nulhomotopic.
  • A space is said to be contractible if the identity map is nulhomotopic.
Two paths and in (with the same domain ) are called path homotopic if there is a homotopy between and such that and are constant.
  • Notation: .
  • is an equivalence relation.
  • represents the path-homotopy equivalence class of .
If and are two paths in such that , then the product of and is the joint path from to .
  • is well-defined.
  • satisfies the groupoid properties: associativity, when applied, the existence of the left and right identities, and the existence of the inverse using the reverse of : .