- #1

Ben2

- 37

- 9

- Homework Statement
- "Show that the composition of homotopy equivalences X-->Y and Y-->Z is a homotopy equivalence X-->Z..."

- Relevant Equations
- F(x,t) = f_t(x)

"[A] map f: X-->Y is called a \mathbf{homotopy~equivalence} if there is a map g: Y-->X such that fg\cong\mathbb{1} and gf\cong\mathbb{1}," where "cong" means "is homotopic." "The spaces X and Y are said to be \mathbf{homotopy~equivalent}..." Additional definitions are in Hatcher, "Algebraic Topology", of which this is part of Exercise 3(a), p. 18. My difficulty is proving that if f\cong\mathbb{1} and h is another homotopy, then f = f\mathbb{1}h\congh. That is, how do we handle an arbitrary homotopy on the RIGHT? Does this involve "associativity" of homotopies? Thanks!