Note Let f X Y and g X Y be continuous maps If f g Y

(Note: Let f : X ? Y and g : X ? Y be continuous maps. If f ? g : Y ? Y is homotopic to id_y and g ? f : X ? X is homotopic to id_x then we say that f (and g) is a homotopy equivalence (abbreviated h.e) and say that g is the homotopy inverse of f)

Solution

Recall that two pointed spaces (X, x0) and (Y, y0) are homotopy equivalent if there exist maps f : (X, x0) Y and g : (Y, y0) (X, x0) where f g 1_Y and g f 1_X. Here the relation ‘\'’ is the appropriate concept of ‘homotopic’ for pointed maps: that is, h \' h 0 if that there exists a pointed homotopy from h to h 0 . We need to check that homotopy equivalence is reflexive, symmetric, and transitive.

• Reflexive - for all X C, 1_X : (X, x0) (X, x0) gives us that 1_X 1_X = 1_X, so X is homotopy equivalent to X.

• Symmetric - Let X, Y C. If X is homotopy equivalent to Y , then by definition of homotopy equivalence, Y is homotopy equivalent to X. This is because the definition is symmetrical

. • Transitive - Let X, Y, Z C where X is homotopy equivalent to Y , and Y is homotopy equivalent to Z. This implies there exist f : X Y and g : Y X with f g 1_Y and g f \' X. Also, there exist h: Y Z and k : Z Y with h k \' 1_Z and k h \' 1_Y . Now consider h f : X Z and g k : Z X.

For each step, we use the fact that composition preserves the relation \'. First h f g k \' h 1_Y k \' h k \' 1_Z Similarly we have the other direction. g k h f \' g 1_Y f \' g f 1_X Thus X is homotopy equivalent to Z.

Therefore homotopy equivalence is an equivalence relation.

Let A be a retract of X, and let X be contractible. This means 1X is homotopic to a constant map, say f(x) = x0. Let H : X × I X be the homotopy from 1X to f with H(x, 0) = x and H(x, 1) = f(x) = x0. If r : X A is the retraction of X to A, then we can consider the homotopy r H|A : A × I A. This is continuous since it is the composition of continuous maps. Also r H|A(x, 0) = r(x) = x , since x A, and r H|A(x, 1) = r(f(x)) = r(x0). This gives us that 1A is homotopic to the constant map g(x) = r(x0), and so A is contractible.