2. Homeomorphisms and homotopy equivalent spaces. (14 October
... f : X → Y and g : Y → X such that g ◦ f is homotopic to the identity map of X and f ◦ g is homotopic to the identity map of X. Problem 6. Prove that the spaces S 1 ∨ I and S 1 are homotopy equivalent. Problem 7. a) Prove that if the image of a map f : X → S 1 is not the whole space S 1 (i.e., the ma ...
... f : X → Y and g : Y → X such that g ◦ f is homotopic to the identity map of X and f ◦ g is homotopic to the identity map of X. Problem 6. Prove that the spaces S 1 ∨ I and S 1 are homotopy equivalent. Problem 7. a) Prove that if the image of a map f : X → S 1 is not the whole space S 1 (i.e., the ma ...
HOMEWORK 6
... 1. If A is a retract of a contractible space X, then A is a deformation retraction of X. 2. Show that a retract of a contractible space is contractible. 3. Find a retraction from the punctured plane R2 \ {0} to the unit circle S 1 . 4. Show that the closed interval [0, 1] is a deformation retract of ...
... 1. If A is a retract of a contractible space X, then A is a deformation retraction of X. 2. Show that a retract of a contractible space is contractible. 3. Find a retraction from the punctured plane R2 \ {0} to the unit circle S 1 . 4. Show that the closed interval [0, 1] is a deformation retract of ...
Topology/Geometry Aug 2011
... GEOMETRY/TOPOLOGY PRELIMINARY EXAM AUGUST 2011 1. Let f : R2 → R be a continuous function. Define an equivalence relation on R2 by x ∼ y if and only if f (x) = f (y). Let X be the quotient space. (a) Show that X is always Hausdorff. (b) Must X be connected? 2. Let D2 denote the unit disc in R2 with ...
... GEOMETRY/TOPOLOGY PRELIMINARY EXAM AUGUST 2011 1. Let f : R2 → R be a continuous function. Define an equivalence relation on R2 by x ∼ y if and only if f (x) = f (y). Let X be the quotient space. (a) Show that X is always Hausdorff. (b) Must X be connected? 2. Let D2 denote the unit disc in R2 with ...