Sheaf Theory in Geometry: Homework 9 for Math 6490, Spring 2009, Assignments of Geometry

Download Sheaf Theory in Geometry: Homework 9 for Math 6490, Spring 2009 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 9 Exercise 1. Let X be the disjoint union of the topological spaces X1 and X2. For any abelian group A, show that Hq(X, A) ' Hq(X1, A)⊕Hq(X2, A). Exercise 2. Let A be a subset of X. We say that A is a retract of X if there is a continuous map r : X → A such that r(a) = a for all a ∈ A. Show that (1) S1 is a retract of D2 − {0}. (2) S1 is not a retract of D2. Exercise 3. Show that the map e : R → S1 given by e(t) = e2πit does not have a continuous section, i.e., there does not exist a continuous map s : S1 → R such that e(s(x)) = x for all x ∈ S1. Exercise 4. Show that Rn is homeomorphic to Rm if and only if n = m. Exercise 5. Let X be the one point union of two circles. (So X is the figure “eight” and geometrically looks like 8.) Compute the cohomology groups Hq(X, Z). (Hint: Let A and B be the two circles and let P = A ∩ B be their common point of intersection in X. Let a : A ↪→ X, b : B ↪→ X and p : P ↪→ X be the inclusion maps. Consider the short exact sequence 0 → ZX → a∗ZA ⊕ b∗ZB → p∗ZP → 0 of sheaves on X.) ∗ ∗ ∗ ∗ ∗ ∗ ∗