HOMEWORK

Assignment 1

Problem 1. Consider the correspondence which associates to an abelian group G its torsion subgroup tor(G).
a).Show tor is a covariant functor from the category of Abelian Groups to itself.
b). If a group homomorphism from G to K is injective, show tor(f) is also injective.
c). Find an example of G and K and f a homomorphism from G to K which is surjective, but tor(f) is not surjective.

Problem 2. Let p be a prime integer. Define for G an abelian group, F_p(G) = G/pG.
a) Show F_p is a functor from the category of abelian groups to itself.
b) If f a homomorphism from G to K is surjective, show that F_p(f) is surjective.
c) Give an example of G, K and f an injective homomorphism from G to K for which F_p(f) is not injective.

Problem 3. Let F be a covariant functor from category C to Category D. Then there exists a functor G from D to C such that F and G define an equivalence of categories if and only if F is fully faithful and for every object A' of D there exists an object A of C such that F(A) and A' are isomorphic in D.

Assignment 2

Problem 1. Clearly S^1 is homeomorphic to the quotient space I/{0,1} where I is the closed unit interval. Prove (S^1)X(S^1) is homeomorphic to the quotient space (I)X(I)/~ where (a,0) ~ (a,1) and (0,a) ~ (1,a) for all a in I.

Problem 2. Corrected definition: A topological space X is locally connected if for every point x in X and every open neighborhood U of x there is a connected open neighborhood V of x with V contained in U. Prove X is locally connected if and only if the connected components of open sets are open.

Problem 3. Prove a locally pathwise connected space is locally connected.

Problems 4-7. Problems 3.7, 3.8. 3.11, 3.12 on pp14-15, Greenberg and Harper.

Problem 8. Definition: A subspace A of a topological space X is called a retract of X if there exists a continuous map r from X to A (called a retraction) such that r(a) = a, for all a in A. Prove that a retract of a Hausdorff space must be a closed subset.

Assignment 3

Problems 1,2: Pages 19-20, Greenberg-Harper, Exercises 4.12 and 4.13.

Problem 3. Let N=(0,...0,1) and S=(0,...0,-1) be the North and South pole of S^n. Show that the equator S^(n-1) of S^n is a deformation retract of S^n - {S, N}, so S^(n-1) and S^n - {S,N} have the same homotopy type.

Problem 4. For fixed t with t in [0,1) prove that the map x into [x,t] defines a homeomorphism from X into CX. Prove that any space can be embedded in a contractible space.

Problem 5. Let X be a space. Show that there is a category C with object(C) = points of X. Let p, q be points of X, then the morphisms from p to q in C all path-classes [f] where f is a path in X from p to q and homotopy is relative homotopy mod {0,1}. Show that every morphism in C is an isomorphism.

Problem 6. If (X,x_0) is a pointed space let the path component C of X containing x_0 be the distinguishd element of (pi_0)(X). Show with this definition pi_0 defines a functor from the category of pointed topological spaces to the category of pointed sets.

Asssignment 4

Problem 1. If f is a closed path in S^1 at 1 and m is an integer, show t goes to (f(t))^m is a closed path in S^1 at 1 and deg(f^m) = m deg(f).

Problem 2. Prove that S^1 is not a retract of the closed unit disk, D^2, in R^2 (or C=complex numbers).

Problem 3. Show that if f is a continuous map from D^2 to D^2 without fixed points then S^1 is a retract of D^2. Conclude (Brouwer fixed point for D^2): every continuous map from D^2 to itself has at least one fixed point.

Problem 4. If f is a closed path in S^1 at 1 which is not surjective, show deg(f) = 0. Find an example of a closed path in S^1 at 1 which is surjective and has deg(f) = 0.

Problem 5. Let (X*,p) be a covering space of X. a). If X is Hausdorff then so is X*. b). If X is locally compact then so is X*.

Problem 6. Let (X*,p) be a covering space of X. Let U be an evenly covered open subset of X by p. Let V be an open subset of U. Show V is evenly covered by p.

Problem 7.Let (X*, p) be a covering space of X. Let f be a homeomorphism from a space Y onto X. Let g be a homeomorphism from a space Z onto X*. Let q be a map from Z to Y such that fq=pg (commutative diagram). Prove that (Z,q) is a covering space of Y.

Problem 8. Let Y be compact, and f a continuous map from Y to X which is a local homeomorphism. Show for any x in X the fiber of f over x (={y in Y such that f(y) = x}) is a finite set.

Asssignment 5

Problem 1. Prove that if X is simply connected and (X*, p) is a covering space of X then p is a homeomorphism from X* onto X.

Problem 2. Determine all covering spaces (up to isomorphism) of the circle S^1 and the real projective plane.

Problem 3. Let G be a connected and locally path connected topological group with unit e. Let (G*, p) be a covering space of G and e* a point in the fiber over e. Prove that there is a unique continuous multiplication defined on G* such that G* is a topological group with unit e* and such that p is a group homomorphism.

Problem 4. In problem 3 above, prove also that the kernel of p is a discrete normal subgroup of G*.

Problem 5. For the covering space R (real numbers) of S^1, find the group of deck transformations Cov(R/S^1). Same question for the covering space the 2-sphere, S^2, over the real projective plane.

Problem 6. Prove the real projective line is homeomorphic to S^1.

Problem 7. Exercise (5.10) page 25. Greenberg and Harper.

Problem 8. Let G be a simply connected topological group and let H be a discrete closed normal subgroup. Replace the exponential map we used in the argument that R/Z (i.e. S^1) has fundamental group = Z, by the natural map here taking G to G/H. Prove that the fundamental group of G/H is H.

Assignment 6

Problem 1. Let (X*, p) be a covering space of X, X connected and locally path connected. Let A be a path connected and locally path connected subset of X. Let A* be a path connected component of the inverse image of A by p. Show (A*, p~) is a covering space of A (here p~ is the restriction of p to A*).

Problem 2. Definition: Let Y be a topological space and G a group of homeomorphism of Y. G is said to act properly discontinuously on Y if every point y of Y has an open neighborhood such that the sets g(U) in Y for distinct g's in G are pairwise disjoint. Now, let Y be a Hausdorff space and let G be a finite group of homeomorphisms of Y such that each element of g has no fixed points. Prove that G acts properly discontinuously on Y.

Problem 3. Let G be a topological group and let H be a normal subgroup. Prove that G/H is a topological group where G/H is regarded as the quotient space of G by the kernel of the natural map.

Problem 4. A discrete normal subgroup H of a connected topological group is contained in the center of G and hence H is abelian. {Hint: Fix h in H. Consider the inverse image of any k in H under the map from G to H given by f(g) = gh(inverse of g)(inverse of h)}.

Assignment 7

Problem 1. Exercise 9.7, page 48. (Greenberg and Harper)

Problem 2. Exercise 9.12, page 50.(G and H).

Assignment 8

Problem 1. Assume f maps A to B, g maps B to C, h maps C to D and A, B, C, D are R-mods. Assume the sequence A to B to C to D is exact at B, and C. Show f is surjective if and only if h is injective.

Problem 2. If we have an exact sequence taking .... C_(n+1) to A_n to B_n to C_n to A_(n-1)..., and every map A_n to B_n is an isomorphism of R-mods, show that C_n = 0 for all n.

Problem 3. Exercise (10.16) on page 58, Greenberg and Harper.

Problem 4. Critique Rotman's proof of the homotopy axiom (Pp 75-77, Rotman)

Problem 5. In assignment 5, problem 3, establish the existence of a multiplication map from G*XG* to G* by showing that the multiplication in the fundamental group of (G,e) agrees with the induced map on fundamental groups coming from the multiplication map GXG to G.

Problem 6. Exercise 9.13 page 50, Greenberg and Harper.

Problem 7. Exercise 10.14, page 57, Greenberg and Harper.

Assignment 9

Problem 1. If f is a path in X (not necessarily closed) show that if f also denotes the corresponding singular 1-simplex in X then f + f* is a 1-boundary (here f* denotes the 1-simplex corresponding to the "reverse" path of f.

Problem 2. If f is a continuous map from S^1 to itself, define degree(f) to be the integer m if the map H_1(f) from H_1(S^1) to itself is multiplication by m. Previously if in addition f(1) = 1, we defined degree(f) to be m in a similar manner according to whether the map induced by f on the fundamental group of S^1, with respect to base point 1, was multiplication by m. Show that when f(1) = 1 the two definitions agree.

Problem 3. If i is less than or equal to n, view S^(i) as contained in S^n as usual. Compute H_m(S^n, S^(i)) for all m.

Problem 4. Write out a proof of Proposition (13.14) on page 73 in Greengerg and Harper.

Problem 5. Exercise (13.16) page 74. Greenberg and Harper

Problem 6 and 7. Exercise (14.15) and (14.16) on page 81 (G and H).

Problem 8. If f is a continuous map from X to Y, show that Sd^Y composed with S(f) is equal to S(f) composed with Sd^X from S_n(X) to S_n(Y) for all non-negative indices n.