2011 Honors Exam in Topology
Point-set topology
(1) Prove that a topological space X is connected if and only if every continuous map from X to the rational numbers Q (topologized as a subspace of the real line) is constant.
(2) Recall that a topology on X is called T1 if points are closed. Find a T1 topology on the real line R in which every subset of R is compact.
(3) Let P2 denote the real projective plane, a point of which is a line through the origin in R3.
(a) Describe P2 as a quotient space of the sphere S2 via a quotient map p: S2 → P2.
(b) Define f: S2 → R4 for
f(x, y, z) = (x2 - y2 , xy, xz, yz).
Prove that there is a continuous map f-: P2 → R4 such that f- o p = f.
(c) Prove that f- is an embedding; that is, prove that P2 is homeomorphic to the subspace f-(P2) of R4.
Fundamental group and surfaces
(4) Suppose X is a path connected, locally path connected topological space with π1(X, x0) = Z/2011Z.
Prove that every map X → S1 is nulhomotopic.
(5) Consider the Klein bottle K and the torus T.
(a) Show that there is a 2-fold covering map T → K.
(b) Show that there is a 3-fold covering map K → K.
(c) Generalize the preceding two parts from 2 and 3 to all positive integers. That is, for every positive integer n, construct a 2n-fold covering map from T to K, and a 2n - 1-fold covering map from K to K.
(6) (a) Describe the fundamental group and the universal cover of the projective plane P2.
(b) Describe the fundamental group and universal cover of P2 ∨P2, the space consisting of two projective planes tied together at one point.
(7) Give a construction of a path connected space whose fundamental group is S3, the group of permutations of a set with 3 elements. It might help to write down generators and relations for S3.
Homology
(8) Find all the homology groups of the 3-skeleton of the 5-simplex. (The 3-skeleton is the subcomplex
consisting of all the simplices of dimension at most 3).
(9) Giving two positive integers n and k, the Moore space Mn(k) is the space Sn ∪f en+1 obtained by gluing the boundary of an n + 1 disk en+1 to Sn via a map f: Sn → Sn of degree k.
(a) Use your knowledge of the homology of spheres to compute the homology of Mn(k).
(b) Prove that any map g: Mn(k) → Mn(k) has a fixed point.
(10) Suppose f: A → B and g: B → C are homomorphisms of abelian groups. Recall that the cokernel of f is coker f = B/f(A). Prove the existence of an exact sequence
0 → ker f → ker(g ? f) → ker g
→ coker f → coker(g ? f) → coker g → 0.