Reference no: EM131106552

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.