# Math 871–872 Qualifying Exam

## June 2011

Solve three problems from Section A and three more from Section B; you may work on any number of the problems, but indicate which six you want graded.
When in doubt about the wording of a problem, ask for clarification.
Do not interpret a problem in such a way that it becomes trivial.
Justify your answers.

## Section A:

(1) Prove that if $f : X \to Y$ is continuous map between topological spaces and C is a compact subset of X, then f (C) is a compact subset of Y .

(2) Suppose $f : X \to Y$ is a quotient map.
Prove that if Y is connected and $f ^ { - 1 } ( \{ y \} )$ is a connected subspace of X for all $y \in Y$ , then X is connected.

(3) Prove that every metrizable space is normal Hausdorff (aka T4).

(4) Suppose A, B are disjoint, compact subspaces of the Hausdorff topological space X. Prove there are open subsets U, V of X such that $A \subseteq U , B \subseteq V$ and $U \cap V = \emptyset$

## Section B:

(5) Let X be the space obtained by deleting three distinct points from $\mathbb { R } ^ { 2 }$ . Compute $\pi _ { 1 } ( X )$

(6) View $S ^ { 3 }$ as the set of unit vectors in $\mathbb { R } ^ { 4 }$ , and consider the equivalence relation on them induced by $u \sim v { \mathrm { ~ i f ~ } } A \cdot u = v$ , where A is the matrix

$$
\left( { \begin{array} { c c c c } { 0 } & { 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 1 } \\ { 0 } & { 0 } & { - 1 } & { 0 } \end{array} } \right)
$$

Compute the fundamental group of the quotient space $S ^ { 3 } / \sim$

(7) (i) Describe a way of identifying pairs of faces of $\Delta ^ { 3 }$ (the standard three simplex) to produce a $\Delta$ complex structure on $S ^ { 3 }$ having a single 3 simplex.

(ii) Write down the chain complex corresponding to the ∆-complex in (i). Be sure to include the differentials of the complex, but you do not need to compute the homology.

(8) Let X be the space obtained from the sphere $S ^ { 2 }$ by joining the North and South poles together with a straight line segment.

(i) Describe a structure of a CW complex on X.

(ii) Compute the homology of X using cellular homology, with decomposition from (i).