# Math 871-872 Qualifying Exam May 2015

Do three questions from Section A and three questions from Section B. You may work on as many questions as you wish, but indicate which ones you want graded.
When in doubt about the wording of a problem or what results may be assumed without proof, ask for clarification.
Do not interpret a problem in such a way that it becomes trivial.

Section A: Do THREE problems from this section.

(1) Let $\begin{array} { r } { X = \prod _ { n \in \mathbb { Z } } [ 0 , 1 ] = \{ ( x _ { n } ) _ { n \in \mathbb { Z } } \ : \ x _ { n } \in [ 0 , 1 ] \} } \end{array}$ , with each [0, 1] having the usual Euclidean topology and X having the product topology, and let

$A = \{ ( x _ { n } ) \in X$ : for some N we have $x _ { n } = 0$ for all $n \geq N \}$

(a) Show that A is a dense subset of X, and is therefore not a closed subset of X.

(b) Show that, as a consequence, A is not a compact subset of X.

(2) If $X = \mathbb { R } ^ { n }$ with the usual Euclidean topology and $\mathcal { U } \subseteq X$ is an open, connected subspace of X, show that U is also path connected.

(3) Let $f : ( X , { \mathcal { T } } ) \to ( Y , { \mathcal { T } } ^ { \prime } )$ be a continuous map.
A section of f is a continuous map $\sigma : ( Y , { \mathcal { T } } ^ { \prime } ) \to ( X , { \mathcal { T } } )$ so that $f \circ \sigma = \operatorname { I d } _ { Y }$ . Show that if f has a section, then f is a quotient map.

(4) Let $f : X \to Y$ be a continuous map, and suppose that there are continuous maps $g , h : Y \to X$ with $f \circ g \simeq \operatorname { I d } _ { Y }$ and h ◦ $f \simeq \operatorname { I d } _ { X }$

(a) Show that $g \simeq h$

(b) Show, moreover, that f is a homotopy equivalence.

Section B: Do THREE problems from this section.

(5) Suppose $X , X _ { 1 } , X _ { 2 }$ are path-connected spaces, $p : X _ { 1 } \to X$ and $q : X _ { 2 } \to X$ are (surjective) covering maps, and $f : X _ { 1 } \to X _ { 2 }$ is a continuous map with $q \circ f = p$ . Show that f is also a covering map.

(6) Show, using covering space theory, that a finitely presented group G has only finitely many distinct subgroups $H \leq G$ of index 4.

(7) Find the (simplicial) homology groups of the space X obtained from the 3-simplex $\Delta ^ { 3 } =$ $[ v _ { 0 } , v _ { 1 } , v _ { 2 } , v _ { 3 } ]$ by gluing the face $[ v _ { 0 } , v _ { 1 } , v _ { 2 } ]$ to the face $[ v _ { 0 } , v _ { 2 } , v _ { 3 } ]$ and the face $[ v _ { 0 } , v _ { 1 } , v _ { 3 } ]$ to the face $[ v _ { 1 } , v _ { 2 } , v _ { 3 } ]$ (respecting the ordering of vertices as written (i.e., the first gluing map sends v1 to v2, and so on)).

(8) Compute the (reduced) singular homology groups of the space $X = S ^ { 1 } \times ( S ^ { 1 } \vee S ^ { 1 } )$ , which can be thought of as two copies of $S ^ { 1 } \times S ^ { 1 }$ glued together along their copies of $S ^ { 1 } \times \{ x _ { 0 } \}$ [You may use your knowledge of the homology groups of $T ^ { 2 } = S ^ { 1 } \times S ^ { 1 }$ in your calculations.]