## Warm-up Problems

• Let $G = \langle x , y \vert x ^ { 2 } y x y = 1 \rangle$ . Describe a space X that has $\pi _ { 1 } ( X ) = G$

• Find two spaces X and Y such that $\pi _ { 1 } ( X ) \cong \pi _ { 1 } ( Y )$ and are nontrivial, and are NOT homeomorphic.

• Find $\pi _ { 1 } ( T ^ { 2 } )$ , where $T ^ { 2 } = S ^ { 1 } \times S ^ { 1 }$ is the standard torus, in two different ways.

1. (Jan 2002, B2) Find the fundamental group of the space X consisting of $\mathbb { R } ^ { 3 }$ with the three coordinate axes removed.

2. (June 2005, B5) Find a cell structure for the quotient space obtained by identifying two distinct points a, b in a 2-torus to a third point c in a 2-sphere, and compute a presentation for the fundamental group of this space.

3. (June 2007, B3) Let X be the surface of genus 2 shown below and let $Y \supseteq X$ be the region of $\mathbb { R } ^ { 3 }$ that it encloses. Show that there is no retraction $r : Y  X$ (that is, there is no continuous map for which $r ( x ) = x$ for all $x \in X )$ .

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4. (June 2008, B5) Let X be the triangular parachute formed from the standard 2-simplex $\dot { \Delta } ^ { 2 }$ by identifying the three vertices with one another.

(a) Compute a presentation for $\pi _ { 1 } ( X )$

(b) Show that $\pi _ { 1 } ( X )$ is isomorphic to a free group $F _ { n }$ for some n; what is n?

5. (June 2010, B5) Show that if A is a path-connected subspace of $X , x _ { 0 } \in A$ , and the homomorphism $i _ { * } : \pi _ { 1 } ( A , x _ { 0 } )  \pi _ { 1 } ( X , x _ { 0 } )$ induced by the inclusion map $i : A \to X$ is surjective, then every path in X with endpoints in A is homotopic relative to the endpoints (ie, via a homotopy that fixes each endpoint) to a path in A.

6. Attach a 2-disk $D ^ { 2 }$ to a torus $S ^ { 1 } \times S ^ { 1 }$ by the attaching map $e ^ { 2 \pi i t } \mapsto ( e ^ { 2 \pi i t } , e ^ { 2 \pi i t } )$ , thinking of the boundary $\partial D ^ { 2 }$ and each $S ^ { 1 }$ factor as the unit circle in the complex plane, and let X be the resulting space. Compute a presentation for the fundamental group of X.

7. (June 2011, B5) Let X be the space obtained by deleting three distinct points from $\mathbb { R } ^ { 2 }$ Compute $\pi _ { 1 } ( X )$ .

Covering space bonus problems:

1. (June 2010, B7) Use covering space theory to show that if H is a subgroup of index 3 in a finitely presented group $G ,$ then H is finitely presented.

2. Let $F ( a , b )$ be the free group of two generators. What is the presentation complex and the Cayley complex for $F ( a , b ) ?$ Find an index 4 subgroup and the covering space that corresponds to it.