## Warm-up Problems:

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

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

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

(1) (Michigan May ’09) Let X be the space obtained from $S ^ { 1 } \times \mathbb { R }$ by removing the interior of k disjoint 2-disks.

(a) Compute the fundamental group $\pi _ { 1 } ( X )$

(b) What would be your answer to part $\mathbf { \Psi } ( \mathrm { a } ) \mathrm { i f } \ S ^ { 1 } \times \mathbb { R }$ is replaced by $S ^ { 2 } \times \mathbb { R }$ and 2-disks are replaced by 3-balls?

(c) Let Y be the union of two copies of the real projective plane R $P ^ { 2 }$ having exactly one point y in common.
Compute $\pi _ { 1 } ( Y , y )$

(2) State the Seifert-van Kampen Theorem and use it to show that $\pi _ { 1 } ( S ^ { n } ) = 0 { \mathrm { f o r } } n > 1$ . Why doesn’t this argument work for $n = 1 2$

(3) (Wisconsin Aug ’10) The graph K has six vertices $a _ { 1 } , a _ { 2 } , a _ { 3 } , b _ { 1 } , b _ { 2 } , b _ { 3 }$ and nine edges $a _ { i } b _ { j }$ for $i , j = 1 , 2 , 3$ . The space X obtained from K by attaching a 2-cell along each loop formed by a cycle of four edges in K. Find $\pi _ { 1 } ( X )$ .

(4) (Jan ’02) Find the fundamental group of the space X consisting of $\mathbb { R } ^ { 3 }$ with the three coordinate axes removed.

(5) (June ’05) 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.

(6) (June ’08) Let X be the triangle parachute formed from the standard 2-simplex $\Delta ^ { 2 }$ by identifying the three vertices with one another.
Compute a presentation for $\pi _ { 1 } ( X )$ and show that $\pi _ { 1 } ( X )$ is isomorphic to a free group $F _ { n }$ (and identify which n!).

(7) Attach a 2-disc $D ^ { 2 }$ to a torus $S ^ { 1 } \times S ^ { 1 }$ by the attaching map $e ^ { 2 \pi i t } \mapsto \left( e ^ { 2 \pi i t } , e ^ { 2 \pi i t } \right)$ , 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.

(8) (June ’11) Let X be the space obtained by deleting three distinct points from $\mathbb { R } ^ { 2 }$ . Compute $\pi _ { 1 } \left( X \right)$

(9) (Jan ’12) Let $X = \mathbb { R } ^ { 3 } \backslash S ^ { 1 }$ , the complement of a single circle in $\mathbb { R } ^ { 3 }$

(a) Describe a deformation retraction of X onto $S ^ { 1 } \vee S ^ { 2 }$

(b) Compute $\pi _ { 1 } \left( X \right)$

(10) (June ’12) Let X be the space obtained from the torus $T ^ { 2 } = S ^ { 1 } \times S ^ { 1 }$ by gluing in a disk $D ^ { 2 }$ along its boundary $\partial D ^ { 2 } = S ^ { 1 }$ using the map $\alpha : S ^ { 1 } \to T ^ { 2 }$ given by $z \mapsto ( z , ( 1 , 0 ) )$ for $z \in S ^ { 1 }$

(a) Find $\pi _ { 1 } \left( X \right)$

(b) Give a CW complex for X.

Covering Space Problems:

• (June ’10) 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.

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