Math 440, Fall 2014, Assignment 13

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Carefully define the following terms, then give one example and one non-example of each:

1. Homotopic (maps from $X$ to $Y$, relative to a subset $A\subseteq X$).
2. Contractible space.
3. Fundamental group (of a space $X$ relative to a point $x_0$).
4. Homomorphism on fundamental groups induced by a continuous map.
5. Homotopy equivalent spaces.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning dependence of the fundamental group on the base point.
2. Theorem concerning the homomorphisms induced on fundamental groups by homotopic maps.
3. Theorem concerning fundamental groups of homeomorphic spaces.
4. Theorem concerning fundamental groups of homotopy equivalent spaces.

Solve the following problems:

1. Prove that for any spaces $X,Y$ and any $x_0\in X$ and $y_0\in Y$, we have $\pi_1(X\times Y, (x_0,y_0)) \simeq \pi_1(X,x_0)\times\pi_1(Y,y_0)$.
2. Suppose $X$ is contractible to $x_0$. Prove that $\pi_1(X,x_0)$ is a trivial group.
3. Prove that $\pi_1(\mathbb{R}^n,\vec{v})$ is trivial, for any $n$ and any base point $\vec{v}$.

--------------------End of assignment--------------------