Topology II -- midterm

1. Show that for any graph $G,$ $v(G)\leq e(G)+1,$ where $v$ and $e$ are the number of vertices and edges in $G$ respectively. Also show that the this is an equality if and only if $G$ is a tree.
2. Suppose $X$ is the plane and $x,y\in X,x\neq y.$ Find a homeomorphism of $X$ that takes $x$ to $y.$ Same problem for $X$ the torus.
3. Prove that the product of two path-connected spaces is path-connected.
4. Suppose $f:\mathbf{S}^{1}\rightarrow\mathbf{S}^{1}$ is a map of the unit circle in the plane that isn't homotopic to the identity. Prove that $f(x)=-x$ for some $x.$
5. Describe the fundamental group and provide its generators of the following spaces (no proof needed): (1) the figure eight; (2) the torus; (3) the sphere; (4) the sphere with a circle attached to the north pole; (5) the "wire-frame" pyramid.
6. The boundary of the Mobius band $M$ is a circle. Let be $f:\mathbf{S} ^{1}\rightarrow M$ the function that wraps the circle onto this boundary. Compute the homomorphism of the fundamental groups generated by this map.