Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this.
Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by \begin{align} A(x,y,z)&=(x+1,y,z)\\ B(x,y,z)&=(x,y+1,z)\\ C(x,y,z)&=(x+y,y,z+1). \end{align} Assume that $M=\mathbb{R}^3/G$ is a three-manifold.
a. Determine $\pi_1(M)$ up to isomorphism by giving a presentation with generators and relations. (note the relations will involve only commutators of generators)
b. Determine $H_{DR}^1(M)$ up to isomorphism and give closed one-forms whose cohomology classes are a basis of $H_{DR}^1(M)$.
Here's all I've got:
We know that $G$ acts properly discontinuously on $\mathbb{R}^3$ so $\pi_1(M)=G$, but I'm not sure how to write $G$.
When we have $G$, we know that $H_{DR}^1(X)=Hom(H_1(X),\mathbb{R})$ where $H_1(X)$ is $\pi_1(X)$ abelianized. I'm also not sure how to find the 1-forms giving a basis.
Thank you.