Many questions about a Klein bottle. 
*

*What are a CW complex structure and a $\Delta$-complex structure on the Klein bottle?

*What is the fundamental group of the Klein bottle?

*What is the simplicial homology of the Klein bottle?

*What is the cellular homology of the Klein bottle?

*What is a description of the isomorphism between the cellular and simplicial homology?

*What is a description of the abelianization map from the fundamental group to $H_1$?

 A: *

*Look at the fundamental polygon for the Klein bottle: $abab^{-1}$.
This is a $\Delta$-complex with one 0-simplex, three 1-simplicies, and two 2-simplicies.  
Note that you have more flexibility in gluing when using a CW structure.  In particular, you can construct the Klein bottle with one 0-cell, two 1-cells, and one 2-cell (think of just using a sheet of paper and gluing the edges as instructed in the fundamental polygon, so you don't have to worry about the edge "$c$" in particular).


*$\pi_1(K) \cong \langle[a],[b] : [a][b] = [b][a]^{-1} \rangle$.  This can be done via Van Kampen and decomposing the polygon (of $K$, without the edge $c$) into two open sets $A$ and $B$ where you can take $A$ to be the union of all edges thickened a bit, and $B$ to be the inside of the polygon with a bit of overlap with $A$ so that $A \cap B$ looks like a frame.

*Look at the above picture and do the computation.  You should get, with coefficients in $\mathbb{Z}$, $H_0(K) = \mathbb{Z}$, $H_1(K) = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, and $H_2(K) = 0$ (and higher groups vanish too).  A sanity check is: $K$ is connected, a surface, and nonorientable, forcing $H_0$ to be $\mathbb{Z}$ and $H_2$ to be $0$.  We calculated the fundamental group and Hurewicz tells us $H_1$ is the abelianization of $\pi_1$.

*The complex looks like (due to the above CW structure)
$$0 \to \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to 0$$
The only nonzero map is the one from $\mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$.  This, from the picture above, is given by (WLOG) multiplication by $2$ in the first coordinate and $0$ in the second, assuming the first $\mathbb{Z}$ is generated by $a$ and the second by $b$.  Now, that you have all the maps, you will get the same answer that you got in $(3)$.

*See Hatcher (for example Theorem 2.35).

*See Hurewicz theorem or Hatcher 2.A.  The basic intuition you should go in with is that $f: I \to X$ can be viewed as both a path and a singular 1-simplex.
