Many questions about a Klein bottle.

Question

1. What are a CW complex structure and a $\Delta$-complex structure on the Klein bottle?
2. What is the fundamental group of the Klein bottle?
3. What is the simplicial homology of the Klein bottle?
4. What is the cellular homology of the Klein bottle?
5. What is a description of the isomorphism between the cellular and simplicial homology?
6. What is a description of the abelianization map from the fundamental group to $H_1$?

Answer 1

1. 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).

2. $\pi_1(K) \cong \langle[a],[b] : [a][b] = [b][a]^{-1} \rangle$. 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.

3. 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 $\pi_1$.

4. 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 $(3)$.

5. See Hatcher (for example Theorem 2.35).

6. 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.