Klein bottle and torus in mod $p$ homology

Question

I read that for mod $2$ homology, the Klein bottle and the torus are indistinguishable.

Q1) How does the second homology work for the Klein bottle? It is $0$ in integral homology. I suppose it is $\mathbb{Z}_2$ in mod $2$ homology so as to match the torus? How is this so?

Q2) Also, can we distinguish the Klein bottle and torus in mod $3$ homology? Is it true that for the torus, its first homology is $\mathbb{Z}_3\oplus\mathbb{Z}_3$, while for Klein bottle its first homology is just $\mathbb{Z}_3$?

Thanks for any help.

Answer 1

Let $M$ be a closed $n$-dimensional manifold and $R$ a commutative ring with identity.

If $M$ is orientable, then $H_n(M; R) \cong R$. If $M$ is non-orientable, then instead the top homology is isomorphic to the $2$-torsion in $R$, i.e. $H_n(M; R) \cong {}_2R = \{r \in R \mid 2r = 0\}$. See Hatcher's Algebraic Topology, Theorem $3.26$.

If we restrict our attention to $R = \mathbb{Z}_p$ for $p$ prime, then $H_n(M; \mathbb{Z}_2) \cong \mathbb{Z}_2$ while for $p \neq 2$ we have

$$H_n(M; \mathbb{Z}_p) \cong \begin{cases} \mathbb{Z}_p & M\ \text{orientable}\\ 0 & M\ \text{non-orientable}.\end{cases}$$

Let $T$ and $K$ denote the torus and Klein bottle respectively.

Note that $H_0(T; \mathbb{Z}_p) \cong \mathbb{Z}_p$ and $H_0(K; \mathbb{Z}_p) \cong \mathbb{Z}_p$ as $T$ and $K$ are path-connected. By the result above, we can know $H_2(T; \mathbb{Z}_p)$ and $H_2(K; \mathbb{Z}_p)$. What about $H_1$? First note that $\mathbb{Z}_p$ is a field, so $H_i(M; \mathbb{Z}_p)$ is a vector space over $\mathbb{Z}_p$ and is therefore determined up to isomorphism by its dimension. Now we use the fact that the alternating sum of the dimensions of the homology groups is the Euler characteristic.

For the torus, we have

\begin{align*} 0 &= \chi(T)\\ &= \dim H_0(T; \mathbb{Z}_p) - \dim H_1(T; \mathbb{Z}_p) + \dim H_2(T; \mathbb{Z}_p)\\ &= \dim\mathbb{Z}_p - \dim H_1(T; \mathbb{Z}_p) + \dim \mathbb{Z}_p\\ &= 1 - \dim H_1(T; \mathbb{Z}_p) + 1\\ &= 2 - \dim H_1(T; \mathbb{Z}_p) \end{align*}

so $\dim H_1(T; \mathbb{Z}_p) = 2$ and therefore $H_1(T; \mathbb{Z}_p) \cong \mathbb{Z}_p\oplus\mathbb{Z}_p$.

For the Klein bottle, we have two cases: $p = 2$ and $p \neq 2$. For $p = 2$, the calculation is analogous to the one above and we see that $H_1(K; \mathbb{Z}_2) \cong \mathbb{Z}_2\oplus\mathbb{Z}_2$. For $p \neq 2$, we have

\begin{align*} 0 &= \chi(K)\\ &= \dim H_0(K; \mathbb{Z}_p) - \dim H_1(K; \mathbb{Z}_p) + \dim H_2(K; \mathbb{Z}_p)\\ &= \dim\mathbb{Z}_2 - \dim H_1(T; \mathbb{Z}_2) + \dim 0\\ &= 1 - \dim H_1(T; \mathbb{Z}_p) + 0\\ &= 1 - \dim H_1(T; \mathbb{Z}_p) \end{align*}

so $\dim H_1(K; \mathbb{Z}_p) = 1$ and therefore $H_1(K; \mathbb{Z}_p) \cong \mathbb{Z}_p$.

In particular, we can distinguish the torus and the Klein bottle using $\mathbb{Z}_p$ homology for $p \neq 2$ (both the first and second homology groups differ).