The Cup Product of the Klein Bottle ($\mathbb{Z}/2$)

I didn’t want to use a brute force method, so I thought for a while about how to compute the cup product of the Klein bottle. This is what I came up with. I thought I’d share it.

Take coefficients in $\mathbb{Z}/2$. We compute the cup product by computing the intersection form (which defines the intersection product), then Poincare duality.

The Klein bottle is compact and $\mathbb{Z}/2$-orientable, so Poincare duality applies: $$a^* \smile b^* = [a.b][K]$$

where $[K]$ is the fundamental class of the Klein bottle, $[a.b]$ is the number of intersections of the cycles $a$ and $b$ (thought of as sub-manifolds embedded in the Klein bottle). Note that $a^*$ and $b^*$ denote the cocycles corresponding to the cycles $a$ and $b$ respectively.

We view the Klein bottle as the fiber bundle $$S^1 \to K \to S^1$$ Fix a fiber circle, $F$, and denote the base circle as $B$.

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By pictoral argument: $F$ has zero transverse intersection with itself, and intersects the base circle $B$ once. By wiggling the base circle by an infinitesimal amount $\epsilon$, we see that $B$ intersects transversely with itself once.

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Another perspective:

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We may more succinctly state the intersection product as the following matrix:

$$\begin{pmatrix} \beta \smile \beta & \beta \smile \gamma \\ \gamma \smile \beta & \gamma \smile \gamma \end{pmatrix} \simeq \begin{pmatrix} [B.B] & [B.F] \\ [F.B] & [F.F] \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$

From the above matrix, we see that $\beta^2 = \beta\gamma = 1[K]$, and $\gamma^2 = 0[K]$. And, of course, $\beta^3 = 0$, because $H^3(K) = 0$. This gives us the ring structure of $H^*(K; \mathbb{Z}/2)$: $$(\mathbb{Z}/2\mathbb{Z})[\beta, \gamma]/(\beta^2 + \beta\gamma, \gamma^2, \beta^3)$$

 

P.S. If we wanted to be totally rigorous, we’d have to show that the cocycles $\beta$ and $\gamma$¬†form the basis of the $H^1(K; \mathbb{Z}/2) \simeq \mathbb{Z}/2 \oplus \mathbb{Z}/2)$. Until then, we don’t quite have that the intersection matrix gives us the cup product of $H^*(K; \mathbb{Z}/2)$.

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