Covering spaces of Klein bottle

hbghlyj

Hatcher 1.3 Problem 20
Give examples of a covering space $Y$ of a Klein bottle $K$ where

• $Y$ is a torus and the cover is normal
• $Y$ is a torus and the cover is not a normal cover
• $Y$ is a Klein bottle and the cover is not a normal cover

hbghlyj

(a) A normal covering space \(Y\) of the Klein bottle \(K\) where \(Y\) is a torus is given by the standard 2-sheeted orienting double cover \(p: T \to K\). This corresponds to the normal subgroup \(H = \langle a, b^2 \rangle \cong \mathbb{Z} \oplus \mathbb{Z}\) of \(\pi_1(K) = \langle a, b \mid abab^{-1} \rangle\), which has index 2 (and is thus normal). The covering map can be constructed by taking two copies of the fundamental square for \(K\) and identifying them appropriately to form the torus, or algebraically via the quotient by the action that flips orientation. To arrive at this, note that the orienting cover of a non-orientable manifold is always a double cover by its orientable counterpart, and index-2 subgroups are always normal.

(b) A non-normal covering space \(Y\) of the Klein bottle \(K\) where \(Y\) is a torus is given by the covering corresponding to the subgroup \(H = \langle a^3, a^2 b^2 \rangle \cong \mathbb{Z} \oplus \mathbb{Z}\) of \(\pi_1(K) = \langle a, b \mid abab^{-1} \rangle\). This subgroup has finite index but is not invariant under the conjugation action \(a \mapsto a^{-1}\) (which arises from the group structure), making the cover non-normal. To arrive at this, classify abelian subgroups of finite index in \(\pi_1(K)\) that are isomorphic to \(\mathbb{Z}^2\); among them, some (like this one) fail to be normal because their conjugates are not contained in themselves, as verified by direct computation of conjugates of generators.

(c) A non-normal covering space \(Y\) of the Klein bottle \(K\) where \(Y\) is a Klein bottle is given by the 3-sheeted cover \(f: K \to K\) corresponding to the subgroup \(H = \langle a^3, b \rangle \cong \pi_1(K)\) of \(\pi_1(K) = \langle a, b \mid abab^{-1} \rangle\), induced by the homomorphism \(\phi: \pi_1(K) \to \pi_1(K)\) sending \(a \mapsto a^3\), \(b \mapsto b\). This homomorphism is injective (yielding the isomorphism) and has image of index 3, but \(H\) is not normal because, for example, \(aba^{-1} = a^2 b \notin H\). To arrive at this, construct an endomorphism of \(\pi_1(K)\) that is injective but whose image is not normal; verify non-normality by checking that some conjugates of elements in \(H\) lie outside \(H\), and confirm the degree via the index.