https://metaphor.ethz.ch/x/2020/fs/401-3002-12L/ex/AlgTopExerciseSh...
... homeomorphically onto U by p. Show that every covering space of an orientable manifold is an orientable manifold. 3 ... . (∗)For a map f : M → N between closed orientable n-manifolds with fundamental classes [M ] and [N ], the ... homeomorphically onto U by p. Show that every covering space of an orientable manifold is an orientable manifold. 3 ... . (∗)For a map f : M → N between closed orientable n-manifolds with fundamental classes [M ] and [N ], the ...
https://metaphor.ethz.ch/x/2019/fs/401-3574-61L/files/Classificatio...
... identified by their orientability and Euler char- acteristic: the Mg are orientable with χ(Mg) = 2 − 2g ... , whereas the Nh are non- orientable with χ(Nh) = 2− h. Definition: The genus g of a closed surface S is ... identified by their orientability and Euler char- acteristic: the Mg are orientable with χ(Mg) = 2 − 2g ... , whereas the Nh are non- orientable with χ(Nh) = 2− h. Definition: The genus g of a closed surface S is ...
https://metaphor.ethz.ch/x/2019/fs/401-3002-12L/ex/ps4.pdf
... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ...
https://metaphor.ethz.ch/x/2019/fs/401-3002-12L/ex/ps5.pdf
... exception: If both M1 and M2 are non- orientable, then Hn−1(M1#M2) is obtained from Hn−1(M1) ⊕Hn−1(M2) by ... that if a closed orientable manifold of dimension 2n has Hn−1(M) torsion-free then Hn(M) is also ... exception: If both M1 and M2 are non- orientable, then Hn−1(M1#M2) is obtained from Hn−1(M1) ⊕Hn−1(M2) by ... that if a closed orientable manifold of dimension 2n has Hn−1(M) torsion-free then Hn(M) is also ...
https://metaphor.ethz.ch/x/2021/fs/401-3002-12L/ex/ps4.pdf
... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ...
https://metaphor.ethz.ch/x/2022/fs/401-3002-12L/ex/ps4.pdf
... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ... covering space of an orientable manifold is an orientable manifold. 2. Show that for a connected non ... - orientable manifold M there is a unique orientable double cover of M . 3. Show that for any connected closed ...
https://metaphor.ethz.ch/x/2019/fs/401-3574-61L/ex/sol10.pdf
... (Y ) = d− b. (b) This surface S is non- orientable, it has one boundary component and Euler ... ) − 2. The Euler characteristic and the genus of an orientable surface F are linked via χ(F ) = 2− 2g(F ... (Y ) = d− b. (b) This surface S is non- orientable, it has one boundary component and Euler ... ) − 2. The Euler characteristic and the genus of an orientable surface F are linked via χ(F ) = 2− 2g(F ...
https://metaphor.ethz.ch/x/2019/fs/401-3002-12L/ex/sol5.pdf
... is orientable (⇔ Hn(M) ∼= Z), the first map is an isomorphism so that we get Hn−1(M ′) = Hn−1(M) in ... this case, whereas if M is non- orientable (⇔ Hn(M) = 0), we end up with a short exact sequence 0→ Hn−1 ... is orientable (⇔ Hn(M) ∼= Z), the first map is an isomorphism so that we get Hn−1(M ′) = Hn−1(M) in ... this case, whereas if M is non- orientable (⇔ Hn(M) = 0), we end up with a short exact sequence 0→ Hn−1 ...
https://metaphor.ethz.ch/x/2020/fs/401-3002-12L/ex/AlgTopExerciseSh...
... subgroup. It can be shown that the torsion subgroup of Hn−1(M ;Z) is trivial if M is orientable and Z/2Z if ... M is nonorientable. Recall from class that if M is orientable and n is odd, then the Euler ... subgroup. It can be shown that the torsion subgroup of Hn−1(M ;Z) is trivial if M is orientable and Z/2Z if ... M is nonorientable. Recall from class that if M is orientable and n is odd, then the Euler ...
Riemann Surfaces and Hurwitz Theory Spring 2019
... orientable manifolds. Comparing to the whole manifold theory, they are not as abstract since they can be ... . But most of what we are dealing with are 2-dimensional orientable manifolds. Comparing to the whole ...
