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What you want is the reason for naming a theorem this way. So the answer has to be some kind of guess and by analogy. It is possible in algebraic topology to find two non-homeomorphic topological spaces having the same fundamental groups (or homology groups). That is within a "collection of topological spaces" of same fundamental groups one can find two ...


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Here is a solution in great generality: Let $X$ be a Riemann surface, $Y$ its universal cover, and $\pi:Y\rightarrow X$ the universal covering map. Let $\Sigma=\big\{\{U_i\},\{\varphi_j\}\big\}$ be the complex structure on $X$. Then, $\Sigma$ may be lifted to $Y$ by way of $\pi$ to induce a complex structure $\Sigma'=\big\{\{\pi^{-1}|_{U_i}\},\{\psi_j\...


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This is an addendum to the question: There exists a 2-generated subgroup $\Gamma$ of isometries of the hyperbolic 5-space ${\mathbb H}^5$ such that every element of $\Gamma$ is elliptic but $\Gamma$ does not fix a point in ${\mathbb H}^5$. (An isometry of a space $X$ of curvature $\le 0$ is elliptic when it fixes a point in $X$. I will use this ...


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Let $$H_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1, 0 < a+b\}$$ And for $n$ odd $$h_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1,0 < a+b< n\}$$ Since $\wp(u) = \wp(-u)$ and $H_n = h_n \cup -h_n \bmod \Lambda$ (disjoint union) we have $$ \prod_{u \in H_n} (\wp(z)-\wp(u))=(\prod_{u \in h_n} (\wp(z)-\wp(u)))^2 =P_n(\...


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Unfortunately You can't just use $D/\Gamma$ because then the boundary $S^1$ will be very badly behaved (e.g., nonexistence of nontangential limit by choosing the appropriate point on each fundamental $4g$-gon). The usual introductory way is to just use Hilbert space theory with minimal amount of Sobolev spaces thrown in. $(I+\Delta)^{-1}\colon L^2(M)\to H^2(...


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They are sometimes called cone points, because they look like taking a piece of paper with a corner of angle $2\pi/m_i$ and folding it over to identify opposite sides of the vertex. This produces something which looks like a cone at the singular point. They are also sometimes called pillowcase points as they look like the corners of a pillow. So, for ...


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This follows straight from the transformations that have been developed, namely $$\left< \hat{e}_i, \hat{e}^{*j}\right> = (\sum_k g_{ik}\hat{e}^{*k}) \cdot \hat{e}^{*j} = \sum_k g_{ik}g^{kj} = \delta_i^j = \begin{cases} 0 \ i\neq j\\ 1 \ i = j \end{cases}$$ As the author states in the text, that the basis $\{ e_i \}$ or $\{ e^{i*} \}$ may not be ...


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Let me first consider the case of nonconstant holomorphic maps $X\to T^2$, from compact connected Riemann surfaces of genus $g\ge 2$ to tori (smooth elliptic curves). Every such map is determined (up to a translation of $T^2$) by the induced map of fundamental groups $G=\pi_1(X)\to \pi_1(T^2)=Z^2$. There are only countably many homomorphisms $G\to Z^2$ and ...


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