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3

The answer to 1 and 3 is yes. Furthermore the formula that lies behind this answer shows that there is no "hole in the middle of it" and that the answer to 2 is no. All these answers are obtained using the inversion formula $$f(z) = \begin{cases} \frac{1}{z} & \quad\text{if $z \not\in \{0,\infty\}$} \\ \infty & \quad\text{if $z=0$} \\ 0 & \quad\...


2

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(...


1

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 ...


1

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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