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 Yearling
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Jan
5
awarded  Yearling
Dec
27
comment Augmentation ideal and the abelianization of $G$
In case it helps, an explicit isomorphism is $g[G,G]\mapsto (g-1)+I^2$. The point here is that the elements $\lbrace g-1\rbrace_{g\in G}$ form a basis for $I$ as a $\mathbb{Z}$-module, noting that $I$ is (by definition) the kernel of $\mathbb{Z}[G]\to\mathbb{Z}$ given by $g\mapsto 1$ for $g\in G$.
Oct
20
comment Open in X (X is a metric space) how can you show or tell if it is open or not in general terms? What does it really mean?
HINT: The typical "ball" in $[0,2]$ is NOT the same as the typical "ball" in $\mathbb{R}$ (near the value 0).
Oct
16
awarded  Notable Question
Sep
8
comment Solutions to Dirichlet problem on manifolds with boundary
And not true (for uniqueness) if $M$ is noncompact. You need a "bounded" assumption on the functions (which the Dirichlet problem typically assumes). Take $f=0$ and solutions $w_1=e^x\sin y$ and $w_2=0$ on the half-space $\lbrace(x,y)\in\mathbb{R}^2\;|\; y\ge0\rbrace$. The analytic function in the background is $e^z$.
Sep
7
accepted Coupled PDE on cylinder
Sep
7
asked Coupled PDE on cylinder
May
21
comment is $dx$ greater than $\frac{dx}{2}$?
$dx$ is not a number, so I don't know what you mean by "greater than". This is the source of your confusion, I believe.
May
11
answered almost complex structures on $R^4$
Mar
20
revised Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?
wrong Thom space
Mar
20
suggested approved edit on Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?
Feb
16
revised Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.
deleted 331 characters in body
Jan
5
awarded  Yearling
Dec
8
awarded  Caucus
Nov
11
comment Is there an interpretation of higher cohomology groups in terms of group extensions?
Read Ken Brown's book Cohomology of Groups, because it explains $H^3$ as crossed-module extensions, and there is an extension-interpretation of $H^1$. In general, the higher cohomologies were described by MacLane (the appropriate references are given in this book) but are essentially intractable.
Nov
9
comment Simple Branched covering over sphere.
Ah you're right, I just realized I'm not answering the real question. I was only finding a complex structure such that the underlying topological surface is realized as a holomorphic cover, and as you point out, this handles all Riemann surfaces of genus 2.
Nov
9
comment Simple Branched covering over sphere.
Huh? The 180 degree rotation quotient map is an example of a simple branched covering for any genus, where the surface has said symmetry.
Nov
8
comment Simple Branched covering over sphere.
Right, and now I don't see how you can deduce anything further. You can use Riemann-Hurwitz to see that it is possible for this branched cover to have degree 2 and 2g+2 simple branch points. And we can construct this explicitly, it's given on pg105 of Donaldson (Figure 7.3), generated by 180 degree rotation.
Nov
6
comment Simple Branched covering over sphere.
Take a divisor of $\text{genus}+1$ points (crucial). Riemann-Roch says that the space of (nonconstant) meromorphic functions with (at worst) simple poles on the divisor is nonempty.
Oct
21
revised Determining an explicit line bundle over surface
added 24 characters in body