JLA
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Smooth map homotopic to Lie group homomorphism
6 votes

I’m not sure if this is what you’re looking for, but the map $\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ given by sending $0\mapsto 1\,, 1\mapsto 0$ isn't homotopic to a homomorphism. More ...

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How to compute the cohomology of a local system?
5 votes

In general you should have $H^1(\pi_1(X,x),V)\cong H^1(X,L)\,,$ where on the left we have the group cohomology of $\pi_1(X,x)$ acting on $V$ according to the representation. You will also have an ...

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How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?
4 votes

$z-\overline{z}=0\iff y=0\,,$ so it would seem infinitely many roots is possible.

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Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations
3 votes

Since $H^2(\mathfrak{g},\mathbb{R})=0$ and since $G$ is simply connected, it follows from the van Est theorems that $H^2(G,S^1)=0,$ which means that all $S^1$- extensions of $G$ are trivial. But a ...

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Link between the hairy ball theorem and the fundamental theorem of algebra
3 votes

This is a (very) partial answer: Suppose the polynomial is real and of odd degree. Then this polynomial is the characteristic polynomial of some matrix acting in an odd number of dimensions. ...

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Group cohomology of $\mathbf{R}^\ast$ acting on $\mathbf{R}$
2 votes

There are van Est theorems that can help with the computation of group cohomology, and in degree one it's pretty simply. One such cocycle seems to be $f(x)=x-1\,,$ and I believe all others are just ...

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Why is generalised complex structure defined to be a reduction of structure group to $O(n,n) \cap Gl(n,\mathbb{C})$?
2 votes

This condition implies that $J $ preserves the natural pairing. This is quite typical; if one puts a metric on a complex manifold one usually requires the analogous condition to hold, making it a ...

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Non-abelian Ext functor and non-abelian $H^2$
1 votes

If you have a morphism $f:G\to N\,,$ then you get a $K$-extension of $G$ by pulling back the $K$-extension of $N\,.$ The morphism $f$ lifts to a morphism into $M$ if and only if this extension is ...

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Variational formulation of second order equations of the divergence form
1 votes

I'll assume the coefficients are independent of $u\,.$ There is no standard variational problem associated with this PDE for nonzero $b_i\,.$ The reason is that for a linear PDE to admit a variational ...

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Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid
0 votes

I'm not sure if this answers your question or not, but it's true that, for a connected manifold $X\,,$ $H^1(X)\cong H^1(\Pi_1(X),\mathcal{O})\,,$ where on the right hand side we are taking groupoid ...

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Convergence series
0 votes

This seems false. As a counterexample, let $$a_n=\frac{1}{n\cosh{(\sqrt{\ln{n}})}}$$ and let $s=0.$

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