archipelago
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Are there geometrically formal manifolds, which are not rationally elliptic?
Many of the connected sums fail to be rationally elliptic because of the bound on the betti number, but this obstruction is also one for geometric formality.
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Are there geometrically formal manifolds, which are not rationally elliptic?
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cohomology of BG, G compact Lie group
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Are there geometrically formal manifolds, which are not rationally elliptic?
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Are there geometrically formal manifolds, which are not rationally elliptic?
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Are there geometrically formal manifolds, which are not rationally elliptic?
Note that examples of these kind cannot be homogeneous spaces or even biquotients, as they are rationally elliptic.
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Are there geometrically formal manifolds, which are not rationally elliptic?
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Is any continuous group homomorphism from R to C* an exponential map?
Even better. Every measurable homomorphism between locally compact groups is continuous, so every measureable homomoprhism between lie groups is smooth.
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Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?
Yes, the class of epimorphisms in an abelian categroy forms a site and the sheaves of abelian groups on site site are exactly the left exact functors. Gabriel's elementary constructed left adjoint I mentioned in my answer then corresponds to the abstractly constructed sheafification-functor.
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Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
A comment on the first part of your question: Every finite dimensional real Lie algebra is a semi-direct product of a solvable and a semi-simple Lie algebra (Levi-Decomposition), so the classifiction of finite dimensional Lie algebras splits up in the (distinct) two cases of solvable and semi-simple lie algebras. In the class of the solvable lie algebras, you have the proper tower of classes: solvable contains nilpotent contains abelian.