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Feb
2
comment Hausdorff limits of fibers of affine maps
Given subsets $X,Y$ of a metric space $Z$, the Hausdorff distance $d_H(X,Y)$ is the infimum number $r$ such that $X$ is contained in the $r$-ball around $Y$, and vice versa. For example, two concentric circles in the plane, of radii $a$ and $b$, have Hausdorff distance $|b-a|$.
Jan
28
asked Hausdorff limits of fibers of affine maps
Feb
22
awarded  Popular Question
Jul
2
awarded  Curious
Mar
23
comment Relative Serre spectral sequences?
Thank you very much for the references! It's exactly what I was looking for.
Mar
23
accepted Relative Serre spectral sequences?
Mar
23
asked Relative Serre spectral sequences?
Feb
1
awarded  Self-Learner
Feb
1
comment Submersions from compact flat manifold
The point is: if $H^a(\tilde{N})$ is the highest nonzero cohomology group of $\tilde{N}$, and $H^b(F')$ the highest cohomology group of $F'$, then in the second page of the spectral sequence the term $E_2^{a,b}$ is nonzero, and no nonzero differentials land on it, or leave it. For this fact to be true it is important that both cohomologies of $F'$ and $\tilde{N}$ have only a finite number of nonzero cohomology group, so this passage does not apply in the situations you described. It follows that $E^{a,b}_2$ survives to the infinity page, and in particular $H^{a+b}(\tilde{M})\neq 0$.
Feb
1
comment Submersions from compact flat manifold
Igor: As you say, you precompose the pullback with the universal cover. By the exact sequence in homotopy the fiber F′ is connected. What is not working in the second paragraph?
Jan
31
answered Submersions from compact flat manifold
Jan
31
asked Submersions from compact flat manifold
Jan
22
awarded  Yearling
Dec
17
revised Dense subgroups of Lie Groups
edited title
Dec
17
comment Dense subgroups of Lie Groups
@Misha: thanks for the explanation. One possible concept of complexity is the growth of H. I am trying to understand what kind of compact manifolds can admit H as fundamental group (or a quotient of it).
Dec
17
comment Dense subgroups of Lie Groups
@Yves: you are right, I meant finitely generated dense subgroup, without "discrete".
Dec
17
comment Dense subgroups of Lie Groups
@Ryan, Yves: I edited my question, I hope it makes more sense now...
Dec
17
revised Dense subgroups of Lie Groups
clearified the question
Dec
17
asked Dense subgroups of Lie Groups
Oct
29
comment (Non)-exoticness of a diffeomorphism of a sphere
Hi Ryan, thanks for the answer first of all. By torus i really meant $(S^1)^n$, even though I didn't say how that came up. I thought it was going to make the question heavier, and all I really wanted was to get to question 3.