# Tagged Questions

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### Leray Spectral Sequence

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$. Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$ be a generic fiber that is a ...
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### Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ ...
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### Cylinders dividing $\mathbb{R}^{3}$

Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$. For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...
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### High Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead d=5,4). Edit: I mistakingly identified my mistake, it is 6,5 but I got the indices shifted by one.] Background: Polygon spaces Given a ...
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### The “grassmannian” of a simplicial complex

This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called. Take an $n$ dimensional ...
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Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help. We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...
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### Program for drawing cobordisms [closed]

Perhaps this is not the right place to ask the following question but I did not find any suitable on the web. So I would be very grateful for sharing your experience. What is a good program to draw ...
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### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
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### Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED]

Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively. Let $G_1,\ldots, G_l$ be their ...
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### What is the “right” deifinition of the homology(cohomology) of an orbifold?

What is the "right" analog in the orbifold case of a singular homology of a topological space? We can not just take the homology of the underlying space, because it does not contain much imformation. ...
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### Mathematics of Doodling and the Winding Number

So I was reading the American Mathematical Monthly Feb 2011 (Volume 118, number 2), and in particular, I was interested in Ravi Vakil's article about mathematics of doodling. There is a question I ...
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### System dynamic of space euclidean and hyperbolic tilings

Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC) tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of $R^{d}$ by translation is on ...
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### Generalization of Moise's theorem

I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it. The claim is ...
Let $P$ be a polyhedron which satisfies the following three conditions: $P$ is built out of regular hexagons and regular pentagons. Three faces meet at each vertex. $P$ is topologically a sphere. ...
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$. Consider a 3-form \$\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 ...