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        Questions tagged [gt.geometric-topology]

        Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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        Trajectory definition using constrained projections on unknown surface

        In $3D$ space where the $Z$ axis is up-down, I have the following: A static camera $A$ at $(x_a, y_a, z_a)$; A laser pointer $B$ at $(x_a, y_a, z_a + b)$ which can yaw or pitch by $1^\circ$ at a ...
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        What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?

        What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$? Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...
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        241 views

        unlinking in 5 dimensions

        If I have a linked pair of circles in $\mathbb{R}^3$, they can be unlinked in $\mathbb{R}^4$. Said differently, there is an isotopy in $\mathbb{R}^4$ between two strands which have been twisted, and ...
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        82 views

        Finitely generated nilpotent groups as cusp groups

        I recently learned about the following question, asked by I. Kapovich : Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
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        120 views

        Distinguishing Square Knot and Granny Knot using Quandles

        It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups. I want to write a work on knot theory and provide these knots as an example ...
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        67 views

        Maximum number of ways of splitting a set of points with an hyperplane

        Given a set $S$ of $n$ points in $\mathbb{R}^d$, let $D_S$ be the set $\{\mathbf{v}=|\mathbf{u}-\mathbf{u'}|: \mathbf{u},\mathbf{u'}\in S\}$ (where $\forall i=1,2,\ldots, d$, $\mathbf{v}_i=|\mathbf{u}...
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        Quadrilateral fundamental domain

        Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find the angle sum ...
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        266 views

        Wildness of codimension 1 submanifolds of euclidean space

        This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the ...
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        268 views

        To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?

        To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in,...
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        473 views

        Simplicial set of permutations

        Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
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        182 views

        Tiling of genus 2 surface by 8 pentagons

        In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons". Question 1: What does this tiling look like? Question 2:...
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        163 views

        Diffeomorphism classification of Grassmannian manifolds

        Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on ...
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        111 views

        Is every $(n-1)$-connected $n$-manifold embeddable in $\mathbb{R}^{n+1}$ homeomorphic to $\mathbb{S}^{n}$? [migrated]

        Let $M^n$ be a compact, topological $n$-manifold which is a subspace of $\mathbb{R}^{n+1}$. If $M^n$ is $(n-1)$-connected (i.e. $\pi_i$ vanishes for $i<n$), does it have to be homeomorphic to the $...
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        98 views

        Is this a typo in Ihara's “On discrete subgroups of the two by two projective linear group over p-adic fields”?

        In Eq. (9'') on p. 227 of Ihara's paper "On discrete subgroups of the two by two projective linear group over p-adic fields" (link), where the second line says $$"\log Z_{\Gamma}(0,\chi)=1",$$ is this ...
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        415 views

        Immersions of surfaces in $\mathbb{R}^3$

        Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere ...

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