Questions tagged [geometric-topology]

The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key example is the literature surrounding the Poincaré Conjecture.

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Definition of uniqueness of tubular neighbourhoods

The tubular neighbourhood theorem states that if $M \subset N$ is an embedding of smooth manifolds without boundary and $\nu: E \to M$ is the normal bundle of $M$ in $N$, then there is a smooth ...
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$T^2 \setminus (\{w\}\times S^1) \text{ is homeomorphic } S^2 \setminus \{ p,q \}$

Let $T^2 = S^1 \times S^1$ be the torus and $p,q \in \mathbb{R}^3, w \in S^1$ Show that $$ T^2 \setminus (\{w\}\times S^1) \text{ is homeomorphic } S^2 \setminus \{ p,q \} $$ Using the ...
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Finding metric projection mapping

Let $X$ = $c_0$, with supremum norm and $W$ = $\{ \{x_n\}_{n \geq 1} \in X$ $|$ $x_1=0\}$. Then how to find $P_W(x)$. Note that, $P_W(x)$ = $\{y_0 \in W : ||x-y_0|| = \inf ||x-y||,$ $ y \in W \}$. In ...
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A list of all surfaces (with or without boundary, orientable or nonorientible etc…) classified according to Euler characteristic, orientibility etc.

Do you know any book, website or any kind of source where I can find the whole list of surfaces classified according to the Euler characteristics. Well, the whole list can’t exists in this universe ...
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The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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generalized Poincare conjecture [duplicate]

How to show that the claim that there exists exactly one differentiable structure on $S^4$ iff smooth four-dimensional Poincaré conjecture is true (homotopy equivalent to S4 implies diffeomorphic to ...
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Reference for an easy lemma on homeomorphisms of connected manifolds

If M is a connected manifold then the set of orientation preserving homeomorphisms of M that are isotopic to the identity acts $n$-transitively on M for all positive $n\in\mathbb{N}$. I know several ...
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Compressible torus in irreducible 3-manifold bounds a solid torus

In his 3-manifolds notes (page 11, item $(4)$), Hatcher shows that a 2-sided compressible torus $T$ in an irreducible 3-manifold $M$ either bounds a solid torus $S^1 \times D^2$ or is contained in a ...
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2 votes
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Non-linearity of Einstein's field equations

How to show that, for two Schwarzschild- metrics, the Ricci tensors of two metric tensors do not sum up linearly: $R_{\mu\nu} (g_1+g_2) \neq R_{\mu\nu} (g_1)+R_{\mu\nu} (g_2)$ while the Ricci tensor ...
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Find in which face lie a point on a hyper polytope, inscribed in an hyper unit-sphere (and how to generate this hyper polytope)

I am working on clustering normalized point in a k dimension space. At first i used Spherical LSH strategy where i used random planes to cut the hyper-sphere but it induces a lot of random (because i ...
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2 votes
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Deforming a circle into closed smooth curve [closed]

If we throw a rubber band on to a smooth table it wiggles and takes variegated shapes before coming to rest. However, the deformed band while taking different shapes remains closed and smooth. I am ...
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A version of Brower's fixed point theorem for contractible sets?

Brouwer's fixed point theorem states that a continuous map $f:B^n\to B^n$ ($B^n\subset\Bbb R^n$ being the $n$-dimensional ball) has a fixed point. It is clear that we can replace $B^n$ with a space $X$...
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Bockstein Homomorphism and real projective space.

Consider the Bockstein Homomorphism $\beta: H^1(\mathbb{R{P}}^2, \mathbb{Z}_2)\rightarrow H^2(\mathbb{R}P^2,\mathbb{Z}_2)$. Now, I was told that For each $\alpha$, $\beta(\alpha) = \alpha \cup \alpha$....
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homogeneous circle bundle over a hyperbolic surface

Let $ M $ be the total space of a circle bundle $ S^1 \to M \to \Sigma_g $ for $ g \geq 2 $. Suppose that there exists a transitive action of $ \widetilde{SL_2(\mathbb{R})} $ on $ M $. Must $ M $ be ...
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Why are closed manifolds defined as they are?

The standard definition of a closed manifold is the following: A closed manifold is a manifold without boundary that is compact. I am wondering, why we request that the manifold is compact. As I ...
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knotted 2-spheres in 4-space?

What would be example of "non-trivially" embedded sphere in 4-space and how to visualize? We could have $S^2 \to \mathbb{R}^4 \subset S^4$ for topologists the 4-space can be "completed&...
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2 votes
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Example of a topological space $X$ and a contractible subspace $A$ such that $X$ and $X/A$ aren't homotopy equivalent. [closed]

Can someone find an example of a topological space $X$ and a subspace $A\subset X$ such that: $A$ is contractible; $X$ and $X/A$ have different homotopy type. I know it exists but I can't find one. ...
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Why is this example (non-simplicial $\mathbb R$-tree) not a simplicial complex?

This is an example of a non-simplicial $\mathbb R$-tree (from Wikipedia): Start with the interval $[0,2]$, for each positive integer $n$, glue an interval of length $1$ to the point $1-1/n$ in the ...
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2 votes
2 answers
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Set of 0-cells of a simplicial complex and discreteness

When given a simplicial complex, does its set of $0$-cells have to be a discrete set? In particular, can the set $\{0, 1/n\}_{n=1,...}$ be a simplicial complex (consisting of $0$-cells)? From the ...
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Fundamental group of Seifert fibered space over $S^2$

I am reading p.3 of Saveliev's Invariants for homology 3-spheres. Here is the construction of a Seifert 3-manifold $X=M(b;(a_1,b_1),\dots,(a_n,b_n))$. Consider the $S^1$-bundle $M\to S^2$ with Euler ...
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4 votes
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Explicit formula of $S^1$-bundle over $S^2$ with euler number $n$

Suppose $M \to S^2$ is an orientable circle bundle with Euler number $n$. Regarding $S^2$ as $D^2_1 \cup_{\text{id}} D^2_2$, since $M$ is trivial over $D^2_i$ ($i=1,2$), we have $M=(S^1\times D^2_1)\...
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1 vote
1 answer
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Inverse operation of Dehn surgery

Suppose we have two closed oriented 3-manifolds $M$ and $N$. Suppose $N$ is obtained by a Dehn surgery operation on a knot $K$ in $M$, so $N=(M-\operatorname{int}\nu K)\cup_\partial (S^1\times D^2)$, ...
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1 vote
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attaching three handles onto three manifold with boundary

Assume $M$ is a three-manifold with a single component of the boundary, if we assume that $S^2$ diffeomorphic to some submanifold $S\subset \partial M$.Prove $S$ has to be the whole boundary. I can ...
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3 votes
1 answer
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List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle [closed]

List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle. State the orientability of the total space, the base and the bundle (orientability of a circle bundle is ...
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1 vote
1 answer
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Can a circle encompassing a continuous, closed plane curve be made to pass through an arbitrary point on the exterior of the curve?

Let $J \subseteq \mathbb{R}^2$ be a continuous, closed curve (closed in the sense that $j(a) = j(b)$ for some continuous parameterization $j:[a,b]\rightarrow\mathbb{R}^2$ of $J$ with $a,b \in \mathbb{...
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2 votes
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Transitive group action on mapping torus of negative Dehn twist

Question: What Lie group acts transitively on the mapping torus of $ \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix} $ (which is exactly the negative of the mapping class of a Dehn twist)? ...
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Examples of Seifert fibered homology spheres

The following is taken from chapter 1 of Saveliev's book Invariants for Homology 3-Spheres, there is the following definition: Let $a_1,\dots,a_n$ be positive integers, $n\geq 3$. Let $B=(b_{ij})$ be ...
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Link of zero set of a quasi homogeneous polynomial in $\Bbb C[x,y,z]$ has a Seifert fibered structure

I have a question about Seifert fibered 3-manifold while reading this survey paper: https://arxiv.org/pdf/math/0602562.pdf. In Example 22 (p.10), it is said that for integers $a,b,c\geq 1$, set $S_{...
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2 votes
1 answer
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If $M$ is the connected sum of $n$ tori and $N$ is the connected sum of $n$ projective spaces, then what surface is $M \# N$?

I don't know if this exercise is correct and I need your opinion: Consider $M$ is the connected sum of $n$ tori and $N$ is the connected sum of $n$ projective spaces. What surface is $M \# N$? And if ...
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5 votes
2 answers
68 views

Is $SL(2, \mathbb{R})$ dense in saturated elementarily extensions of the reals?

Suppose $\mathcal{A} = \langle A, <, +, \cdot \rangle$ is a $\aleph_1$-saturated elementarily extension of the real field. Is $SL(2, \mathbb{R})$ dense in $SL(2,A)$? In compact groups one can find ...
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How can I prove that the following function is a homeomorphism? $f_T:\mathbb{S}^2 \to \mathbb{S}^2$

I need to prove the following $\pi_1\left(f_T(\mathbb{S}^2), f_T(\mathbf{x}_0)\right)\simeq \pi_1\left(\mathbb{S}^2, \mathbf{x}_0\right)$. My idea is to prove that the function $f_T$ is an ...
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Is every coarse map between proper geodesic spaces a quasi-isometric embedding?

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. Is every coarse map $f: S\rightarrow S'$ a quasi-isometric embedding?. Just to recall, a coarse map $f: S\rightarrow S'$ between ...
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1 vote
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"Tightening up" a map in the fibre of a principal torus bundle

Suppose $p_1:E_1\to B$ and $p_2: E_2\to B$ are two compact principal $\mathbb{T}^d$-bundles over the same base $B$. Suppose there exists a fibre-preserving map $F:E_1\to E_2$ that covers the identity $...
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Seifert surface with components

I was wondering what the knot of a Seifert surface with 3 or more boundary components would be. Doesn’t every Seifert surface have one boundary component by definition ?
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Smooth homogeneous implies universal cover Riemannian homogeneous

Let $ M $ be smooth homogeneous. That is, there exists a finite dimensional Lie group $ G $ which acts smoothly transitively on $ M $. Let $ M' $ be the universal cover of $ M $. Does there always ...
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1 vote
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Is Homeo(M) locally path-connected for a general topological manifold?

I am wondering if there exists a closed topological manifold for which Homeo$(M)$ is not locally path-connected. If $M$ admits a smooth structure, then one can prove that Homeo$(M)$ is in fact locally ...
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2 votes
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Inverse limit in category of graphs

Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
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3 votes
1 answer
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Base and fiber Riemannian homogeneous implies total space smooth homogeneous

Let $ F,B $ be Riemannian homogeneous manifolds. Suppose that $ E $ is the total space of a fiber bundle $$ F \to E \to B $$ Is $ E $ always smooth homogeneous (admits a smooth transitive action by a (...
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1 answer
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Is a mapping torus of a solvmanifold always a solvmanifold?

Is it true that the mapping torus of a solvmanifold is always a solvmanifold? Some relevant facts, many from Flat 3 manifolds and mapping tori of flat surfaces : Every mapping torus of a compact ...
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Flat 3 manifolds and mapping tori of flat surfaces

Let $ M $ denote the Moebius band and $ K^2 $ denote the Klein bottle. Mapping tori of the cylinder (thanks to comment from Michael Albanese): $ (t,z) \mapsto (t,z) $ is $ \mathbb{R} \times T^2 $ ...
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4 votes
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Three Manifold with (almost) all the Thurston Geometries

Here $ L(S^1) $ is the unique nontrivial line bundle over the circle (the mobius strip). The manifold $ \mathbb{R}^3 $ (with infinite volume) admits all six of the aspherical Thurston geometries (so ...
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5 votes
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Inscribed square problem/Toeplitz' conjecture: how to progress from 3blue1brown video argument?

I was watching the video "who cares about topology?" by 3blue1brown proving the inscribed rectangle, and started from there to see if I could say something about the inscribed square too. ...
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1 vote
1 answer
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Are loops not allowed when gluing together simplices in a simplicial complex?

I'm taking a graduate geometric topology class and our professor made a quick remark that we're not allowed to make stuff like these by gluing together simplexes in a simplicial complex. I didn't get ...
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3 votes
1 answer
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Real line bundles on the Klein bottle

I am trying to better understand real line bundles. The real line bundles on a compact manifold are classified by $ H^1(M,C_2) $. See https://mathoverflow.net/a/113944/387190 So, for example, every ...
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1 vote
0 answers
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Heegaard Floer homology of a genus two diagram of $S^3$

I am reading the introductory paper "Heegaard diagrams and holomorphic disks" by Ozsváth and Szabó (https://arxiv.org/abs/math/0403029v1, Section 2.2), and I do not understand one of the ...
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1 vote
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What is the geometry of the twisted I-bundle over the Klein bottle?

Let $K^2$ be the Klein bottle and $M = K^2\widetilde{\times}I$ be the twisted, orientable $I = [0,1]$-bundle over $K^2$. So, $M$ is geometrically atoroidal (there is only one $\mathbb{Z}\times\mathbb{...
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1 vote
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Intersection number of two simple closed curves on a torus

The isotopy classes of oriented simple closed curves on the torus, are classified by primitive vectors in $\mathbb{Z}^2$, namely, $\{(p, q)| gcd(p,q) = 1 \}$. Prove that for two simple closed curves, $...
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43 views

Is there a notion like topological equivalence but more finely distinguishes between shapes?

A circle is topologically equivalent to a triangle, because there is a continuous deformation to transform one into the other. But this misses out on a key fact of the triangle, which is that it has 3 ...
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0 votes
1 answer
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Can I find the topology according to certain definition of convergence?

For instance, consider the space $C[0,1]$, I hope $f_n \rightarrow f$ means that $\sup_{[0,1]} |f_n - f| \rightarrow 0$. I know the convergence is meaningful only by specifying the topology, and I am ...
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  • 161
1 vote
1 answer
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Atoroidal closed 3-manifold

A 3-manifold $N$ is called atoroidal if any incompressible torus is boundary parallel, i.e. can be isotoped into the boundary. To me, this definition assumes that $N$ has boundary, but I have read a ...
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