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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Intuitive crutches for higher dimensional thinking
I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...
128
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10
answers
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Are there any very hard unknots?
Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
127
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2
answers
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What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
114
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3
answers
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The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
103
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5
answers
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Independent evidence for the classification of topological 4-manifolds?
Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...
95
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4
answers
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Which manifolds are homeomorphic to simplicial complexes?
This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The ...
92
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1
answer
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The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
90
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4
answers
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Can every manifold be given an analytic structure?
Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...
80
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12
answers
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Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
79
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1
answer
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Topological cobordisms between smooth manifolds
Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--...
77
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3
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Gromov's list of 7 constructions in differential topology
At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order ...
76
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4
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What are good mathematical models for spider webs?
Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. ...
75
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5
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When the automorphism group of an object determines the object
Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
74
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29
answers
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Proofs where higher dimension or cardinality actually enabled much simpler proof?
I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
71
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10
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Nice proof of the Jordan curve theorem?
As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive ...
70
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10
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Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
70
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3
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The story about Milnor proving the Fáry-Milnor theorem
This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, ...
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What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets ...
64
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4
answers
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Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of ...
62
votes
9
answers
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Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
61
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14
answers
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What are some of the big open problems in 3-manifold theory?
From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...
61
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answers
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Intuitively, what does a graph Laplacian represent?
Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. ...
61
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4
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Drawing of the eight Thurston geometries?
Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries?
I am imagining something akin to the standard picture (of a sphere, plane,...
61
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2
answers
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The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many ...
59
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7
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Status of PL topology
I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
58
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6
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Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
55
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6
answers
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Poincaré Conjecture and the Shape of the Universe
Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?
55
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3
answers
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Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
54
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8
answers
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Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
54
votes
5
answers
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Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
53
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2
answers
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How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
52
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1
answer
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Open map D⁴ → S²
Is it possible to construct an embedding $D^4\hookrightarrow S^2\times
\mathbb R^2$
such that the projection $D^4\to S^2$ is an open map?
Here $D^n$ denotes closed $n$-ball.
An open map D⁴ → S².
It ...
50
votes
7
answers
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Triangulating surfaces
I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's ...
50
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4
answers
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To which extent can one recover a manifold from its group of homeomorphisms
Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One ...
49
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8
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Classification problem for non-compact manifolds
Background
It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic).
I'm also under the impression that there is ...
49
votes
4
answers
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Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
49
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3
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Explicit metrics
Every surface admits metrics of constant curvature, but there is usually a disconnect between
these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces.
Can ...
49
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2
answers
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Can knot diagrams be monotonically simplified using under moves?
It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger ...
48
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3
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Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2)...
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3
answers
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A metric characterization of the real line
Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...
45
votes
1
answer
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Exotic $R^4$ as the universal covering space
Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$?
Remark. I am aware of examples (due to Mike Davis) of compact $...
44
votes
1
answer
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Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
43
votes
9
answers
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Applications of knot theory
An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...
43
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7
answers
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Why should I care about Heegaard-Floer theory?
I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these ...
43
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2
answers
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Meaning/origin of Seiberg-Witten equations/invariants
Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take ...
41
votes
9
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In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that ...
41
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5
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Can cotangent bundles see exotic smooth structures?
I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school.
Caveat: I'm not a ...
40
votes
2
answers
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Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
40
votes
1
answer
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Are there only countably many compact topological manifolds?
Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
40
votes
1
answer
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Four circles on the sphere
Consider generic configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to ...