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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Does every orientable surface embed in $\mathbb{R}^{3}$

A topological surface can be pretty strange (consider, for instance, covers of $S^{2}-K$, where $K$ is a Cantor set.) Can every orientable topological surface be topologically embedded in $\mathbb{R}^...
David Cohen's user avatar
12 votes
3 answers
2k views

Why are $S^3-K$ and $\operatorname{SL}(2,\mathbb R)/{\operatorname{SL}(2, \mathbb Z)}$ diffeomorphic? Here $K$ is a trefoil knot in $S^3$

I've heard this result from my differential manifold class, and I don't know how to prove it. Does anyone know how to construct such diffeomorphism? Please tell me, thanks a lot. Any comments are ...
Yuchen Liu's user avatar
  • 1,038
12 votes
3 answers
833 views

Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
Joe's user avatar
  • 535
12 votes
2 answers
1k views

regular tiling of a surface of genus 2 by heptagons

Let $S$ be a Riemann Surface of genus 2. Is there a picture in the literature for a regular tiling of $S$ by 12 heptagons (where three heptagons meet at each vertex)? Also, apart from the obvious ...
Hacon's user avatar
  • 2,372
12 votes
2 answers
645 views

Vector bundle for prescribed Stiefel-Whitney classes

I hope this is not trivial. Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice) For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
Oliver Straser's user avatar
12 votes
3 answers
2k views

To what extent is convexity a local property?

A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
Nathan Reading's user avatar
11 votes
1 answer
2k views

Seiberg-Witten theory on 4-manifolds with boundary

What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist? I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...
J Fabian Meier's user avatar
11 votes
2 answers
5k views

What is parameterization of the trefoil knot surface in R³?

What is a parameterization, say (x(u,v),y(u,v),z(u,v)), of the trefoil knot surface in R³ whose cross-section can be circular or, in general, elliptic? Thanks!
QHLIU's user avatar
  • 199
11 votes
2 answers
2k views

Retraction of a Riemannian manifold with boundary to its cut locus

This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus. Recall that the cut locus of a set $S$ in a ...
Dmitri Panov's user avatar
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11 votes
2 answers
751 views

Knot groups with big number of generators

I start by saying that I am not an expert in this field and I apologize if the question is too elementary. Let $K$ be a knot in $S^3$. I denote by $\pi_1(K)$ the knot group, which is the fundamental ...
P. Tolo's user avatar
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10 votes
1 answer
984 views

Acyclic Finite Groups

A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also ...
Nicolas Boerger's user avatar
10 votes
1 answer
790 views

3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
Neil Hoffman's user avatar
  • 5,221
10 votes
2 answers
1k views

The boundary of a domain whose interior is diffeomorphic to the ball

We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors). My question is about a very ...
Entaou's user avatar
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9 votes
3 answers
3k views

Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?

A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector ...
Theo Johnson-Freyd's user avatar
9 votes
4 answers
2k views

How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?

The intuitive idea is that the sphere connected the two manifolds is not contractible, which implies the (n-1)th homotopy group is not zero. Another argument, which I am not totally understand, uses ...
Xiaolei Wu's user avatar
  • 1,588
9 votes
1 answer
730 views

On the topology induced by a Lorentzian metric

Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread: https://physics.stackexchange.com/questions/228669/why-pseudo-riemannian-metric-cannot-...
Bilateral's user avatar
  • 3,064
9 votes
0 answers
366 views

Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$

This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal. Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
Francesco Polizzi's user avatar
8 votes
1 answer
506 views

Triangulations of submanifolds of smooth manifolds

Every smooth manifold $M$ has a PL structure, and therefore a triangulation. Given a submanifold $N$ of $M$, does anyone know some nice conditions for $N$ to be the subcomplex of some triangulation of ...
overworkedmule's user avatar
8 votes
2 answers
972 views

rational cohomology of finite real grassmannian

Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
QSR's user avatar
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7 votes
1 answer
719 views

More on completion/compactification of open manifolds

This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact ...
Igor Khavkine's user avatar
7 votes
1 answer
172 views

Stability Question for Isotopies Between Compact Sets

Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$. Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
John Samples's user avatar
7 votes
1 answer
472 views

How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?

I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference ...
Math Diego's user avatar
7 votes
2 answers
1k views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
wonderich's user avatar
  • 10.3k
7 votes
2 answers
1k views

All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$. If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex? Can we know the cell structure of $\...
Jino's user avatar
  • 699
7 votes
2 answers
2k views

Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected. Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...
Kathrin L.'s user avatar
6 votes
0 answers
207 views

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold? Let $X_d$ be a $d$-manifold which is NOT a ...
wonderich's user avatar
  • 10.3k
5 votes
1 answer
953 views

Bott's Formula for Grassmannians

Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space $\mathbb{P}_{\...
theStudent's user avatar
5 votes
5 answers
2k views

A simple closed curve on a surface

How to describe a simple closed curve on an oriented surface of genus g? I know the answer only for the torus. It would be nice to find an article or a book where proof can be found.
Andrew's user avatar
  • 192
5 votes
1 answer
485 views

Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
wonderich's user avatar
  • 10.3k
5 votes
3 answers
625 views

Is it possible to connect every compact set?

Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set. Is there a always a compact connected $L\subset X$ such that $K\subset L$? This ...
erz's user avatar
  • 5,385
4 votes
1 answer
243 views

Homogeneous manifold deformation retracts onto compact submanifold

Let $G$ be a connected Lie group. Then by a theorem of Cartan there is a diffeomorphism $$ G \cong K \times \mathbb{R}^n $$ where $K$ is a maximal compact subgroup of $G$. Now, let $M$ be a ...
Ian Teixeira's user avatar
4 votes
1 answer
682 views

Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$) $$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
Tom Copeland's user avatar
  • 9,937
4 votes
2 answers
534 views

Is every group object in TopMan a Lie group?

Recall that a Lie group is a group object in the category of C∞ manifolds. If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure ...
Theo Johnson-Freyd's user avatar
2 votes
1 answer
160 views

Under which conditions is it possible to find points with same distances under bi-Lipschitz map [closed]

Given two metric spaces $(X,d_X), (Y, d_Y)$, a bi-Lipschitz map $f:X \to Y$ and a finite set of points $\{x_1, \ldots, x_n\} \in X$. Consider in addition, that $X$ is a vector space over $\mathbb{R}$, ...
CAT0's user avatar
  • 177
2 votes
0 answers
82 views

Enveloping a Jordan curve with a trace of another one

This question is inspired by this one, or rather the way I understood it. Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
erz's user avatar
  • 5,385
2 votes
2 answers
354 views

Algorithm for identifying reducible braids

If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism $B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$ where $N = \sum n_i$ and $...
Just Me's user avatar
  • 343
2 votes
1 answer
350 views

Induced Map on Sp(2g,Z) is surjective

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map ...
Larry's user avatar
  • 105
1 vote
1 answer
658 views

Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi! I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
Malte Muth's user avatar
127 votes
2 answers
16k views

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 ...
Bill Thurston's user avatar
114 votes
3 answers
5k views

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 $$\...
Nikita Kalinin's user avatar
92 votes
1 answer
11k views

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 ...
algori's user avatar
  • 23.2k
76 votes
4 answers
14k views

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”. ...
Claus's user avatar
  • 6,777
74 votes
29 answers
7k views

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 ...
65 votes
4 answers
4k views

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 ...
61 votes
2 answers
3k views

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 ...
algori's user avatar
  • 23.2k
55 votes
3 answers
6k views

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 ...
algori's user avatar
  • 23.2k
53 votes
2 answers
3k views

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 ...
Morteza Azad's user avatar
51 votes
7 answers
10k views

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 ...
Andy Putman's user avatar
  • 43.4k
47 votes
3 answers
3k views

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 ...
Taras Banakh's user avatar
  • 40.8k
45 votes
1 answer
2k views

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 $...
Moishe Kohan's user avatar
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