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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6
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0answers
72 views

Surjective homomorphisms between braid groups

There cannot be a surjective homomorphism $B_2 \to B_n$ for any $n > 2$ because $B_2$ is commutative and $B_n$ is not. It seems plausible that if $m < n$, there cannot be a surjective ...
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0answers
49 views

Nullity of the linking matrix of a framed link $L$ equals the first betti number of the manifold obtained by surgery on $L$

I have asked this on mathstackexchange as well. I'm not necessarily asking for a proof, just a hint or a point to the right direction (although I'm not saying that a proof isn't welcome). I'm studying ...
4
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0answers
91 views

How to perturb the Chern-Simons functional

Let $Y$ be a $3$-manifold. Suppose that $P \rightarrow Y$ is a principle $SU(2)$-bundle with a choice of trivialization. Then the space of connections over $P$ is identified with $1$-forms taking ...
9
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148 views

Oriented cobordism classes represented by rational homology spheres

Any homology sphere is stably parallelizable and hence nullcobordant. Rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}({\...
4
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1answer
63 views

Is there any maximal 1-planar or 2-planar graph that is not 3-connected

A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. A $k$-planar graph $G$ is maximal if $G+uv$ is not $k$-planar for any non-adjacent vertices $u,v\...
4
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0answers
83 views

Hyperbolicity of twist knots

In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
5
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0answers
197 views

Can a knotted sphere isometrically embed into $\mathbb R^3$?

All smooth simple closed curves in $\mathbb R^3$ (knotted or not) can be isometrically embedded into $\mathbb R^2$ as a circle of equal arclength. The situation for knotted spheres seems more ...
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179 views

$SU(n)$ character variety of integral homology spheres

Suppose that $Y$ is an integral homology $3$-sphere. Is it true that, for any $n \in \mathbb{N}$, there is always an irreducible representation $\pi_{1}(Y) \rightarrow SU(n)$? I'm also happy to see ...
4
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0answers
166 views

Instanton numbers for diverse gauge bundles on diverse manifolds — their relations to characteristic classes

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$ $$ n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
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0answers
102 views

Embedded surfaces in pseudo-Anosov mapping tori

Given a genus $g$ (where $g \geq 2$) closed surface $S$, and a pseudo-Anosov map $\varphi: S \to S$, one can construct the mapping torus, which turns out to be a hyperbolic $3$-manifold, which we'll ...
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1answer
464 views

Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere. Is there an example of a finite covering $\pi : N \to M$ of closed ...
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5answers
572 views

Example of homeomorphism of $3$-manifolds

How can we see that the following $3$-manifolds are homeomorphic? I couldn't use the moves of Kirby calculus. EDIT: An homeomorhism between $(-1/3)$-surgery $4_1$ and $(+1)$-surgery on $8_1$.
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1answer
64 views

Fixed points of diffeomorphisms of tori isotopic to identity and their traces under isotopies

Suppose $T^n$ is the $n$-dimensional torus ($n\geq 2$) and $f: T^n\to T^n$ is a diffeomorphism isotopic to the identity and fixing points $x_1,\ldots,x_k\in T^n$. Does there exist an isotopy $\{ f_t: ...
6
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2answers
501 views

“Well-known fact” that every irreducible 3-manifold with non-empty boundary has an incompressible surface

I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one? Also, it would be great if someone could provide me with a ...
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0answers
82 views

Knot traces and $4$-dimensional analogue of theorem of Lickorish and Wallace

We have the classical theorem of Lickorish and Wallace: Theorem: Any closed oriented smooth $3$-manifold can be obtained integral surgery on links $\mathcal L $ in $S^3$. On the other hand, the ...
5
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2answers
359 views

3-colored triangulations of the sphere $S^2$, and Sperner's Lemma

I noticed something about colored triangulations of the topological sphere $S^2$ and have a question about this. Observation. If you triangulate the sphere $S^2$ and color the vertices with three ...
22
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0answers
585 views

Curves on potatoes

On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler: The puzzle is attributed to the book "The mathemagician and the pied piper", and ...
5
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1answer
135 views

How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?

Chen and Yang have a more general version of the volume conjecture that they state for all hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to ...
6
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1answer
93 views

Subdivision of closed homology manifold reference request

I am interested in the barycentric subdivision of closed homology manifolds. Definition A (finite) simplicial complex $K$ is a closed homology manifold of dimension $n$ if for every $k$-simplex, its ...
6
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1answer
130 views

Constructing exotic $\mathbb{R}^4$'s using vector fields on $\mathbb{R}^5$

I was reading a paper of Arnol'd ("Topological Properties of Eigenoscillations in Mathematical Physics") where he gives the following claim (hopefully I am stating it correctly). One way to ...
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0answers
53 views

Fundamental class of products of spaces [migrated]

Let $M$ be smooth oriented manifold where $M=X\times F$, $X$ and $F$ smooth oriented manifolds. We note by $[M]$ the fundamental class of $M$. Is this equality true: $$[X\times F] = [X]\times [F]?$$
3
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1answer
149 views

Geometric content of area of a word in geometric group theory?

Where does the idea of 'area' come from in Geometric Group Theory? The wikipedia article states that this definition was 'inspired' from Riemannian geometry: Gromov's proof was in large part informed ...
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0answers
97 views

Mapping class groups of $T^2 \times [0, 1]$ and $T^2 \times S^1$

Are the mapping class groups of $T^2 \times [0, 1$] and $T^2 \times S^1$ explicitly known?
8
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3answers
469 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 $...
7
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0answers
95 views

Small flag triangulations

In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...
8
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0answers
180 views

Rational cobordism classes of manifolds fibered over spheres

Fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$. The signature ...
4
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0answers
210 views

Enlarging a compact set in order to improve its shape

In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\...
15
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1answer
565 views

Sperner's Lemma implies Tucker's Lemma - simple combinatorial proof

Sperner’s Lemma is often called the "combinatorial analog" of Brouwer’s Fixed Point Theorem, and similarly Tucker’s Lemma is often called the combinatorial analog of Borsuk–Ulam’s Theorem. We can ...
5
votes
1answer
203 views

The metric difficulty of unknotting unknots

Despite the catastrophe for the world and the many victims, at least the lockdown is favorable to two activities: Mathematics and Gardening. The difficulty of handling hedge trimmer wires and garden ...
3
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0answers
84 views

(symplectic) $h$-cobordism from $S^1\times S^2$ to itself

I ran into an oriented smooth $h$-cobordism from $S^1\times S^2$ to itself in my project. I wish to argue that it is diffeomorphic/homeomorphic to the product. From this question 4-dimensional h-...
4
votes
1answer
140 views

“Basic” loops on standardly embedded surfaces

Take a genus $g$ surface $S$ standardly embedded in $\mathbb{R}^3$, by which I mean it is unknotted. Surface $S$ bounds a volume $V$ that deformation retracts on a standardly embedded planar graph $G$ ...
4
votes
2answers
260 views

Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-...
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0answers
65 views

Convex surfaces with transverse boundary (contact geometry)

Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...
10
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0answers
165 views

Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...
6
votes
1answer
156 views

Cobordism monopole Floer homology

From the famous book: Monopole and three manifold, Kronheimer and Mrowka(https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf). It is known that: Let $Y$ be a closed oriented $3$ manifold, ...
6
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0answers
190 views

Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
3
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0answers
143 views

Cobordism theory of some weird space

Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$. The $W$ is a homogeneous space (also a quotient space), but not a group. Previously, I am aware of the ...
6
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2answers
371 views

Books on foliations

I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints. ...
6
votes
1answer
166 views

Mean curvature flow and knot theory

I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-...
5
votes
1answer
257 views

Whitney sum formula for topological Pontryagin classes

Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.
1
vote
2answers
113 views

Transverse invariant measures to vector fields

Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure? I've seen examples of higher-...
64
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30answers
6k 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 ...
4
votes
0answers
164 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
3
votes
1answer
109 views

Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
1
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0answers
120 views

When a free action gives rise to a $G$-principal bundle

When a free action gives rise to a $G$-principal bundle Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that $G \backslash X$ is Hausdorff. (equivalently the image of ...
4
votes
0answers
63 views

Implicit function theorem for PL maps

Let $K$ be a PL triangulation of a closed manifold and $f: K\to\mathbb R^k$ be a PL map. Equivalently, $f$ is a map that becomes linear on every simplex after subdividing. Suppose $v$ is a vertex in ...
6
votes
0answers
114 views

Why does the inverse Alexander polynomial appear in the MMR conjecture?

In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
6
votes
1answer
390 views

Solutions of PDE under changing topology

Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we ...
3
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0answers
149 views

Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
7
votes
1answer
211 views

Retracting off a compact set

Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected. Can we always find an open $V$ such that $K\subset V\subset\...

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