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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10
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1answer
267 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 ...
6
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1answer
142 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.
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1answer
60 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
480 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
72 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
286 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 ...
21
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0answers
556 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
131 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
84 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 ...
5
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1answer
122 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
147 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
75 views

Is there an analogue of the Thurston norm for curves in an an arbitrary oriented 3-manifold?

Is there a similar function on H_1(Y) measuring the "complexity" of curves representing a class in H_1? Apologies in advance if this is a silly question. Thanks for your time!
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0answers
93 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?
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3answers
465 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
94 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
178 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 ...
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0answers
200 views
+50

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
499 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
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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
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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
259 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
64 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
163 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
154 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
188 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
142 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
367 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
160 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
255 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
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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
162 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
107 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
119 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
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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
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0answers
112 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
388 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
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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\...
4
votes
1answer
222 views

Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...
4
votes
3answers
319 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 ...
1
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0answers
51 views

Spaces that are comparable with their compacts

This is an outgrowth of this question. For a (metrizable) space $X$ consider the following increasingly strong properties: (i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
1
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0answers
52 views

Spherical space-form as the boundary of an Euclidean ball

Let $M^n$ be a smooth compact manifold such that the boundary $\partial M$ is diffeomorphic to a spherical space-form $S^{n-1}/\Gamma$, where $\Gamma \subset O(n)$ is a finite subgroup acting freely ...
2
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0answers
173 views

Retracting to a bigger compact

Consider the topological spaces $X$ with the following property: For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$. Let ...
12
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1answer
197 views

Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
3
votes
1answer
117 views

Unknotting algorithm in higher dimensions?

Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be ...
3
votes
2answers
121 views

Homology torsion in the double branched cover of a tangle?

Let $T$ be a locally unknotted $2$-tangle in $B^3$ and $\Sigma(T)$ be its double branched cover. Can $H_1(\Sigma(T))$ have a non-trivial torsion? (Obviously, not for rational tangles.)
2
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0answers
151 views

cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be. For example, say $T^4$ is regarded as a complex ...

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