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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15
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1answer
2k views

How well can we localize the “exoticness” in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, Exotic Structures on smooth 4-manifolds by Selman Akbulut and Localized Exotic Smoothness by ...
7
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2answers
2k views

Ideals in the ring of single-variable Laurent polynomials with integer coefficients

I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...
3
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2answers
915 views

exotic smooth structure clarification

Does the existence of exotic smooth structure in $\mathbb{R}^4$ imply the existence of an atlas which has a $C^0$ mapping to the Cartesian atlas, but not a $C^k$ mapping (for some finite $k$)? Does ...
3
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1answer
259 views

Equivariant Surgery problem

I have a question about surgery. Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.(...
10
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1answer
439 views

4-manifolds in the 4-sphere such that it, *and* its complement have unsolvable word problem

In an earlier thread I had asked whether or not one can find a smooth 4-dimensional submanifold of $S^4$ whose fundamental group has an unsolvable word problem. The answer is yes, and the reference ...
14
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6answers
3k views

Geometric flavored textbook on algebra

I am interested in topology, while I am not so comfortable with some algebraic flavored textbook on algebra. Actually, it was not until I learned some topology that I began to understand some ...
6
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1answer
910 views

Simply-connected domain around a curve

In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this ...
8
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2answers
542 views

Small neighborhoods of singularities on varieties

In Singular points of complex hypersurfaces, John Milnor proves the following theorem: Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...
11
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3answers
2k views

distance formula in Farey graph?

Consider the usual "Farey graph", i.e. the 1-skeleton of the (essentially unique) triangulation of the disk by ideal triangles. If one insists that 1/0, 0/1, and 1/1 are vertices of a triangle then ...
2
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2answers
213 views

Lipschitz orthonormal frames on submanifolds of $\mathbf{R}^n$ ?

Suppose we are given a $d-$dimensional submanifold of $\mathbf{R}^n$ with a trivial normal bundle, whose $d-$dimensional volume is $V$ and has a non-self-intersecting tube of radius $r$ around it. Can ...
21
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1answer
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Word problem for fundamental group of submanifolds of the 4-sphere

Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-...
6
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3answers
617 views

Jordan Curve Homotopy

Does there exist a notion of Jordan curve homotopy? In particular, suppose we have two Jordan curves $C_0 : S^1 \rightarrow \mathbb{R}^2$ and $C_1 : S^1 \rightarrow \mathbb{R}^2$. When does there ...
5
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1answer
575 views

Recognizing regular neighborhoods

In a Riemannian manifold consider two compact smooth submanifolds $S$, $S^\prime$ that intersect transversely. It seems intuitively obvious that for a sufficiently small number $r$, the union of $r$-...
58
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14answers
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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 ...
7
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1answer
571 views

Intersection product in a manifold, taking values in one factor

In a joint paper that I am working on, we are interested in taking the intersection product $[X] \cap [Y]$ of the fundamental classes of two compact, oriented pseudomanifolds $X$ and $Y$ in a compact, ...
9
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3answers
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Closed 3-manifolds with free abelian fundamental groups

Which free abelian groups can be realized as the fundamental group of a closed 3-manifold? The only one I can come up with is $\mathbb{Z}$, which is the fundamental group of $S^1 \times S^2$. For ...
27
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2answers
702 views

What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
10
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2answers
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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 ...
3
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3answers
349 views

Collapsing contractible subsets of the two-disk.

This question is quite specific, but it may admit answers in more general contexts. Consider a subset $\Lambda \subset D^2$ where $D^2$ is the two dimensional disk. We consider in $\Lambda$ an ...
3
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2answers
485 views

A question about the Kakeya problem

Besicovich proved a long time ago that a straight line segment of fixed length could be rotated 360 degrees within a subset S of the Euclidean plane such that $M(S)$ is arbitrarily small-where M is ...
24
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4answers
3k views

Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle

Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way. I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent ...
32
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1answer
1k views

“Affine communication” for topological manifolds

There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this: Prove ...
2
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1answer
148 views

Some more questions about regularity of homeomorphisms of foliations

This is a continuation of A question about regularity of foliations>this question answered by Dmitri. Let $F$ and $F'$ be smooth ($C^\infty$) foliations of a manifold $M$. Assume that there is a ...
5
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3answers
326 views

A question about regularity of foliations

Let $F$ be a smooth foliation of a torus. Assume that $F$ can be mapped by a homeomorphism to an irrational-straight-line foliation $L$. Does it follow that $F$ can be mapped to $L$ by a ...
52
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3answers
4k 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 ...
20
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1answer
2k views

How are the Conway polynomial and the Alexander polynomial different?

Background story: I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he ...
5
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1answer
398 views

Rotation part of short geodesics in hyperbolic mapping tori

Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
25
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2answers
3k views

Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon. Does there exist a ...
2
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1answer
417 views

When can a folded polygon be isometrically (locally) embedded into R^3 ?

I am interested in 3-D representations of various things that naturally live in a non-simply-connected compact surface. There is the usual way of producing a compact surface of any orientable or non-...
4
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3answers
212 views

What kind of 3-manifolds arise has hypersurfaces in R^4?

What kind of 3-manifolds can arise as hypersurfaces $\{ f(x,y,z,w) = 0\} \subset \mathbb{R}^4$? Can they have nontrivial H1 or H2?
12
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5answers
1k views

Which manifolds admit a diffeomorphism of order $n$?

Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$? For a closed orientable surface $S_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S_g$ can be ...
11
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2answers
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Constructing 4-manifolds with fundamental group with a given presentation.

Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold. When starting to write this answer I found this related MO ...
7
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4answers
1k views

To differently gluing of two Riemann surfaces with boundary we get different surfaces

If $M,N$ are two Riemann surfaces with boundary, then we can glue them along one of each of their boundary component, which is $S$, to form a new Riemann surface with boundary, but for different ...
7
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3answers
386 views

An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?

Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the plain sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are ...
6
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1answer
343 views

Coordinates on Flag Manifolds

Suppose you want to work with complete flags $\mathbb{F}_3$ on $\mathbb{C}^3$. Given a flag $$ \{0\}\leq V_1\leq V_2 \leq \mathbb{C}^3$$ you can think of $V_1$ as the span of a vector $\vec{u}$, ...
13
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4answers
533 views

Minimal-length embeddings of braids into R^3 with fixed endpoints

(Apologies in advance for any imprecision in the following; I am a computer scientist and regret never having taken an actual course on topology.) One way to define the pure braid group $P_n$ is as ...
5
votes
2answers
233 views

Symmetric colorings of regular tessellations

Given a regular tessellation, i.e. either a platonic solid (a tessellation of the sphere), the tessellation of the euclidean plane by squares or by regular hexagons, or a regular tessellation of the ...
13
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5answers
854 views

A characterization of convexity

While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here. Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\...
73
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3answers
4k views

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 ...
5
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1answer
278 views

Solenoid of a continuous map of a ball, is it contractible?

Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map. Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid $$ S_f=\...
1
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0answers
345 views

Riemann surfaces

In Hatcher's book on Algebraic Topology, a lot of coverings of $S^1 \vee S^1$ has given. From these coverings, can we get different coverings of double torus (genus 2, compact surface)?
21
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3answers
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“Largest” finite-dimensional Lie subgroups of Diff(S^n), are they known?

The group $Diff(S^n)$ ($C^\infty$-smooth diffeomorphisms of the $n$-sphere) has many interesting subgroups. But one question I've never seen explored is what are its "big" finite-dimensional ...
7
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2answers
956 views

Osculating conics and cubics and beyond

The osculating circle at a point of a smooth plane curve can be obtained by considering three points on the curve and the circle defined by them. When the three points approach $P$, the circle becomes ...
19
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2answers
4k views

De Rham decomposition theorem, generalisations and good references

De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$ that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
2
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2answers
315 views

Name for the motion of an immersion?

I have an immersion of a 2-simplicial complex S in $\mathbb{R}^3$, and then a piecewise linear motion of that immersion over an interval of time [0,1]. Is there an existing name for the map $f:S\...
5
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1answer
507 views

Is a simple loop in a spine of a strongly irreducible Heegaard splitting primitive in the fundamental group?

Let $\gamma$ be a simple loop in a spine of a strongly irreducible Heegaard splitting of a closed 3-manifold $M$ with torsion-free fundamental group. Does $\gamma$ necessarily correspond to a ...
9
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1answer
955 views

How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?

I'm wondering if anyone can point me to a reference on how the various Lefschetz-Poincare dual torsion pairings of a manifold with boundary fit together. To explain in more detail, consider a ...
4
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3answers
689 views

About the proof of Wajnryb's finite presentation of Mod(S)

I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't ...
19
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0answers
541 views

Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
18
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4answers
1k views

Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it ...

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