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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71
votes
12answers
10k views

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 ...
21
votes
2answers
1k views

Uniqueness of compactification of an end of a manifold

Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
51
votes
5answers
5k views

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 ...
68
votes
4answers
7k views

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? ...
62
votes
3answers
9k views

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 ...
50
votes
9answers
6k views

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 ...
25
votes
5answers
2k views

Compactification theorem for differentiable manifolds ?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
30
votes
1answer
2k views

Not all manifolds can be triangulated: In which dimensions?

I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail ...
12
votes
2answers
2k views

Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented

A Noetherian group (also sometimes called slender groups) is a group for which every subgroup is finitely generated. (Equivalently, it satisfies the ascending chain condition on subgroups). A ...
8
votes
1answer
285 views

Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)

From "The multiple facets of the associahedra" by Loday: Let us consider the formal power series $$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$ and let $$ g(x) = x+b_1 ...
25
votes
6answers
3k views

Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
24
votes
2answers
4k views

Are non-PL manifolds CW-complexes?

Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex? I'm pretty sure that the answer is yes. However, I have not managed to find a reference for this....
26
votes
4answers
2k views

Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
20
votes
3answers
2k views

Cohomology of fibrations over the circle: how to compute the ring structure?

This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
18
votes
3answers
3k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
17
votes
6answers
2k views

Diffeomorphism of 3-manifolds

Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder ...
12
votes
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 ...
12
votes
3answers
818 views

Linking topological spheres

Is there a simple proof of the fact that: If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are ...
9
votes
3answers
604 views

Homeomorphic but Non-Conjugate Mapping Tori

Suppose we fix a genus $g$ closed surface $S$. Let $f, g \in Map(S)$ be conjugate, for $Map(S)$ the mapping class group of $S$. Then I know that $M_f$ (the mapping torus of $M$ with monodromy $f$) is ...
6
votes
2answers
1k views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
10
votes
1answer
473 views

CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
52
votes
8answers
6k views

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
7answers
5k views

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. ...
57
votes
10answers
16k views

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 ...
50
votes
7answers
6k views

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello, I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ : For example : 1) Are all ...
46
votes
4answers
3k views

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 $M$? ...
30
votes
4answers
2k views

How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he ...
18
votes
7answers
4k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
77
votes
1answer
2k views

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--...
16
votes
3answers
2k views

When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$? Edit: Formerly, I asked about a flat connection on the frame ...
51
votes
4answers
2k views

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 ...
46
votes
3answers
5k views

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)...
26
votes
3answers
3k views

finite generated group realized as fundamental group of manifolds

This is discussed in the standard textbooks on algebraic topology. Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$ where $g_i$ are generators and $r_j$ are ...
23
votes
3answers
3k views

Complete knot invariant?

I've seen a couple papers (that I now can't find) that say that in his paper "On irreducible 3-manifolds which are sufficiently large" Waldhausen proved that the data $\pi_1(\partial (S^3\setminus K)) ...
20
votes
1answer
2k views

connectivity of the group of orientation-preserving homeomorphisms of the sphere

In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written: Is the group of orientation-preserving ...
32
votes
1answer
3k views

Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
29
votes
1answer
824 views

Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
16
votes
3answers
2k views

When are (finite) simplicial complexes (smooth) manifolds?

Hi, is there an algorithm that determines if a given simplicial complex is a.) a manifold b.) a smooth manifold c.) homotopy equivalent to a manifold d.) a real algebraic variety ?
31
votes
3answers
2k views

Why are there no wild arcs in the plane?

On math.stackexchange it was asked whether all arcs in the plane are ambient-isotopic. I suggested that one could prove this by appealing to the Schönflies theorem, which you can do as long as you can ...
22
votes
5answers
4k views

Proof of the Reidemeister theorem

While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
31
votes
1answer
773 views

Classifiying sphere eversions

For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as ...
26
votes
1answer
2k views

Fake versus Exotic

Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic? Terminology (perhaps non-...
22
votes
0answers
753 views

Monoid structure of oriented manifolds with connect sum

Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation ...
15
votes
3answers
1k views

Is it true that all sphere bundles are boundaries of disk bundles?

Let $E$ be the total space of the sphere bundle $S^k\to E\to M$, is it true that there exists a disk bundle $D^{k+1}\to N\to M$ such that $E=\partial N$? (where $D^{k+1}$ is the unit disk in $\mathbb ...
10
votes
1answer
1k 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 ...
20
votes
2answers
956 views

Manifolds with homeomorphic interiors

Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
21
votes
1answer
1k views

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-...
26
votes
5answers
1k views

Complexity of random knot with vertices on sphere

Connect $n$ random points on a sphere in a cycle of segments between succesive points:       I would like to know the growth rate, with respect to $n$, of the crossing ...
23
votes
1answer
2k views

Building a genus-$n$ torus from cubes

I wonder if this has been studied: What is the fewest number of unit cubes from which one can build an $n$-toroid? The cubes must be glued face-to-face, and the boundary of the resulting object ...
15
votes
10answers
2k views

Orbifold fundamental group in terms of loops?

In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...

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