All Questions
Tagged with gt.geometric-topology reference-request
361 questions
64
votes
7
answers
7k
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. ...
62
votes
9
answers
9k
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 ...
- 44.8k
53
votes
7
answers
11k
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 ...
- 44.8k
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
- 151k
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
39
votes
10
answers
4k
views
Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
36
votes
2
answers
5k
views
Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
- 151k
34
votes
1
answer
4k
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 ...
- 6,210
29
votes
5
answers
7k
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 ...
- 23.5k
28
votes
3
answers
3k
views
Kirby's torus trick
My basic question is: What is Kirby's torus trick and why did it solve so many problems?
I can get a glimmer of it from looking at Kirby's original paper, "Stable Homeomorphisms and the Annulus ...
- 281
26
votes
9
answers
13k
views
Teichmuller Theory introduction
What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
26
votes
5
answers
2k
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 number
(the minimal number of ...
- 151k
24
votes
3
answers
3k
views
Is there a combinatorial analogue of Ricci flow?
The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing.
However I have ...
- 9,132
24
votes
1
answer
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 ...
- 151k
24
votes
2
answers
1k
views
SnapPea for the uninitiated
SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...
- 56.9k
24
votes
0
answers
1k
views
Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
23
votes
1
answer
3k
views
The cone on a manifold
I believe that I have run across the statement that if $X$ is a compact smooth manifold and $CX$ is the cone on $X$, i.e. $[0,1] \times X$ modulo $(0,x)\sim(0,y)$ for all $x,y \in X$, then $CX$ admits ...
- 26.3k
23
votes
2
answers
1k
views
fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
- 593
23
votes
2
answers
850
views
Classification of fake (quaternionic, octonionic) projective spaces
If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
- 9,580
22
votes
4
answers
2k
views
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology:
Let f be a $C^\infty$ function on a closed manifold ...
- 22.1k
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
21
votes
2
answers
875
views
Do Betti numbers beyond the first have a "number of cuts" interpretation?
I have heard stated the following
Theorem. If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you ...
- 23.3k
21
votes
0
answers
861
views
A mysterious paper of Stallings that was supposed to appear in the Annals
In Stallings's paper
Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389.
he briefly discusses how to prove "several generalizations" of Brown's ...
- 353
21
votes
0
answers
777
views
Is the mapping class group of $\Bbb{CP}^n$ known?
In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...
- 9,580
21
votes
0
answers
344
views
Are there exotic $\mathbb{R}^4$'s that are products with $\mathbb{R}$? [duplicate]
This came up in a conversation:
Question: Is there an exotic $\mathbb{R}^4$ that smoothly splits off an $\mathbb{R}$ factor? More precisely, suppose that $\mathcal R$ is a smooth 4-manifold which ...
- 724
20
votes
4
answers
2k
views
Topological spaces made by identifying opposite faces of a cube?
My bashful, nameless, colleague asked me:
When you identify opposite faces of a square, then depending on where you twist or not, you get a torus, Klein bottle, or projective plane.
What spaces ...
- 27.8k
20
votes
2
answers
2k
views
Simple curves on non-orientable surfaces.
Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
- 32.1k
19
votes
7
answers
6k
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 ...
- 23.5k
19
votes
1
answer
901
views
Locus of equal area hyperbolic triangles
Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
- 565
19
votes
1
answer
843
views
Vector field on a K3 surface with 24 zeroes
In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...
- 35.5k
19
votes
0
answers
575
views
The oriented homeomorphism problem for Haken 3-manifolds
Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
- 25k
18
votes
1
answer
2k
views
Unique smooth structure on 3-manifolds
Do you know a good reference for the existence and uniqueness of a smooth structure on $3$-manifolds?
As far as I understand topological $3$-manifolds admit a unique smooth structure.
I could find ...
- 28k
18
votes
3
answers
1k
views
Classification of knots by geometrization theorem
I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does this ...
- 1,314
18
votes
2
answers
1k
views
Hyperbolic Volume and Chern-Simons
In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
- 183
18
votes
2
answers
1k
views
formula for Eta invariant
Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...
- 259
17
votes
6
answers
2k
views
On the number of Archimedean solids
Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, ...
- 273
17
votes
3
answers
2k
views
What is the state of the art for algorithmic knot simplification?
Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
- 22.1k
17
votes
1
answer
526
views
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...
- 151k
17
votes
2
answers
1k
views
What is the homotopy type of the space of the homeomorphisms of the n-ball whose restriction to the boundary is isotopic to the identity?
Consider the set of homeomorphisms of the topological n-ball to itself with the compact open topology. Sitting inside this space of homeomorphisms are particular subspaces. The first subspace is those ...
- 1,325
16
votes
3
answers
942
views
Maximal degree of a map between orientable surfaces
Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.
What is an elementary proof of ...
- 1,095
16
votes
2
answers
820
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 45k
16
votes
1
answer
891
views
Compact contractible topological manifold with boundary=sphere is a ball
(this question is joint with Steven Karp and Thomas Lam) We need to use the following fact in our paper:
Theorem 1. Let $M^n$ be a compact contractible topological manifold with boundary, such that ...
- 436
16
votes
1
answer
521
views
Extending a map from $S^n\to M^n$ to a nice map from $B^{n+1}\to M^n$
Let $S^n$ and $B^{n+1}$ be the unit sphere and unit ball in $\mathbb{R}^{n+1}$, and let $M^n$ be a contractible space of dimension $n$.
If necessary, assume that $M^n$ is a contractible simplicial $n$-...
- 368
16
votes
1
answer
569
views
Shortest Casson tower containing a slice disk for the attaching curve
A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk (...
- 711
16
votes
0
answers
438
views
Survey of known results on equivariant transversality
Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant ...
- 9,580
15
votes
5
answers
3k
views
Generalization of winding number to higher dimensions
Is there a natural geometric generalization of the winding number to higher dimensions?
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theorem),
...
- 151k
15
votes
3
answers
1k
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 ...
- 28k
15
votes
3
answers
2k
views
moduli spaces are kahler?
I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see (...
- 829
15
votes
2
answers
3k
views
Covering the space by disjoint unit circles
Andrzej Szulkin [MR0719756] (thanks to Alexey Ustinov for the reference) proved the following two interesting theorems.
Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate ...
- 3,104
15
votes
1
answer
954
views
Extending diffeomorphisms
Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...
- 28k