All Questions
Tagged with gt.geometric-topology reference-request
80 questions with no upvoted or accepted answers
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
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
776
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
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
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
12
votes
0
answers
383
views
Is the quotient map of the action of homeomorphisms on embeddings well-behaved?
It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the $C^\...
- 8,167
11
votes
0
answers
269
views
Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
- 111
11
votes
0
answers
379
views
Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
- 525
10
votes
0
answers
455
views
Exotic analytic triangulations of $S^5$?
I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie
https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf
http://www-math.mit.edu/~...
- 14.3k
10
votes
0
answers
415
views
Lipschitz homotopy groups
There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find
...
- 28k
10
votes
0
answers
303
views
When were bordered Heegaard Floer homology's DA bimodules invented?
This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston (...
- 1,474
8
votes
0
answers
222
views
references on categorification of knot invariants
I am extremely sorry if this is not the right place for this kind of question.
I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
- 81
8
votes
0
answers
200
views
Disciplining dunce hats
I'm wondering if anyone has a copy of a preprint by Charles Giffen from 1977, with the enjoyable title, Disciplining dunce hats in 4-manifolds. I've seen it referred to in various places, including ...
- 19.4k
8
votes
0
answers
502
views
Reference request: Mapping class group action on homology of surface with boundary
This is a request for a reference to a proof of a result. The result is not very hard, but I'd rather cite than reprove.
I'm looking for a generalization of the following result (Farb and Margalit, ...
- 366
8
votes
0
answers
251
views
Exact triangle for monopole Floer homology with $\mathbb{Z}$-coefficient
Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained ...
- 1,069
8
votes
0
answers
193
views
PL surface projections - is there a theory of folds and cusps?
For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...
- 81
7
votes
0
answers
407
views
Understanding that a simplicial complex is a PL manifold via links
Suppose $X$ it a simplicial complex homeomorphic to a topological $n$-manifold. Suppose we know that the link of each $k$-simplex $\Delta^k$ is homeomorphic (as a topological space) to the sphere $S^{...
- 14.3k
7
votes
0
answers
156
views
Two papers on surface diffeomorphisms
The following two papers appeared in the reference of a paper i was reading.It seems that neither is published formally.Is there a website where i could find them?
A. Casson, Cobordism Invariants of ...
- 103
7
votes
0
answers
1k
views
Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
- 10.5k
7
votes
0
answers
279
views
Relations between Betti numbers for clique complex
Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ ...
- 8,071
7
votes
0
answers
355
views
Making diffeomorphism of submanifolds boring
This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds"...
- 1,981
7
votes
0
answers
1k
views
A "direct" proof that hyperbolic groups are not amenable
I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible.
Here are the two proofs I am aware ...
- 4,432
7
votes
0
answers
448
views
Reference Request: Topological h-cobordism theorem in higher dimensions
I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in ...
- 71
7
votes
0
answers
208
views
How do metrics behave under joining along a manifold embedded in the boundary?
How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
- 1,897
6
votes
0
answers
399
views
Borel conjecture and arbitrary surface
Before starting my question I want to write something that I already know.
Borel Conjecture: Any homotopy equivalence between two closed
aspherical manifolds is homotopic to a homeomorphism.
Now, my ...
- 632
6
votes
0
answers
132
views
Is there any work in topological data analysis on something like "Voronoi complexes"?
Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...
- 15.4k
6
votes
0
answers
321
views
Elementary questions about Morse-Bott functions
Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...
- 14.3k
6
votes
0
answers
163
views
Reference to the theorem about linear bundles
The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof?
Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed ...
- 1,095
6
votes
0
answers
172
views
Does Novikov additivity hold for topological manifolds?
Recall that Novikov additivity of signature of compact oriented smooth $4k$-manifolds is the following statement :
If two manifolds are glued by an orientation-preserving diffeomorphism of their ...
- 14.3k
6
votes
0
answers
812
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
6
votes
0
answers
217
views
What is the state of the art in 4-manfold 2-types?
In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
- 35.5k
5
votes
0
answers
289
views
A certain kind of proof of the Hairy Ball Theorem
I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.
From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
- 17.6k
5
votes
0
answers
131
views
Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"
I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
- 1,097
5
votes
0
answers
272
views
When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
- 1,763
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
5
votes
0
answers
233
views
Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
- 18.4k
5
votes
0
answers
1k
views
Prerequisites for reading Gregory Perelman's work
What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.
- 1,594
5
votes
0
answers
461
views
When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?
Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a ...
- 722
5
votes
0
answers
265
views
Quotienting disk inside sphere result in sphere
Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...
- 2,023
4
votes
0
answers
154
views
Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
4
votes
0
answers
183
views
In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
- 13.6k
4
votes
0
answers
177
views
Ping pong with parabolic isometries on Gromov hyperbolic spaces
For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
- 41
4
votes
0
answers
158
views
Postnikov square explicitly on a simplicial complex
$\DeclareMathOperator\Z{\mathbb{Z}}$
Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by ...
- 10.5k
4
votes
0
answers
88
views
What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?
Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
- 2,533
4
votes
0
answers
79
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 ...
- 29.1k
4
votes
0
answers
188
views
Mapping class group of $\mathbb{S}^3$
If I recall correctly from a lecture I attended the last year we have that
$MCG(\mathbb{S^2})\simeq\frac{\mathbb{Z}}{2\mathbb{Z}}$ by Smale in the 60' and $MCG(\mathbb{S^3})\simeq\frac{\mathbb{Z}}{2\...
- 2,533
4
votes
0
answers
290
views
Generalized Postnikov square
Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
- 1,329
3
votes
0
answers
227
views
Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
- 803
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 613