The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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4
votes
2answers
131 views

Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$

Let $n$ be a nonnegative integer and $x\in S^n$ a point in the n-sphere. Combining the map $\alpha\colon SO_{n+1}\longrightarrow S^n$ induced by matrix multiplication with $x$ and the connecting ...
2
votes
1answer
105 views

Homeomorphism type of the cone on a cylinder

Let $X$ be a topological space. The cone $CX$ on $X$ is the cylinder $X \times I$ with the top $X \times 1$ identified to a point. Clearly for every $X$, $CX$ is contractible. Looking at the ...
15
votes
1answer
253 views

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ...
3
votes
0answers
182 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
5
votes
2answers
85 views

Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
2
votes
1answer
63 views

Isotopic tori in $\mathbb{R}^4$

Intuitively it seems to me that two tori in $\mathbb{R}^4$ are isotopic to each other. By isotopic, I mean a smooth family of deformations beginning in one and ending in the other, and each member of ...
1
vote
1answer
67 views

Does a closed surface in the 3-sphere bound a handlebody? [closed]

If a closed surface is embedded in the 3-sphere, then does it bound a handlebody?
0
votes
1answer
73 views

Is “small disk” well-defined?

I saw the notion "small disk" very frequently used in literature. For example, in Brunnian braids on surfaces by V. G. Bardakov, R. Mikhailov, V. V. Vershinin, J. Wu, one line reads: Let $P_n(M)$ ...
-2
votes
1answer
66 views

Check the relation between the topology

Let $\mathbb Z_+ = \{1, 2, 3, .....\}$ be the set of positive integer . Let $\tau_1 := $ subspace topology on $\mathbb Z_+$ induced from the usual topology on $\mathbb R ,$ $\tau_2 :=$ ...
1
vote
1answer
52 views

Geometric Intuition for “Right-Veering” Property of $f$ in MCG(S)?

let $S$ be a compact surface with non-empty boundary, let $\alpha : [0,1] \rightarrow S$ be a Properly-embedded arc (meaning both endpoints of the arc are in $\partial S$) and let $f$ be an element ...
2
votes
1answer
369 views

The notion of a right-angled hexagon in hyperbolic geometry

I was hoping someone would help me understand better what a "right-angled hexagon" is in hyperbolic geometry. I know these are glued together somehow to produce hyperbolic pairs-of-pants. The only ...
14
votes
3answers
396 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
0
votes
1answer
68 views

Are there “interesting” examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?

Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a ...
2
votes
0answers
46 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
0
votes
1answer
44 views

Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
3
votes
1answer
132 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...
1
vote
1answer
74 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
0
votes
1answer
52 views

How can I show that if a set is bounded, then it's contained in a k-cell?

The set is a bounded subset of R (under the Euclidean metric), and a k-cell is a set of points {x_1...x_k} such that a_j < x_j < b_j for j=1...k. Any ideas on how to show this?
0
votes
0answers
46 views

Classification of surface with 18-gon planar diagram

For starters, this is a problem from L. Christine Kinsey's "Topology of Surfaces." The problem is to classify the surface using cut and paste arguments on polygons. However, between my limited ...
-1
votes
1answer
139 views

Proof of Jordan curve theorem

Is it possible for the following to be proof for Jordan curve theorem: Given the distance function on $\mathbb{R}^2$ ($d((x_1,y_1),(x_2,y_2))=\sqrt{ |x_2-x_1|^2 + |y_2-y_1|^2}$), and $\varepsilon ...
2
votes
1answer
102 views

Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
4
votes
0answers
66 views

Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, ...
7
votes
0answers
84 views

Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
3
votes
2answers
248 views

Proof of Brouwer's Fixed Point Theorem.

What is the simplest way to prove Brouwer's Fixed Point Theorem in three dimensions?
3
votes
2answers
94 views

Is there only one free (continuous) action of $\mathbb{Z}_2$ on $S^2$?

We all know that the antipodal map is a free action of $\mathbb{Z}_2$ on $S^2$. Considering $\mathbb{Z}_2 = \{1, -1\}$, a free action may be viewed as a map $f : S^2 \rightarrow S^2$, i.e. the action ...
10
votes
2answers
180 views

Partitioning the plane into three sets each intersecting the vertices of every square with side 1?

Q1. Is it possible to partition the plane into three sets such that each of them contains at least one vertex of every square with side 1 ? (I mean all squares of side-length 1, not just those with ...
1
vote
0answers
84 views

A Mistake in GTM 247 (Braid Groups)?

I am reading Braid Groups (GTM 247) by Kassel Christian and Turaev Vladimir and am puzzled by a detail in the proof of a theorem: I do not quite see the reason of the inequality sign in the ...
2
votes
2answers
195 views

isotopy of homeomorphisms of a torus

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy? Edit: It's well-known statement that a homoemorphism of a torus is ...
0
votes
1answer
24 views

Gluing of two geodesic space along a proper space is geodesic.

Let $X_1$ and $X_2$ geodesic metric spaces glued along $A$ a proper subspace of both and then given the pseudo metric. Why is the glued space geodesic? Any hint ? For notation and details one can ...
0
votes
1answer
66 views

Discontinuity of the identity function in topology

According to a theorem I was taught, the identity function $id(x)=x$ from $(\mathbb{R}, \tau_1)$ to $(\mathbb{R}, \tau_2)$ is continuous if $\tau_1 = \tau_2$. Are there any examples of topologies ...
1
vote
1answer
95 views

Plane models from the “word”

I have a "word" for a plane model $abacdc^{-1}db^{-1}$. From what I reckon, it's a torus. But I am not too sure of it. I sketched it up and did some "adjustments". Could it be a projective plane ...
3
votes
1answer
84 views

Forgetting a Strand in Braid Groups

Let $B_n$ be the braid group of $n$ strings over the unit disk $D$. Let $$d_i:B_n\to B_{n-1}$$ be the operation which is obtained by forgetting the $i$-th strand, $1\leq i\leq n$. Geometrically this ...
7
votes
1answer
339 views

Does there exist homeomorphism without fixed points?

Does there exist a homeomorphism of the unit disk with two holes $$\left\{(x,y):x^2+y^2 \le 1\right\} \setminus \left (\left \{(x,y):\left(x+ \frac 1 2 \right)^2+y^2 < \frac 1 {10} \right \} ...
0
votes
3answers
55 views

Is the closure of an open connected subset of $\mathbb{R}^{n}$ a topological manifold?

If we remove the connectedness restriction, there are easy counter examples, as in: $\left(\frac{1}{2}, \frac{1}{1}\right) \cup \left(\frac{1}{4}, \frac{1}{3 }\right) \cup \left(\frac{1}{6}, ...
0
votes
1answer
68 views

Braid Groups on Manifolds

I am studying braid groups on manifolds and am getting confused. In a geometric definition, one needs to first choose a simple curve $\theta$ on a given manifold $M$ and well-ordered points ...
2
votes
1answer
63 views

Are there PL-exotic $\mathbb{R}^4$s?

The title may or may not say it all. I know that there are examples of topological 4-manifolds with nonequivalent PL structures. In some lecture notes, Jacob Lurie mentions that not every PL manifold ...
0
votes
1answer
47 views

Non-punctual Boundary

In the book of Bill Thurston, Three dimensional geometry and topology, there is an exercise to show torus can be partitioned into 7 countries, each on one piece and has common (non-punctual) ...
4
votes
2answers
98 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
0
votes
0answers
491 views

Topology of biological compartments

In the field of cell biology, there is a general sub-field concerned with the topology of organelle membranes, and a key focus remains on how these dynamic membranes deform and interact with cellular ...
2
votes
0answers
38 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
0
votes
0answers
41 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
2
votes
1answer
61 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
0
votes
1answer
48 views

Definition of a metric space with bounded growth

Does anyone know the definition of a metric space with bounded growth? I was reading a paper by Roe titled Hyperbolic groups have finite asymptotic dimension, where he writes a definition, but I ...
3
votes
0answers
53 views

References on the relations between Top, Diff and PL

I have heard many times informal statements like "differentiable and pl manifolds are essentially the same for such and such dimensions", but I would like to know what they mean exactly and how such ...
5
votes
0answers
56 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
6
votes
1answer
123 views

Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that ...
3
votes
1answer
115 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
1
vote
2answers
42 views

Are topologically well-behaved measure 0 subsets of $\Bbb R^2$ finite graphs?

Conjecture: If $X\subseteq \Bbb R^2$ is locally simply connected (hence locally path connected), compact and Lebesgue measure $0$ then $X$ is homeomorphic to a finite graph. It is clear that ...
2
votes
0answers
87 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
1
vote
0answers
69 views

An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...