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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1answer
91 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
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0answers
12 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
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1answer
26 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
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0answers
14 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 ...
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1answer
48 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
42 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 ...
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0answers
27 views

An integrality theorem for immersions of complex projective spaces in the euclidean space

There are two questions: Please let me know your proof of the following theorem: If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...
3
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0answers
29 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, ...
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0answers
39 views

What is the definition of boundary-parallel Dehn twist?

I have not been able to find a working definition for the term: "boundary-parallel Dehn twist ". I know what a boundary-parallel surface is, and what a parallel surface is, but I have not been able to ...
7
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0answers
48 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. ...
2
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2answers
70 views

Proof of Brouwer's Fixed Point Theorem.

What is the simplest way to prove Brouwer's Fixed Point Theorem in three dimensions?
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0answers
38 views

How Fundamental Polygons Work

I am having trouble understanding how to represent a closed surface by a fundamental polygon. Wikipedia (http://en.wikipedia.org/wiki/Fundamental_polygon) gives the following examples, but I have no ...
2
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2answers
61 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 ...
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2answers
142 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 ...
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0answers
52 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 ...
0
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2answers
105 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 ...
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0answers
33 views

Show that S is homeomorphic to a Klein Bottle

I've been struggling quite a bit with this question. Any hints/help would be greatly appreciated! Consider the quotient S = R^2/G where G = Z^2 acts by (n, m) • (x, y) = ((−1)mx + n, y + m) on R^2 , ...
0
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1answer
47 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 ...
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1answer
25 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
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1answer
72 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
307 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 \} ...
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3answers
43 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
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0answers
30 views

Generalized Heegaard splittings and the rank of the fundamental group

Let $M$ be a compact, connected, orientable 3-manifold. Then M possesses a Heegaard splitting into two handle bodies of equal genus $g$. By looking at the Heegaard diagram and using Seifert ...
0
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1answer
43 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 ...
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0answers
28 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
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1answer
36 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
43 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
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0answers
71 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 ...
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0answers
17 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
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0answers
32 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 ...
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0answers
18 views

Isotopy of a graph extending to an ambient isotopy

Let $\Gamma$ be a graph and $f: \Gamma \rightarrow \mathbb{R}^{2}$ be its embedding into the plane. Suppose I deform $f$ with an isotopy, ie. I have a map $F: \Gamma \times I \rightarrow ...
2
votes
1answer
52 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
23 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
41 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 ...
3
votes
0answers
32 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 ...
4
votes
1answer
74 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
93 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 ...
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2answers
37 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
77 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 ...
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0answers
51 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 ...
1
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1answer
47 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
5
votes
1answer
283 views

Russian Texts on Geometry

I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing ...
5
votes
1answer
64 views

The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
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0answers
72 views

What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
3
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1answer
28 views

Extensions of diffeomorphisms of $S^2$ and the connectedness of $\text{Diff}^+(S^2)$

In this MO question by Daniel Moskovich, he claims that the fact that every diffeomorphism of $S^2$ extends to a diffeomorphism of $D^3$ implies that $\text{Diff}^+(S^2)$, the group of ...
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2answers
32 views

Questions about the Nature of Chirality (with some focus on dimensionality)

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in ...
3
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0answers
40 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
1
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1answer
80 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed (or just compact) 4-manifolds. Are there any closed (or compact) ...
9
votes
2answers
226 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
3
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0answers
64 views

Homeomorphisms of product spaces: an example [duplicate]

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...