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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137 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, $T_{...
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0answers
19 views

How to got there are only two kinds of 1-dim manifold without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
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0answers
38 views

The Fundamental Semigroups of a Topological Object?

I had the following idea for a generalization of the "fundamental group" of a manifold. So the idea basically was that we can consider a manifold $O$ which has boundary $\partial O$, and instead of ...
2
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1answer
47 views

Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
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52 views

Is Perelman comparable with Cantor? [closed]

Well, it seems Cantor is fundamental in set theory whereas Perelman extended the topological ideas of Poincaré. But set theory and topology are intimately connected, so that my question makes some ...
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2answers
513 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
1
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1answer
29 views

Solid tori, meridians, and longitudes

I am working through some of Rolfsen's "Knots and Links" and I have needed to go back and take a more careful look at the first few sections where he carefully discusses curves on solid tori. Let $V$ ...
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1answer
51 views

Proving Hopf degree theorem using Pontrjagin-Thom isomorphism

Does anyone know a good reference which proves Hopf degree theorem using Pontrjagin-Thom theorem, that is passing to the determination of framed bordism classes of 0-manifolds? Many thanks! Hopf ...
2
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0answers
63 views

Commensurability of two groups

If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
6
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2answers
110 views

Is Floer homology always isomorphic to the singular homology of some space?

After I studied Morse homology, I'm now studying the following Floer homology theories : 1) Symplectic Floer homology ; 2) Floer homology of lagrangians ; 3) Heegard-Floer homology ; ...
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0answers
30 views

Using the general ham sandwich theorem to proof Hobby-Rice

Matousek mentions that you can proof the continuous necklace theorem known as Hobby-Rice theorem via the continuous ham sandwich theorem. The continuous ham sandwich states: Let $\mu_1,\mu_2,...,\...
2
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0answers
33 views

geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
2
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0answers
45 views

singular homology of a simplicial complex [duplicate]

On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...
3
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2answers
158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
8
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1answer
123 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
4
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1answer
77 views

Homology 4-balls with boundary $S^3$

Are there interesting homology 4-balls with boundary $S^3$? Going the other way, must any homology 4-ball with boundary $S^3$ be homotopy equivalent/homeomorphic/diffeomorphic to $B^4$?
2
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0answers
74 views

How can we assume the first homology group of the complement is zero when constructing a Casson handle?

I am currently working through Scorpan's Wild World of 4-Manifolds specifically the section on Casson Handles. On page 78, he says if $D$ is the core of the handle after $n$ stages we may assume $\...
2
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1answer
148 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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0answers
52 views

Jones polynomial invariance

I'm studying the Jones polynomial and I know that it is a knot invariant. I saw that a possible way to define the Jones polynomial is to set the Jones polynomial of the unknot to be 1 and then use the ...
0
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2answers
59 views

submanifold with same homology

Suppose $M$ is a manifold without boundary, and $N\subseteq M$ is any submanifold, possibly with boundary. If $H_*(N)\cong H_*(M)$, is it necessarily true that $N\cong M$?
70
votes
1answer
2k views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
3
votes
1answer
73 views

What is the topology of an infinite cylinder?

Consider an infinitely long straw. This is a genus 1, orientable manifold. It is not closed because it is infinitely long. Is there a way I can describe the property that it is "partially closed" or ...
4
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2answers
102 views

Current research on inverse knot equivalence?

What is the current status of the open problem in knot theory 'When is a knot equivalent to its inverse?' Additionally, I would like to know what work has been done on this problem (I cannot find ...
4
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2answers
56 views

What is the relationship between diffeomorphisms of the sphere modulo isotopy and exotic spheres?

In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical ...
3
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1answer
62 views

Trying to understand Heegaard diagrams

I have been looking through Rolfsen's "Knots and Links" and I have come across some questions that I am confused about regarding Heegaard diagrams. Let $H_1$ and $H_2$ be genus $g$ handlebodies and ...
6
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0answers
128 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
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1answer
59 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
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2answers
903 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
3
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2answers
113 views

Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
2
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1answer
16 views

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $3$ cutpoints of order $3$

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $2$ cutpoints of order $3$ Definition: A cut point of order $k$ is a point $a \in X$ whose complement $X-\{a\}$ consists of $k$ ...
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votes
0answers
52 views

What is $S^3/S^1$?

I have been given this space in a question but I am unsure what it means I know that $S^3=\{(z_1, z_2) \in \mathbb{C^2}\mid |z_{1}|^2 + |z_{2}|^2=1 \}$ Could you help me understand what set of ...
5
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2answers
86 views

Klein bottle in $\mathbb{R}^4$ does not have a couple of normal vector fields

For an orientable 2-manifold in $\mathbb{R}^4$, it's somewhat obvious that if the manifold has a normal vector field then it has a pair of normal vector fields. I am trying to understand why it is ...
14
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3answers
465 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 ...
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1answer
33 views

invariants of knots that are invariants under band move.

I am asking whether there are known knot invariants which are invariants under band move. Note that band move operation is similar to a connected sum of two knots except that the projections of two ...
1
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1answer
45 views

Open set in $\mathbb{C}$ with non-trivial boundary

When studying complex analysis - or even real analysis for that matter - we most times consider open sets $\Omega \subset \mathbb{C}$ (or $\Omega \subset \mathbb{R}^2$) having smooth curves as its ...
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2answers
51 views

Why do equatons of two variables specify curves in $\mathbb{R}^2$?

I suppose to more formally characterize the question more formally, why are all points of the set $\{ (x,y) \mid F(x,y) = 0 \}$ always boundary points (and I believe also never isolated points) in the ...
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0answers
87 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
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1answer
25 views

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse theory....
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0answers
40 views

Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
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0answers
46 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
3
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2answers
71 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times [0,\...
6
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1answer
276 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...
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0answers
30 views

the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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0answers
30 views

Euler characteristic and Phase rule? Is there a connection between them?

Eulers characteristic states $$Vertices+Faces=Edges+2$$ Gibbs' phase rule states $$ (\text{degrees of Freedom}) + (\text{no. of Phases}) = (\text{no. of Components}) + 2$$
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1answer
82 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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1answer
22 views

Construction of an $n$-Sphere

I have been thinking about various ways to construct an $n$-sphere. Starting with $S^2$, we can construct it by taking two disks, lifting the "meat" of the disks into a third dimension and then ...
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68 views

Graph theory: creating surfaces

If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside. On the following surfaces, determine the maximal number of circles that you can draw on the surface ...
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0answers
56 views

Compute Euler characteristic $\chi(S^2 \times S^3)$

This is a question from an undegraduate Topology course Compute the Euler characteristic $\chi(S^2 \times S^3)$ For topological polyhedra $\chi(X \times Y)=\chi(X) \times \chi(Y) \implies \chi(S^...
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0answers
34 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
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0answers
11 views

Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})...