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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6
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2answers
100 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 ; ...
0
votes
0answers
24 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
votes
0answers
30 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
votes
0answers
41 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
votes
2answers
152 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
votes
1answer
118 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 ...
3
votes
1answer
59 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 $B^4$?
1
vote
0answers
70 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
votes
1answer
145 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 $C^2$ ...
1
vote
0answers
49 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
votes
2answers
58 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
votes
2answers
97 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
votes
2answers
54 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
votes
1answer
53 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
votes
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 ...
1
vote
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 ...
16
votes
2answers
895 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
votes
2answers
109 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
votes
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 $k$ ...
0
votes
0answers
51 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
votes
2answers
83 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
votes
3answers
458 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 ...
1
vote
1answer
31 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
vote
1answer
44 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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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....
2
votes
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 ...
8
votes
1answer
468 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 ...
0
votes
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
votes
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
votes
1answer
275 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 $g$ ...
1
vote
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 $A$ ...
4
votes
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 $$ (\text{degrees of Freedom}) + (\text{no. of Phases}) = (\text{no. of Components}) + 2$$
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
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^...
1
vote
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 ...
1
vote
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\})...
4
votes
1answer
49 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
0
votes
0answers
59 views

Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ($...
0
votes
1answer
49 views

Homeomorphism between $\Bbb S$\ $\lbrace-1\rbrace$ and $\left(-1,1\right)$

I Know that $\Bbb S$\ $\lbrace-1\rbrace=\lbrace e^{i\theta}:\theta\in \left(-\pi,\pi\right)\rbrace=\lbrace e^{i\pi t}: t\in\left(-1,1\right)\rbrace$ Let $f:\Bbb S$ \ $\lbrace-1\rbrace\longrightarrow\...
0
votes
2answers
40 views

Is an isometric and bijective mapping between two metric spaces complete?

If I have the two metric spaces $(X,d_x)$ and $(Y,d_y)$ with the mapping $f : X \to Y$ that is both an isometry and bijection between X and Y. How do I show that $(Y,d_y)$ is complete iff $(X,d_x)$ ...
1
vote
1answer
22 views

Is the fundamental group of a retract a subgroup of the original space?

Let $X$ be a topological space and $A$ a retract of $X$. Is the fundamental group of $A$ a subgroup of the fundamental group of $X$?
0
votes
1answer
25 views

Euler's formula about graphs embedded in $\mathbb{R^2}$

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$ I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the complement ...
0
votes
1answer
63 views

if M is compact and N is connected, then M=N …?

Let M and N be surfaces in $R^3$ such that M is contained in N. If M is compact and N is connected, prove that M=N. ================================= I thought intuitively the compactness means ...
2
votes
0answers
38 views

stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...