Tagged Questions

2
votes
1answer
45 views

question from hatcher basic 3 manifolds

The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M? I had this problem reading ...
0
votes
1answer
35 views

Embedded 2-Submanifold

Can you help for solving this problem ı am triying to understand embedded 2 submanifold please help me.
0
votes
1answer
93 views

Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...
2
votes
0answers
55 views

Topological Manifold with ball, removed and antipodal points identified orientable?

Suppose you have a compact, orientable $(2n+1)-$manifold $M$, as in $H_{2n+1}=\mathbb{Z}$. You take a neighborhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So ...
3
votes
2answers
106 views

Orientability of projective space

Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even. First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
3
votes
0answers
82 views

Simple exercise in cohomology

I know this is a simple exercise but I am stuck unfortunately. Question: Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
1answer
59 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
2
votes
0answers
56 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
2
votes
1answer
50 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
1
vote
2answers
111 views

Help understanding manifolds and topological spaces

I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a ...
8
votes
0answers
110 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
5
votes
0answers
57 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
0
votes
0answers
63 views

necessary condition for subset of manifold to be manifold

It is clear that a subset of the manifold $\mathbb R^n$ is not always a manifold: for example the irrationals is a subset of $\mathbb R$ that is not a manifold itself. it is known that if a subset ...
6
votes
2answers
207 views

This set is a manifold

let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on ...
1
vote
1answer
32 views

Orientation of closed combinatorical surfaces

I have some problems with the equivalence of definitions of orientation. I know two definitions of orientation, namely: A surface is orientable if it contains no 1-sided curves (a 1-sided curve in a ...
3
votes
1answer
71 views

Topology of manifolds

Where can I find a stricter presentation of topology of manifolds, then in section 0.4 in Griffiths-Harris? For example, they define the map $H_k \times H_{n-k}$ by presenting a cycle by a submanifold ...
3
votes
0answers
78 views

Construct a space with free involution and homological restriction

I'm looking for a space $X$ which satisfies the following conditions: $X$ is a compact manifold. $H_\ast (X;\mathbb Z)$, the integral homology groups $X$, are torsion free. There is a free ...
1
vote
1answer
71 views

Submanifolds of a space of functions?

Let $f:\mathbb{R}\rightarrow(\mathbb{R}\rightarrow\mathbb{R})$ be a function mapping a real number uniquely into the set $\mathbb{F}$ of total functions from $\mathbb{R}$ to $\mathbb{R}$. $\mathbb{F}$ ...
0
votes
0answers
58 views

How to parametrize the torus with discs removed?

I am trying to glue a handle to a torus with two open discs removed. But I fail because I can't parametrize the torus. My try: Parametrize the cylinder (handle ) $C$ as $(h,\theta)$ and the torus ...
5
votes
0answers
79 views

Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
4
votes
1answer
113 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
2
votes
1answer
82 views

hairy ball thm. and projective space

Is it possible to find $n>1$ such that $\mathbb{R}P^{2n+1}$ doesn't have smooth non vanishing vector field? I know it is not true for $S^{2n+1}$ and $\mathbb{R}P^{2n+1}$ is a sphere modolu antipod ...
2
votes
0answers
50 views

To what extent is the global angular form well-defined?

I am reading Differential Forms in Algebraic Topology by Bott & Tu. The authors constructed the global angular form $\psi$ for an oriented $k$-sphere bundle $E$ over a smooth manifold $M$. It has ...
2
votes
1answer
139 views

Orientation of manifold in topological sense

What do we mean by orientability of a topological manifold? How do we orient two dimensional Euclidean space and why is Moebius band non-orientable? And it would be a great favour to me if you can ...
2
votes
1answer
64 views

Ramified coverings of a manifold

Maybe someone could help me with a bit of alebraic topology. Take $M$ a $n$-manifold with $n \geq 3$ , and $V$ a submanifold of codimension $2$ in $M$. Assume $H_{n-2}(V) = 0$. I've read that under ...
0
votes
1answer
60 views

Homology of a $3$-manifold obtained by rational surgery on an $m$-component link

I was trying to understand homology of a $3$-manifold $M$ obtained by rational surgery on an $m$-component oriented link $L$. I have a few questions regarding the following paragraph in the book ...
1
vote
1answer
173 views

Prove that $[0,\infty)$ is not a manifold.

Prove that $[0,\infty)$ is not a manifold. Using diffeomorphisms and the implicit function theorem perhaps.
3
votes
2answers
135 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
5
votes
2answers
138 views

Why this topological space is not a topological manifold?

I'm having troubles to prove that the following space is not a topological manifold: Let $r:S^1\to S^1$ be a rotation of $\frac{2\pi}{3}$, i. e., ...
1
vote
1answer
73 views

homotopy type of the manifold minus the boundary

Let $X$ be a topological manifold with boundary.What is the idea behind the fact that emoving the boundary doesn't change the homotopy type of the manifold; i.e.,that is the manifold $X$ has the same ...
2
votes
1answer
148 views

Existence of tubular neighborhood

Let $X$ be a topological space and $A$ a subset of $X$. My understanding of a tubular neighborhood $N$ of $A$ is that $N$ is an open set containing $A$ such that $\bar N$ is a manifold with boundary ...
0
votes
0answers
72 views

Deformation retract of a product with a CW structure.

Let $X = M \times [0,1]$, where $M$ is an $n$-manifold. Suppose $X$ is given some CW structure. I know that there is a deformation retract $f:X\times [0,1] \to X$ of $X$ to $M \times \{ 0 \}$. ...
4
votes
1answer
115 views

Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
2
votes
1answer
130 views

Trivialisation of the normal bundle of $S^1$

I'd like to show that the normal bundle of $S^1$ is trivial. That is, I want to find a homeomorphism $\varphi : NS^1 \to S^1 \times \mathbb R$ such that $\varphi \mid_{N_s S^1}$ is linear for every $s ...
1
vote
1answer
115 views

Framed manifolds and framed knots

I've been looking at intersection forms and the Arf invariant recently and I got a comment to one of my previous questions related to this. So I looked at framed manifolds. There seems to be quite a ...
7
votes
2answers
237 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
3
votes
1answer
180 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
1
vote
1answer
87 views

Why is the fundamental group of a prime, reducible 3-manifold $\mathbb{Z}$?

I've read in a paper that if $M$ is a prime, reducible $3$-manifold, then $\pi_{1}(M) \cong \mathbb{Z}$. Can anyone explain why this is true? Thanks in advance.
4
votes
1answer
145 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
2
votes
1answer
80 views

How to detect a twist or framing in a 3-manifold.

This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside. If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the ...
0
votes
3answers
329 views

Why isn't $\mathbb{RP}^2$ orientable?

How to prove that $\mathbb{RP}^2$ isn't orientable? My book (do Carmo "Riemannian Geometry") gives a hint: "Show that it has a open subset diffeomorphic to the mobius band", but I don't know even who ...
0
votes
0answers
48 views

Condition for a continuous Vector field not to be surjective

Let's say we have a continuous vector field $f: \mathbb R^2 \to \mathbb R^2$ such that $f(x) \neq 0,\forall x$. Then we have $f$ not surjective. Let us normalize $f$ so that $|f(x)|=1,\forall x$ and ...
0
votes
0answers
85 views

Given Poincare Polynomial find the manifold.

Suppose we have a polynomial, is it always the Poincare polynomial of some manifold? I guess the answer is no, but don't know any example. Even more, if we have a ring, is it the cohomology ring of ...
5
votes
3answers
99 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
2
votes
2answers
146 views

Heegaard splitting of a 3-manifold with boundary

A Heegaard splitting of a closed orientable 3-manifold $M$ is $M=H \cup H'$, where $H$ and $H'$ are handlebodies. Is there any similar concept for orientable 3-manifolds with boundaries?
4
votes
2answers
347 views

De Rham cohomology of $S^2\setminus \{k~\text{points}\}$

Am I right that de Rham cohomology $H^k(S^2\setminus \{k~\text{points}\})$ of $2-$dimensional sphere without $k$ points are $$H^0 = \mathbb{R}$$ $$H^2 = \mathbb{R}^{N}$$ $$H^1 = \mathbb{R}^{N+k-1}?$$ ...
6
votes
1answer
212 views

De Rham cohomology of $\mathbb{RP}^{n}$

Consider map from $S^{n}$ to $\mathbb{RP}^{n}$ $$\varphi:S^{n}\to\mathbb{RP}^{n}$$ which maps point $x\in S^{n}$ to corresponding direction in $\mathbb{R}^{n+1}$. This map induces map ...
2
votes
2answers
158 views

De Rham cohomology of $S^n$

Can you find mistake in my computation of $H^{k}(S^{n})$. Sphere is disjoint union of two spaces: $$S^{n} = \mathbb{R}^{n}\sqcup\mathbb{R^{0}},$$ so $$H^{k}(S^n) = H^{k}(\mathbb{R}^{n})\oplus ...
3
votes
1answer
346 views

Manifold embedded in euclidean space with nontrivial normal bundle

Let $X$ be a differentiable $n$-manifold embedded in some $\mathbb{R}^{n+1}$. I have two questions. I have read that if $X$ is compact and orientable, then the normal bundle of the embedding is ...
2
votes
0answers
112 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...

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