# Tagged Questions

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 ...
1answer
35 views

### Embedded 2-Submanifold

Can you help for solving this problem ı am triying to understand embedded 2 submanifold please help me.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\mathbb{R}$. $\mathbb{F}$ ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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., ...
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 ...
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 ...
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 $M \times \{ 0 \}$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 $$H^0 = \mathbb{R}$$ $$H^2 = \mathbb{R}^{N}$$ $$H^1 = \mathbb{R}^{N+k-1}?$$ ...
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 ...
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 ...
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 ...
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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