Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

30
votes
2answers
516 views

Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...
23
votes
3answers
749 views

Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
19
votes
1answer
490 views

Can the interior of a manifold be orientable but not its boundary?

Suppose $M^m$ is a manifold with boundary. If we are given an orientation for $M$, we can then derive an orientation for $\partial M$ by considering the orientation of $TM$ at $\partial M$ and then ...
9
votes
1answer
662 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold $...
9
votes
2answers
325 views

A topological group which is also a (not necessarily smooth) manifold is orientable

I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for smooth manifolds, in ...
9
votes
0answers
372 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ which ...
8
votes
1answer
90 views

Applied math: analyse cell orientations

I'm analyzing the orientation of cells and I stumbled over a peculiarity when I try to make a statement about the main direction of the cells and how many cells are oriented along this main direction. ...
7
votes
2answers
217 views

Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
7
votes
2answers
2k views

Meaning of the expression “orientation preserving” homeomorphism

The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a ...
7
votes
2answers
564 views

Partition of unity and volume form on a manifold

A smooth manifold $M$ is orientable iff there exists a nowhere-vanishing top form (i.e. volume form). In a coordinate chart $U\subset M$ we can find a volume form over $U$ that corresponds to the ...
7
votes
1answer
883 views

Proving that the quotient manifold is orientable if and only if the group action is orientation-preserving

I'm trying to solve the following exercise in Lee's book. Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is ...
7
votes
1answer
290 views

Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed ...
7
votes
1answer
297 views

A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts ...
7
votes
1answer
114 views

A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing ...
6
votes
2answers
670 views

Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
6
votes
1answer
106 views

Expressing invertible maps $\bigwedge^{d-1} V \to \bigwedge^{d-1} V$ as $\bigwedge^{d-1}A$ for some $A$

Let $V$ be a real $d$-dimensional vector space, let $\bigwedge^{d-1} V$ be its exterior power. Consider the following claim: Proposition: If $d$ is even, then every invertible linear map $\bigwedge^...
6
votes
1answer
821 views

$\mathbb{R}P^n$ is orientable iff $n$ is odd, without homology, without differential geometry

I am trying to prove that $\mathbb{R}P^n$ is orientable iff $n$ is odd. One way to do that is to calculate the homology of the space, and then use the (heavy?) theorem that states that a $n$-...
6
votes
1answer
89 views

Computing the “limit” of a SSeq with $E_{\infty}^{*,*}$ a free graded $\Gamma$-module.

I'm trying to prove the following proposition from Kochman's book. For completion I will write it here the relevant part: Let $E$ be an oriented spectrum with orientation class $x\in E^2(\mathbb{C}...
6
votes
1answer
224 views

Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis of ...
6
votes
1answer
362 views

Is there a orientable surface that is topologically isomorphic to a nonorientable one?

Is there a surface that is orientable which is topologically homeomorphic to a nonorientable one, or is orientability a topological invariant.
6
votes
1answer
8k views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
6
votes
0answers
81 views

What would be the concept of “Infinite Dimension Orientability”?

We have the following definition of orientability for finite dimensional vector spaces: Definition: Let $\Bbb{E}$ be a finite dimension vector space and $\mathscr{B}(\Bbb{E})$ be the set of all ...
5
votes
2answers
1k views

Understanding the orientable double cover

Definition: if $M$ is a smooth manifold, define the orientable double cover of $M$ by: $$\tilde{M}:=\{(p, o)\mid p\in M, o\in\{\text{orientations on }T_pM\}\}$$ together with the function $\...
5
votes
1answer
1k views

The Antipodal Map is Orientation Preserving iff $n$ is Odd

The following result is well-known. Theorem. The antipodal map on $S^n$ is orientation preserving if and only if $n$ is odd. Below I provide a proof in which there must be an error since I reach ...
5
votes
2answers
103 views

Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
5
votes
1answer
305 views

$M \times N$ orientable iff both $M$ and $N$ are orientable proof in terms of volume forms [duplicate]

I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable. I want to show this ...
5
votes
1answer
53 views

If $\omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$.

Let $M$ be a compact, oriented, smooth manifold of dimension $n$. I have to show that if $ \omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$. My first attempt was to use the ...
5
votes
0answers
136 views

A vector bundle which has an orientation-reversing isomorphism has a subbundle of rank $1$?

Let $E$ be a smooth real vector bundle of even rank, over a smooth manifold $M$. Suppose there exist an orientation-reversing vector bundle isomorphism $\Phi:E \to E$. Is it true that $E$ has a ...
5
votes
0answers
103 views

Why all the alternating signs?

For instance: $ |A \cup B \cup C|=(|A|+|B|+|C|)-(|A \cap B| + |A \cap C| + |B \cap C|)+|A \cap B \cap C|$ $\chi(X) = F- E + V = \sum_{i} (-1)^i \text{rank}(H_i(X)) =\sum_{i} (-1)^i \text{rank}(C_i(X))...
5
votes
0answers
64 views

Can the concept of orientability be applied to more general spaces?

Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in ...
5
votes
0answers
70 views

Topological invariants that detects change of orientation for an odd dimensional manifold.

Assume all manifolds involved to be closed and orientable. During my studies I learnt about the signature of a $4k$-manifold, and it turned out that (when it is non zero) it can be use to make a ...
5
votes
0answers
607 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let $\pi_M:M\...
4
votes
1answer
141 views

Every oriented vector bundle admits an orientation reversing isomorphism?

Let $E$ be a real oriented vector bundle over a smooth manifold $M$. Is there a vector bundle isomorphism $\Phi:E \to E$ which reverses the orientation? (I know there are oriented manifolds with no ...
4
votes
2answers
75 views

Immersion of non-orientable manifold in a small orientable one

I was trying to prove the following fact: given a non orientable manifold $M$ of dimension $m$, $M$ is always contained in an orientable manifold of dimension $m+1$. I have gotten nothing out of it, ...
4
votes
2answers
253 views

How does one orient a simplicial complex?

I have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...
4
votes
2answers
90 views

$\chi(X\times Y)=\chi(X)\cdot\chi(Y)$

For a compact oriented manifold $X$ define $\chi (X)=I(\Delta,\Delta)$ where $\Delta$ is the diagonal in $X\times X$ and $I$ is the intersection number. How do I show that $\chi(X\times Y)=\chi(X)\...
4
votes
1answer
649 views

Geometrical meaning of orientation on vector space [closed]

Can any one explain, the geometrical meaning of orientation in a vector space?
4
votes
2answers
162 views

Curl of a vector field.

Let S be a piecewise smooth oriented surface in $\mathbb{R}^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
4
votes
1answer
188 views

Outward-pointing vector field on Projective space

Lee Smooth Manifolds problem 8-4 says that for every manifold with boundary there exists a smooth vector field that is outward-pointing when restricted to the boundary. Now if our manifold is $M=\...
4
votes
1answer
60 views

Sign of integral of a 2 form

Given a 1-form $\omega$ define on $\mathbb{R}^2$ such that $$ d\omega = f \; dx\wedge dy $$ is a 2-form with $f>0$ and consider $D\subset \mathbb{R}^2$ How do I know the orientation of $ D$ (or $\...
4
votes
1answer
379 views

If $\phi: M_1 \to M_2$ a diffeomorphism between diff. manifolds, prove that if $M_2$ is oriented then so is $M_1$

Let $\phi: M_1 \to M_2$ a local diffeomorphism between two differentiable manifolds $M_1,M_2$. I want to prove that if $M_2$ is orientable so is $M_1$. Attempt: In order a manifold to be orientable ...
4
votes
1answer
353 views

How to explicitly perform the circle eversion in the $3$-dimensional space?

The following claim is a well-known consequence of the Whitney-Graustein theorem: Claim. It does not exist $H\colon\mathbb{S}^1\times[0,1]\overset{C^1}{\rightarrow}\mathbb{R}^2$ such that for all $...
4
votes
1answer
188 views

Direct sum of non-orientable bundles is orientable?

Let $M$ be a smooth manifold, and let $E_1,E_2$ be two non-orientable vector bundles over $M$. Is $E_1 \oplus E_2$ orientable? I am sure there is an easy answer, but somehow my search didn't result ...
4
votes
0answers
154 views

How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
4
votes
0answers
142 views

Orientability of variety

Consider the real projective variety on $\mathbb{R}P^n \times \mathbb{R}P^n$ defined by the bihomogeneous polynomial $x_0y_0+...+x_ny_n$, i.e. $V=\{([x_0:...:x_n],[y_0:...:y_n])|x_0y_0+...+x_ny_n=0\}$....
4
votes
0answers
133 views

$\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity

Let $M$ be a connected oriented smooth n-dimensional manifold-with-boundary, and let $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity. If $\omega$ is any smooth n-...
4
votes
0answers
263 views

Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
4
votes
0answers
361 views

Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
4
votes
1answer
39 views

What is the simplest way to extract a rough orientation statistics from images

Which is the fastest method to extract a rough orientation statistics from images. I think the most precise way is the scanning with local Gabor filters, but its very time consuming. Is it possible to ...
3
votes
2answers
123 views

Every smooth map $S^4 \to\mathbb{CP}^2$ has degree zero

I'm trying to show that every smooth map $f: S^4 \to \mathbb{CP}^2$ has degree zero. I'm not sure how to go about this, but I know that the oriented intersection number between any two closed 2-...