# Questions tagged [orientation]

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### Finding a volume form on a fibre of a submersion between oriented manifolds

Let $f:X\to Y$ be a submersion between orientable smooth manifolds of respective dimensions $n,p$ and let $j:M=f^{-1}(y)\hookrightarrow X$ denote the inclusion of some fibre of $f$. My naïve (I am ...
2answers
117 views

### On the “Non-conservation of parity in weak interactions”

Kostrikin and Manin, in their Linear Algebra and Geometry, state that: (The excerpt is on pp. 42-43.) The statement comes after a proof of general linear group over reals having two connected ...
0answers
90 views

### Invariance of $\mathrm{SO}(3)$ dynamics when expressed via Euler angles

$\DeclareMathOperator{\SO}{\operatorname{SO}}$I am trying to understand the properties of the $\SO(3)$ Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am ...
0answers
94 views

### A query on Galvin's theorem for bipartite graphs

The Galvin's theorem is the generalized version of Dinitz conjecture that states that if the maximum degree of any bipartite graph is $\Delta$, then its edges are colorable properly if each of its ...
0answers
109 views

### Nontrivial integer homology class implies orientability

I posted this question on MSE and I would like to see if my reasoning is correct. Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and ...
1answer
352 views

### An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success. By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, Wikipedia states in this ...
0answers
403 views

### When does a graph have a minimally strong orientation?

Given an asymmetric relation $A\subseteq V^2$ a digraph $D=(V,A)$ is minimally strong iff $D$ is strongly connected and for all arcs $\alpha\in A$ the digraph $D−\alpha=(V,A\setminus\{\alpha\})$ is ...
1answer
382 views

### Reference on complex cobordism

I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism. ...
2answers
275 views

### Is there a notion of a connection for which the horizontal lift of a curve depends on its orientation?

Given a fiber bundle $\pi:E\to M$, a curve $\gamma:[0,1]\to M$, and a point $p \in \pi^{-1}(\gamma(0))$, a connection on the bundle allows us to uniquely lift $\gamma$ to a horizontal curve in E ...
0answers
57 views

1answer
460 views

### Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?

Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle? There is a similar theorem ...
2answers
2k views

### Examples of interesting non orientable closed 3-manifolds

In dimension 2, there are two remarkable non-orientable closed manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and ...
1answer
270 views

### Is there a well-known notion of orientability for finite geometries?

I'm wondering if the notion of an orientable/non-orientable manifold has any reasonable extension that allows for a similar classification of finite geometries. For example, the real projective plane ...
1answer
494 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 ...
1answer
121 views

### Orientability and higher dimensional Moebius strip

A compact surface is non-orientable if and only if there is a Moebius strip on it. Is there a similar result in higher dimension? More specifically, at least in the smooth setting, we can define a ...
0answers
261 views

### Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
1answer
199 views

### Orientability of orbit type strata of Lie group actions

Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...
1answer
160 views

### non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle  \xi:\mathbb{R}^k\longrightarrow M\times_{\...
0answers
163 views

### Constrained absolute orientation of 3D point sets

Let us assume we have two 3D point sets, $P=\{p_i\}$ and $Q=\{q_i\}$, and that we need to recover the transformation that takes $P$ as close to $Q$ as possible. In particular, I am interested in roto-...
1answer
253 views

### Petersen 2-factor decomposition theorem for directed graphs

Petersen proved that every 2k-regular graph can be decomposed into k disjoint 2-factors. I would like to know that is it true that if G is a directed regular graph (d_out(v)=d_in(v)=k), then can G be ...
1answer
764 views