# Questions tagged [orientation]

The orientation tag has no usage guidance.

**10**

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### 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.
...

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**2**answers

146 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 ...

**3**

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**0**answers

49 views

### Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...

**16**

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464 views

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

I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...

**0**

votes

**1**answer

49 views

### Orientations in connected bridgeless graphs

Let $n\geq 3$ be an integer, set $[n] = \{1,\ldots,n\}$ and let $G=([n],E)$ be an undirected connected bridgeless graph. Is there an orientation (explanation below) of $G$ such that for all $a\neq b\...

**8**

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**1**answer

277 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 ...

**14**

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**2**answers

1k 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 ...

**11**

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**1**answer

254 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 ...

**8**

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**1**answer

396 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 ...

**0**

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**1**answer

81 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 ...

**4**

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207 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 ...

**5**

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**1**answer

133 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 $\mu_g$ ...

**6**

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**1**answer

155 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_{\...

**2**

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**0**answers

100 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-...

**0**

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**1**answer

218 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 ...

**5**

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**1**answer

579 views

### Fundamental class in K-theory and orientability

In ordinary homology, the classical results give the following situation:
for a compact, connected, topological manifold $M$ of dimension $n$ we have, for each ring $R$, that $H_n(M,M \setminus \{x\};...

**0**

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**1**answer

125 views

### Orientability of Stiefel manifold V2(R4) [closed]

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups ...

**2**

votes

**1**answer

152 views

### A self-homeomorphism of $L_{p,q}$ is isotopic to one which preserves heegaard splitting

Consider the lens space $L_{p,q}$, which we can describe using its standard heegaard splitting, i.e. define $L_{p,q}$ as a quotient of two solid tori, identifying meridians on the boundary of one with ...

**14**

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**1**answer

411 views

### Does the signature admit a homotopy coherent refinement?

Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;
2) the $\widehat A$-genus ...

**4**

votes

**4**answers

699 views

### Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ...

**2**

votes

**1**answer

375 views

### Lefschetz Fixpoint theorem for non-orientable manifolds

The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows:
the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the ...

**0**

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**0**answers

385 views

### Orientation predicate CG

Shewchuk 97 gives me the orientation of 4 points, by finding the sign of a determinant, where the matrix is composed of the coordinates of the points. So, the signed volume of a tetrahedron, or which ...

**2**

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**1**answer

1k views

### Orientation Sheaf and Double Cover

The orientation sheaf of an $n$-manifold $M$ is $\mathcal{O}_n=Sheaf(U\mapsto H_n(M,M-U;\mathbb{Z}))$, with stalks given by $(\mathcal{O}_n)_x = lim H_n(M,M-U)=H_n(M,M-x)=\mathbb{Z}$ (the limit is ...

**12**

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**3**answers

1k views

### Orientation and generalized cohomology theories

Let h* be a multiplicative generalized cohomology theory and $E \rightarrow X$ a real vector bundle.
Is it true that, if $E$ is orientable with respect to h*, then it is also orientable with respect ...