23 votes

Examples of interesting non orientable closed 3-manifolds

One gets non-orientable closed 3-manifolds by taking a non-orientable surface, and crossing with $S^1$, such as $P^2\times S^1$. In fact, the geometrization theorem hasn't been proven completely for ...
Ian Agol's user avatar
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17 votes
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

The answer is already given in the comments (by Ryan Budney and Mizar). But I think it makes sense to clear this confusing point. The classical Gauss-Bonnet formula is [e.g. https://en.wikipedia....
Alex Gavrilov's user avatar
15 votes
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Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?

No. (The main idea here is present in Dylan Wilson's comment.) Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $...
Tom Goodwillie's user avatar
11 votes
Accepted

Reference on complex cobordism

This is worked out in part 2 of Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ...
Denis Nardin's user avatar
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11 votes
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A topological group which is also a (not necessarily smooth) manifold is orientable

In fact beside your question there is a beautiful theorem in homotopy theory. This theorem due to T. Bauer, N. Kitchloo, D. Notbohm and E. K. Pedersen guarantees that any loop space $X=\Omega B$ where ...
David C's user avatar
  • 9,792
11 votes

Examples of interesting non orientable closed 3-manifolds

A curiosity is obtained as follows: take a solid cube $[-1,1]^3$ and identify one pair of two opposite faces by a symmetry with respect to a coordinate axis, while identifying the other two pairs of ...
Benoît Kloeckner's user avatar
8 votes
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Is there a well-known notion of orientability for finite geometries?

To do that you would need a notion of a non-orientable and an orientable linear transformation, i.e., essentially a notion of a "positive" and "negative" determinant, where "positive" determinants ...
Mikhail Katz's user avatar
  • 15.1k
4 votes

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

I don't think the mathematics needs to be more involved than to appreciate the difference between an axial vector (or pseudovector) and a polar vector. An axial vector does not change sign upon ...
Carlo Beenakker's user avatar
3 votes

Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime

Chirality is related to orientation via spin-momentum locking. The projector $P_\pm=\tfrac{1}{2}(1\pm\gamma_5)$ projects a spinor onto two subspaces that are decoupled in the evolution equation of a ...
Carlo Beenakker's user avatar
3 votes
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Orientation bundle and its flat connection

There is a different construction of orientation bundles. One considers the $\{\pm1\}$-principal bundle $o(TM)$ of fibrewise orientations of $TM$. The associated real line bundle $o(TM)\times_{\pm 1}\...
Sebastian Goette's user avatar
3 votes

An orientable surface that cannot be embedded into $\Bbb R^3$?

It seems to me that every orientable surface is indeed embeddable in $\mathbb{R}^3$. By Ian Richards' classification theorem (https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143186-0/...
Benoît Kloeckner's user avatar
3 votes
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Associativity of orientations of determinant bundles in Floer homology

The issue here are the isomorphisms in your suggested proof. There are choices and conventions involved. To give names to the isomorphisms clarifies what needs to be checked. $$ \iota_{K,L}:Det(K\...
Tom Mrowka's user avatar
  • 3,004
3 votes
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Orientations in connected bridgeless graphs

Yes. This is known as Robbins' theorem.
Peter Heinig's user avatar
  • 6,001
3 votes

Orientability of orbit type strata of Lie group actions

In your question, you may as well take $M=M_j=$ a ($G$-invariant) tubular neighborhood of a single orbit $G/S$. Near the point $S/S$, the space looks like $\mathfrak g/\mathfrak s \times N$, where $N$ ...
Allen Knutson's user avatar
2 votes
Accepted

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

Yes if $M$ is connected, no otherwise. If $M$ is connected, the covering gives a surjective homomorphism $\pi_1(M/\Sigma_k)\to \Sigma_k$. The vector bundle comes from a representation of $\pi_1$ ...
Will Sawin's user avatar
  • 135k
2 votes

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

If you look at the Ehresmann connection in T(TM) (or T(TM\0)) associated with a second order differential equation or a Finsler metric (which is not necessarily reversible), then this is orientation ...
alvarezpaiva's user avatar
  • 13.2k
2 votes

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

Take a unit tangent p-vector $a$ based at $y \in Y$ (you need the volume form on $Y$ to say what a unit p-vector is). At each point $x$ of the fiber $f^{-1}(y)$ consider the contraction of your volume ...
alvarezpaiva's user avatar
  • 13.2k
1 vote

Exponential trigonometric integral

Here are my comments as an answer: Based on my short study as well as also on the comment by @TheSimpliFire, I am quite sure that the best one can get is the integral representation below, or a ...
Fred Hucht's user avatar
  • 2,705
1 vote
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Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable

For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well. The idea is as you suggested. Maximal-rank timelike (or ...
Ryan Budney's user avatar
  • 42.8k
1 vote

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

If some interaction acts differently on the right-handed and left-handed components of a particle, the interaction is automatically parity violating. So you just need to look at the Lagrangian and ...
Daniel Castro's user avatar
1 vote

Orientability and higher dimensional Moebius strip

Yes. If all loops give rotations as holonomy, then the holonomy group lies in the rotation group, so the manifold is orientable.
Ben McKay's user avatar
  • 25.4k

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