The tag has no usage guidance.

learn more… | top users | synonyms

0
votes
1answer
30 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
votes
0answers
147 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 ...
4
votes
1answer
73 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
votes
1answer
137 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
votes
0answers
36 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
votes
1answer
127 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
votes
1answer
385 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
votes
1answer
98 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
1answer
128 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 ...
13
votes
1answer
355 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 ...
2
votes
4answers
456 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
1answer
304 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
votes
0answers
216 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
votes
1answer
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
votes
3answers
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 ...