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4
votes
1answer
64 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
134 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 ...
2
votes
0answers
34 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 ...
0
votes
1answer
117 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
374 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 ...
0
votes
1answer
96 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
126 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
349 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
455 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 ...
-1
votes
1answer
166 views

How to define an “anisotropic vector” for a given object? [closed]

Dear experts, I am looking for a way to define an "alignment vector" (or anisotropy or orientation vector?) for a given geometrical object. I am not sure how to put this into correct technical terms, ...
2
votes
1answer
295 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
210 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 ...
11
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 ...