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) th …

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 …

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 corr …

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 $I(\Delta, \mathrm{graph}(f))$ …

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 te …

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 ori …

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 …