# Tag Info

Accepted

• 7,450
Accepted

### How does one orient a simplicial complex?

As Eric Wofsey says, there are two notions here: An orientation of an $n$-dimensional simplicial complex is a choice of orientation for each $n$-dimensional simplex. The easiest way to find one is to ...
• 61.5k
Accepted

### Is orientability needed to define volumes on riemannian manifolds?

You don't need orientability to define the volume of a parametrized region. Your value $vol(R)$ is completely well-defined in terms of a parametrization, and doesn't even depend on the parametrization ...
• 34.1k
Accepted

• 99.1k
Accepted

### Every oriented vector bundle admits an orientation reversing isomorphism?

In general no, since such an isomorphism would reverse the sign of the Euler class. Consider $TS^2$ for a simple example where this yields two different isomorphism classes of vector bundles.
• 15.9k

### How does one orient a simplicial complex?

Let $S$ be the set of all the vertices of your simplicial complex. If you choose a total order $<$ on the set $S$, then $<$ is also a total order on any subset of $S$. In particular, the set ...
• 329k

• 54.1k

### A topological group which is also a (not necessarily smooth) manifold is orientable

The solution by JHF seems correct to me; here is an alternative using the same core idea of paths in the space $M$ yielding homotopies between multiplication maps. The "top then right" leg of the ...
• 592
### Orientation twisted on the $2$-sphere
Actually, it's not that simple. Consider a similar example: Let $M$ be the $xy$-plane in $\mathbb R^3$ -- that is, $M$ is the set of all points of the form $(x,y,0)$ for $(x,y)\in \mathbb R^2$. Even ...