# Orientable manifold

*This article defines a property of manifolds and hence also of topological spaces*

## Contents

## Definition

A manifold is said to be **orientable** if it possesses an orientation, viz there exists a global section of the orientation-generator sheaf (the subsheaf of the orientation sheaf whose fibre at every point is the set of generators of the stalk at that point).

By default *orientable* means orientable with integer coefficients. Orientability over any ring is equivalent to orientability with integer coefficients if in the ring; if then any manifold is orientable with respect to that ring.

However, the number of possible orientations depends on the choice of ring; for a connected orientable manifold, the number of possible orientations equals the number of invertible element in the ring of coefficients.

## Relation with other properties

### Stronger properties

## Facts

- The map from the orientation-generator sheaf to the manifold is a double cover; hence if the fundamental group of the manifold does not possess a normal subgroup of index two, the manifold must be orientable. In particular, any simply connected manifold is orientable. However, the converse is not true: real projective space in odd dimensions has fundamental group of order two, but is orientable.
- Every manifold is either orientable or has an orientable double cover: this double cover is the orientation-generator sheaf itself

## Metaproperties

### Products

*This property of topological spaces is closed under taking finite products*

A direct product of two orientable manifolds is again orientable. In fact, a converse statement is true: a direct product of manifolds is orientable iff each is manifold.

### Covering spaces

*This property of topological spaces is closed under passing to covering spaces; viz if a topological space has this property, so does any covering space of it*

Any covering space of an orientable manifold is again an orientable manifold (the manifold structure is given in the usual way.

### Fiber bundles

This property of topological spaces is a fiber bundle-closed property of topological spaces: it is closed under taking fiber bundles, viz if the base space and fiber both satisfy the given property, so does the total space.

Manifold, Orientable manifold

If is a fiber bundle with base space and fiber space , and if and are both orientable manifolds, then so is . Note that this covers the case of direct products and covering space.

The converse is not necessarily true, but some weaker variants thereof are. *Fill this in later*