Compact connected orientable manifold

Definition

A compact connected orientable manifold is a manifold which is compact, connected and orientable. Note that compactness and connectedness are purely topological properties while orientability is a property that makes sense only in the context of a manifold.

The collection of all compact connected orientable manifolds upto homotopy is an important object of study, as are questions like: how many different compact connected orientable manifolds are there of a particular homotopy type? How many different possible differential structures are there on such manifolds?

Given a compact connected orientable manifold of dimension $n$, the $n^{th}$ homology is isomorphic to $\mathbb {Z}$, and choosing a generator is tantamount to choosing an orientation. A generator for this is termed a fundamental class for the manifold, and maps between compact connected orientable manifolds are often studied in terms of their degree, which is the integer $d$ such that the fundamental class of the manifold on the left goes to $d$ times the fundamental class on the right.