Compact connected orientable manifold: Difference between revisions
No edit summary |
No edit summary |
||
| Line 2: | Line 2: | ||
A '''compact connected orientable manifold''' is a [[manifold]] which is [[compact space|compact]], [[connected space|connected]] and [[orientable manifold|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. | A '''compact connected orientable manifold''' is a [[manifold]] which is [[compact space|compact]], [[connected space|connected]] and [[orientable manifold|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 <math>n</math>, the <math>n^{th}</math> homology is isomorphic to <math>\mathbb{Z}</math>, 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 of a map|degree]], which is the integer <math>d</math> such that the fundamental class of the manifold on the left goes to <math>d</math> times the fundamental class on the right. | |||
Revision as of 21:04, 2 December 2007
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 , the homology is isomorphic to , 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 such that the fundamental class of the manifold on the left goes to times the fundamental class on the right.