Homology Theory - Applications

Applications

Notable theorems proved using homology include the following:

• The Brouwer fixed point theorem: If is any continuous map from the ball to itself, then there is a fixed point with .
• Invariance of domain: If U is an open subset of and is an injective continuous map, then is open and is a homeomorphism between and .
• The Hairy ball theorem: any vector field on the 2-sphere (or more generally, the -sphere for any ) vanishes at some point.
• The Borsuk–Ulam theorem: any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)