Fixed points

If you stretch a rubber band by moving one end to the right and the other to the left, then some point of the band will end up in its original position. It is called a fixed point. This fact can be proved by tools of Calculus I (how?). However a similar problem about stretching a disk or a ball requires more advanced techniques. This result is called the Brouwer fixed point theorem.

Algebraic topology deals with this and similar problems by assigning groups (and their homomorphisms) to manifolds (and their maps) to account for loops, holes, voids, and twists.

More:

• Banach fixed point theorem
• Borsuk-Ulam theorem
• Lefschetz fixed point theorem