Theorems with balls

Giovanni Viglietta

Brouwer's fixed point theorem, the hairy ball theorem, the Borsuk-Ulam theorem and the Lusternik-Schnirelmann theorem are among the cornerstones of algebraic topology. Standard proofs are non-constructive applications of homology theory.

We present self-contained combinatorial proofs of the above theorems, based on Sperner's lemma and Tucker's lemma, which are constructive. This approach naturally leads to algorithms to compute approximations of fixed points or zeros of certain functions, or antipodal points on spheres with certain properties, whose existence is asserted by the theorems.