Antipodal coincidence for maps of spheres into complexes

Abstract:

This paper gives a partial answer to the question of whether there exists a Borsuk-Ulam type theorem for maps of ${S^n}$ into lower-dimensional spaces, which are not necessarily manifolds. It is shown that for each k and $n \leq 2k - 1$, there exists a map f of ${S^n}$ into a contractible k-dimensional complex Y such that $fx \ne f( - x)$, for all $x \in {S^n}$. In particular, there exists a map of ${S^3}$ into a 2-dimensional complex Y without an antipodal coincidence. This answers a question raised by Conner and Floyd in 1964. The complex Y provides also an example of a countractible k-dimensional complex whose deleted product has the Yang-index equal to $2k - 1$.