On the Topology of the Set of Nonnegative Forms That Are Not Sums of Squares

Abstract

We study the topology of the normalized space of real nonnegative forms of degree $2d$ in $n$ variables that are not sums of squares. Whenever it is nonempty, we prove that it is path-connected but not acyclic, and in particular not contractible. We give explicit bounds on the degrees in which its reduced homology can be nonzero. In the two minimal nonempty cases, we show an embedding of an exterior algebra into the cohomology ring, and prove containment of the binary octahedral and binary icosahedral groups, respectively, in its fundamental group.