Resumen de Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem

Saugata Basu, Deepam Patel

• In this article, we use cohomological techniques to obtain an algebraic version of Toda’s theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. This result follows from a general ‘connectivity’ result in cohomology. More precisely, given a closed subvariety X⊂Pn over an algebraically closed field k, and denoting by J[p](X)=J(X,J(X,…,J(X,X)⋯) the p-fold iterated join of X with itself, we prove that the restriction homomorphism on (singular or ℓ-adic etale) cohomology Hi(PN)→Hi(J[p](X)), with N=(p+1)(n+1)−1, is an isomorphism for 0≤i