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Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda's theorem

Let $X \subset \mathbb{P}^{n}$ be a non-empty closed subscheme over an algebraically closed field $k$, and $\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\cdots,\mathrm{J}(X,X)\cdots)$ denote the $p$-fold iterated join of $X$ with itself. In this article, we prove that the restriction homomorphism on cohomology $\mathrm{H}^{i}(\mathbb{P}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))$, with $N = (p+1)(n+1)-1$, is an isomorphism for $0 \leq i < p$, and injective for $i=p$, for any good cohomology theory. We also prove this result in the more general setting of relative joins for $X$ over a base scheme $S$, where $S$ is of finite type over $k$. We give several applications of these results including a cohomological version of classical quantifier elimination in the first order theory of algebraically closed fields of arbitrary characteristic, as well as an algebraic version of Toda's theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. We also apply our results to obtain effective bounds on the Betti numbers of image of projective varieties under projection map.