  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.
