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

## TL;DR

This paper establishes cohomological properties of iterated joins of algebraic varieties, leading to applications in quantifier elimination, complexity theory, and bounds on Betti numbers over algebraically closed fields.

## Contribution

It proves isomorphism and injectivity of cohomology restriction maps for iterated joins, extending classical results to a broader algebraic and cohomological context.

## Key findings

- Cohomology restriction maps are isomorphisms for degrees less than p
- Injectivity of cohomology maps at degree p for iterated joins
- Derived applications in quantifier elimination and complexity bounds

## Abstract

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.