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We present new upper bounds for the height of elements in the
cohomology of the unordered configuration space
\(H^*(\mathrm{Conf}_n(\mathbb{R}^d)/\mathfrak{S}_n;\mathbb{F}_p)\) with
coefficients in the field \(\mathbb{F}_p\).
In the special case when \(d\) is a power of \(2\) and \(p=2\) we settle the
original Vassiliev conjecture by proving that
\(\mathrm{height}(H^*(\mathrm{Conf}_n(\mathbb{R}^d)/\mathfrak{S}_n;\mathbb{F}_2))=d\).
As applications of these results we obtain new lower bounds for the
existence of complex \(k\)-regular maps as well as for complex
\(\ell\)-skew maps \(\mathbb{C}^d\rightarrow\mathbb{C}^N\).
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