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$Z_2$-bordism and the Borsuk-Ulam Theorem

The purpose of this work is to classify, for given integers $m,\, n\geq 1$, the bordism class of a closed smooth $m$-manifold $X$ with a free smooth involution $τ$ with respect to the validity of the {\it Borsuk-Ulam property} that for every continuous map $ϕ: X \to R^n$ there exists a point $x\in X$ such that $ϕ(x)=ϕ(τ(x))$. We will classify a given free $Z_2$-bordism class $α$ according to the three possible cases that (a) all representatives $(X , τ)$ of $α$ satisfy the Borsuk-Ulam property; \ (b) there are representatives $(X_ 1, τ_1)$ and $(X_2, τ_2)$ of $α$ such that $(X_1, τ_1)$ satisfies the Borsuk-Ulam property but $(X_2, τ_2)$ does not; \ (c) no representative $(X , τ)$ of $α$ satisfies the Borsuk-Ulam property.