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Let p be a prime number and X a simply connected Hausdorff space equipped with a free ${\mathbb{Z}}_p$-action generated by $f_p:X\to X$. Let $\alpha :S^{2n-1}\to S^{2n-1}$ be a homeomorphism generating a free ${\mathbb{Z}}_p$-action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map $F:(S^{2n-1},\alpha )\to (X,f_p)$. As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

Let N be a closed orientable n-manifold, n ≥ 3, and K a compact non-empty subset. We prove that the existence of a transversally orientable codimension one foliation on N∖K with leaves homeomorphic to ${ℝ}^{n-1}$, in the relative topology, implies that K must be connected. If in addition one imposes some restrictions on the homology of K, then N must be a homotopy sphere. Next we consider C² actions of a Lie group diffeomorphic to ${ℝ}^{n-1}$ on N and obtain our main result: if K, the set of singular points of the...