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A note on the theorems of Lusternik-Schnirelmann and Borsuk-Ulam

100%

Barros T. , Biasi C.

Colloquium Mathematicum

2008

tom 111

nr 1

35-42

Let p be a prime number and X a simply connected Hausdorff space equipped with a free $ℤ_{p}$-action generated by $f_{p}:X → X$. Let $α:S^{2n-1} → S^{2n-1}$ be a homeomorphism generating a free $ℤ_{p}$-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 $F:(S^{2n-1},α) → (X,f_{p})$. As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

Foliations by planes and Lie group actions

80%

López J. , Arraut J. , Biasi C.

2003

tom 82

nr 1

61-69

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 action, is a finite non-empty subset, then K contains only one point and N is homeomorphic to Sⁿ.