On the existence of Borsuk-Ulam theorems

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A Borsuk-Ulam theorem for the finite group G consists of finding a function b:N → N with b(n)→ ∞ as n→ ∞ and such that the existence of a G-map SV→SW between representation spheres without fixed points implies dim W≥b(dim V). We show that such a function b exists iff G is a p-group. We also prove that a G-map SV→SW as above with W{subset not double equals}V exists. iff G is not a p-group. Similar results hold for compact Lie groups. As a corollary we obtain an algebraic characterization of p-groups. If there exists an element a ∈ A (G) of the Burnside ring of G with |aG|=1 and a representation V of G with |aH| ·dim VH=0 for all subgroups H of G then G is not a p-group and vice versa. © 1992.




Bartsch, T. (1992). On the existence of Borsuk-Ulam theorems. Topology, 31(3), 533–543. https://doi.org/10.1016/0040-9383(92)90048-M

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