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Journal of the London Mathematical Society 2000 61(3):950-960; doi:10.1112/S0024610799008522
© 2000 by London Mathematical Society
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© The London Mathematical Society

Essential Equivariant Maps and Borsuk–Ulam Theorems

Mónica Clapp and Waclaw Marzantowicz

Instituto de Matemáticas, Universidad Nacional Autónoma de México 04510 México DF, Mexico, mclapp{at}math.unam.mx
Faculty of Mathematics and Computer Science, A Mickiewicz University Poznan, Poland, marzan{at}math.amu.edu.pl

Received 8 October 1998. Revision received 18 January 1999.

A full characterization is given of those compact Lie groups G with the property that every G-map X->X on a finite-dimensional G-complex X of finite orbit type, XG = Ø, is (non-equivariantly) essential. For arbitrary G, conditions are given on the G-space X which guarantee this property. Finally, conditions are given for the non-existence of a G-map X->Y inducing a homotopy equivalence XG~=YG on the fixed point sets. These results have applications to critical point theory of almost G-invariant functionals.


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