Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ N. Mramor Kosta 14 February 2006 2006 2 14 7 15 721-5017-Kosta-2005 10.wxyz/C2005V72P1p7 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5017-Kosta/ We consider the analogue of the fixed point theorem of A. Borel in the context of Tate cohomology. We show that for general compact Lie groups $G$ the Tate cohomology of a $G$-CW complex $X$ with coefficients in a field of characteristic $0$ is in general not isomorphic to the cohomology of the fixed point set, and thus the fixed point theorem does not apply. Instead, the following excision theorem is valid: if $X'$ is the subcomplex of all $G$-cells of orbit type $G/H$ where $\dim H > 0$, and $V$ is a ring such that for every finite isotropy group $H$ the order $|H|$ is invertible in $V$, then $\widehat {H}^*_G(X;V)\cong \widehat {H}^*_G(X';V)$. In the special cases $G=\TT $, the circle group, and $G=\UU $, the group of unit quaternions, a more elementary geometric proof, using a cellular model of $\widehat {H}^*_\UU $ is given. 57Q91, 57S99 MR2162289 02212181 Supported in part by the Ministry for Education, Science and Sport of the Republic of Slovenia Research Program No. 101-509. K.S. Brown Springer-Verlag MR672956 K.S. Brown; \textit{Cohomology $pf$ groups} (Springer-Verlag, New York, 1982). M. Cencelj Quart. J. Math. Oxford (2) MR1366613 M. Cencelj; Jones-Petrack cohomology, \textit{Quart. J. Math. Oxford (2)} \textbf{46} (1995), pp.~409--415. M. Cencelj and N. Mramor Kosta Bull. Austr. Math. Soc. MR1889377 M. Cencelj and N. Mramor Kosta; CW decompositions of equivariant complexes, \textit{Bull. Austr. Math. Soc.} \textbf{65} (2002), pp.~45--53. M. Cencelj, N. Mramor Kosta and A. Vavpetič J. Math. Kyoto Univ MR2028668 M. Cencelj, N. Mramor Kosta and A. Vavpeti\vc; $G$-complexes with a compatible CW structure, \textit{J. Math. Kyoto Univ} \textbf{43} (2003), pp.~585--597. T.G. Goodwillie Topology MR793184 T.G. Goodwillie; Cyclic homology, derivations, and the free loop space, \textit{Topology} \textbf{24} (1985), pp.~187--215. J.P.C. Greenlees and J.P. May Memoirs of the American Mathematical Society 113 American Mathematical Society MR1230773 J.P.C. Greenlees and J.P. May; \textit{Generalized Tate cohomology}, Memoirs of the American Mathematical Society 113 (American Mathematical Society, Providence, R.I., 1995). W.Y. Hsiang Springer-Verlag MR423384 W.Y. Hsiang; \textit{Cohomology theory of topological transformation groups} (Springer-Verlag, Berlin, Heidelberg, New York, 1975). J.D.S. Jones Invent. Math. MR870737 J.D.S. Jones; Cyclic homology and equivariant homology, \textit{Invent. Math.} \textbf{87} (1987), pp.~403--423. J.D.S. Jones and S.B. Petrack C.R. Acad. Sci. Paris Ser. I MR929113 J.D.S. Jones and S.B. Petrack; Le th\'eor\`eme des points fixes en cohomologie \'equivariante en dimension infinite, \textit{C.R. Acad. Sci. Paris Ser. I} \textbf{306} (1988), pp.~75--78. J.D.S. Jones and S.B. Petrack Trans. Amer. Math. Soc. MR1010411 J.D.S. Jones and S.B. Petrack; The fixed point theorem in equivariant cohomology, \textit{Trans. Amer. Math. Soc.} \textbf{322} (1990), pp.~35--50. I. Suzuki Springer-Verlag MR648772 I. Suzuki; \textit{Group theory I} (Springer-Verlag, Berlin, Heidelberg, New York, 1982). T. tom Dieck Lecture Notes in Mathematics 766 Springer-Verlag MR551743 T. tom Dieck; \textit{Transformation groups and representation theory}, Lecture Notes in Mathematics 766 (Springer-Verlag, Berlin, Heidelberg, New York, 1979).