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pp. 7–15 : N. Mramor KostaA strong excision theorem for generalised Tatecohomology.

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@article {Kosta2005, author="N. Mramor Kosta", title="{A strong excision theorem for generalised Tate cohomology}", journal="Bull. Austral. Math. Soc.", fjournal={Bulletin of the Australian Mathematical Society}, volume="72", year="2005", number="1", pages="7--15", issn="0004-9727", coden="ALNBAB", language="English", date="13th January, 2005", classmath="57Q91, 57S99", publisher={AMPAI, Australian Mathematical Society}, MRID="MR2162289", ZBLID="02212181", url="http://www.austms.org.au/Publ/Bulletin/V72P1/721-5017-Kosta/index.shtml", acknowledgement={Supported in part by the Ministry for Education, Science and Sport of the Republic of Slovenia Research Program No. 101-509. }, abstract={ 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. }}

Bull. Austral. Math. Soc., Vol 72, No 1Metadata in BibTeX format

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/ A strong excision theorem for generalised Tate cohomology N. Mramor Kosta 7 June 2008 2008 6 7 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 Cohomology pf groups Springer-Verlag MR672956 K.S. Brown; \textit{Cohomology $pf$ groups} (Springer-Verlag, New York, 1982). M. Cencelj Jones-Petrack cohomology 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 CW decompositions of equivariant complexes 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 G -complexes with a compatible CW structure J. Math. Kyoto Univ MR2028668 M. Cencelj, N. Mramor Kosta and A. Vavpeti\v{c}; $G$-complexes with a compatible CW structure, \textit{J. Math. Kyoto Univ} \textbf{43} (2003), pp.~585--597. T.G. Goodwillie Cyclic homology, derivations, and the free loop space 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 Generalized Tate cohomology 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 Cohomology theory of topological transformation groups Springer-Verlag MR423384 W.Y. Hsiang; \textit{Cohomology theory of topological transformation groups} (Springer-Verlag, Berlin, Heidelberg, New York, 1975). J.D.S. Jones Cyclic homology and equivariant homology 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 Le théorème des points fixes en cohomologie équivariante en dimension infinite 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 The fixed point theorem in equivariant cohomology 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 Group theory I Springer-Verlag MR648772 I. Suzuki; \textit{Group theory I} (Springer-Verlag, Berlin, Heidelberg, New York, 1982). T. tom Dieck Transformation groups and representation theory 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).

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A strong excision theorem for generalised Tate cohomologyN. Mramor Kosta . . . . . . . . . . . . . . . . . . . . 7

Linear geometries on the Moebius strip: a theorem of Skornyakov typeRainer Löwen and Burkhard Polster . . . . . . . . . . . . . . 17

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Bull. Austral. Math. Soc. 57q91, 57s99

Vol. 72 (2005) [7–15]

A STRONG EXCISION THEOREM FOR

GENERALISED TATE COHOMOLOGY

N. Mramor Kosta

We consider the analogue of the fixed point theorem of A. Borel in the context of Tatecohomology. We show that for general compact Lie groups 𝐺 the Tate cohomologyof a 𝐺-CW complex 𝑋 with coefficients in a field of characteristic 0 is in generalnot isomorphic to the cohomology of the fixed point set, and thus the fixed pointtheorem does not apply. Instead, the following excision theorem is valid: if 𝑋 ′ isthe subcomplex of all 𝐺-cells of orbit type 𝐺/𝐻 where dim𝐻 > 0, and 𝑉 is a ringsuch that for every finite isotropy group 𝐻 the order |𝐻| is invertible in 𝑉 , then�̂�*𝐺(𝑋; 𝑉 ) ∼= �̂�*𝐺(𝑋 ′; 𝑉 ). In the special cases 𝐺 = 𝕋, the circle group, and 𝐺 = 𝕌,the group of unit quaternions, a more elementary geometric proof, using a cellularmodel of �̂�*𝕌 is given.

1. Introduction

The fixed point theorem of A. Borel states that for Abelian groups 𝐺 and for suit-

able coefficients 𝑉 the localised Borel equivariant cohomology 𝑆−1𝐻*𝐺(𝑋; 𝑉 ) of a finite𝐺-CW complex is isomorphic to the tensor product 𝐻*(𝐹 ; 𝑉 ) ⊗ 𝑆−1𝐻*(𝐵𝐺; 𝑉 ) of thecohomology of the fixed point set 𝐹 and the localised equivariant cohomology of a point

[7, Proposition 1]. Thus, in this case, the cohomology of the fixed point set 𝐻*(𝐹 ) iscompletely determined by the localised equivariant cohomology of 𝑋. For nonabelian

groups this is far from true since any finite complex 𝐾 can be the fixed point set of a

compact finite-dimensional 𝐺-space 𝑋 with trivial equivariant cohomology [7].

On infinite dimensional spaces it was shown by Goodwillie [5] that for 𝐺 = 𝕋,the circle group, the localised 𝕋-equivariant Borel cohomology does not satisfy the fixedpoint theorem. Jones-Petrack [9], [10] and Cencelj [2] constructed a completed localised

𝕋-equivariant cohomology which coincides with the classical Borel one on finite dimen-sional spaces, and for which the fixed point theorem holds also for infinite dimensional

spaces and suitable coefficient rings. This cohomology essentially coincides with the spe-

cial case of 𝐺 = 𝕋 of the generalised Tate cohomology �̂�*𝕋 of Greenlees and May [6] (with

Received 13th January, 2005Supported in part by the Ministry for Education, Science and Sport of the Republic of Slovenia ResearchProgram No. 101-509.

Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/05 $A2.00+0.00.

7

8 N. Mramor Kosta [2]

underlying Borel equivariant cohomology), which is defined generally, for any compact

Lie group 𝐺. In this note we study the fixed point theorem for this cohomology, which we

denote by �̂�*𝐺. We show that, as in the case of localised Borel equivariant cohomology,the fixed point theorem does not hold for infinite groups 𝐺, different from 𝕋. Instead,we prove the following excision theorem:

Theorem 1. Let 𝐺 be a compact Lie group, 𝑋 a 𝐺-CW complex, and 𝑋 ′ thesubcomplex containing all cells of type 𝐺/𝐻, where dim 𝐻 > 0. If 𝑉 is a ring such

that for every finite isotropy group 𝐻 the order |𝐻| is invertible in 𝑉 , then �̂�*𝐺(𝑋; 𝑉 )∼= �̂�*𝐺(𝑋 ′, 𝑉 ).The proof depends on the fact that the cohomology �̂�*𝐺(𝒪; 𝑉 ) of any orbit𝒪 = 𝐺/𝐻,

where 𝐻 is finite and |𝐻| is invertible in 𝑉 , is trivial. We first prove this for the circlegroup 𝐺 = 𝕋, and for the group 𝐺 = 𝕌 of unit quaternions (Proposition 1). These twocases have an additional geometric side since, for a smooth manifold 𝑋, the cohomology

�̂�*𝐺(𝑋; ) is expressible in terms of invariant differential forms on 𝑋. In addition, Tatecohomology �̂�*𝐺(𝑋) of a 𝐺-CW complex 𝑋 can be, in these two cases, computed froma nonequivariant cellular decomposition of 𝑋 by [6, 10.3, 14.9]. This follows from two

properties of these two groups. First, by [6, 14.1] and [3, Theorem 1], every 𝐺-CW

complex 𝑋 has a 𝐺-homotopy equivalent CW complex 𝑌 with an action of 𝐺 such that

the action map 𝜇 : 𝐺×𝑌 → 𝑌 is cellular, which implies that 𝐺𝑌 𝑛 ⊂ 𝑌 𝑛+𝑑 for all 𝑛, where𝑑 = 1 in the case of 𝕋 and 𝑑 = 3 in the case of 𝕌. It is not known which compact Liegroups have this property. Apart from finite groups and the groups 𝕋 and 𝕌 it has beenshown for all toral groups [4]. Second, in these two cases the classifying space 𝐸𝐺 has

a CW decomposition with cells only in dimensions apart by 𝑑, and therefore any 𝐺-CW

complex is calculable in the sense of [6, 10.1]. In Section 2 we use nonequivariant cellular

decompositions to compute the cohomology �̂�*𝐺(𝒪) of the orbits 𝒪 = 𝐺/𝐻, where 𝐺 iseither 𝕋 or 𝕌. In the case 𝐺 = 𝕋 this leads to a new proof of the fixed point theorem of[2], and in the case 𝐺 = 𝕌 it follows that the fixed point theorem is not valid, and theexcision theorem 1 follows.

In Section 3 we prove that for any compact Lie group 𝐺 and any finite subgroup

𝐻 such that the order |𝐻| is invertible in the coefficient ring 𝑉 the Tate cohomology�̂�*𝐺(𝐺/𝐻; 𝑉 ) = 0. The proof follows from general properties of Tate cohomology. InSection 4, the proof of the excision theorem is given.

The author wishes to thank Matija Cencelj for his help in writing this paper.

2. Tate cohomology of orbits of 𝕋 and 𝕌

Equivariant cohomology for the circle group 𝕋 and the group of unit quaternions𝕌 has several geometric properties which justify a special treatment. Throughout thissection, 𝐺 will denote one of the groups 𝕋 and 𝕌.

[3] Generalised Tate cohomology 9

It is known [6] that, up to minor differences, �̂�𝕋(𝑋) coincides with localised cyclic

cohomology of Jones [8], [9]. If 𝑋 is a smooth 𝕋-manifold, then this cohomology withcoefficients in a field of characteristic 0 (for example ) is expressible in terms of invari-ant differential forms on 𝑋. More precisely, �̂�*𝕋(𝑋; ) is the cohomology of the complexΩ*𝕋(𝑋)

[[𝑢, 𝑢−1]

]of invariant differential forms on 𝑋 and 𝑢 is of degree 2 with the differ-

ential 𝑑𝕋 = 𝑑 + 𝑢𝑗, where 𝑑 is the exterior derivative and 𝑗 : Ω𝑛𝕋 → Ω𝑛+1𝕋 is integration of

differential forms along orbits of the action [9]. It was shown in [2] that this cohomology

satisfies the fixed point theorem. Similarly, �̂�*𝕌(𝑋; ) where 𝑋 is a smooth 𝕌-manifoldcoincides with the cohomology of the complex Ω*𝕌(𝑋)

[[𝑢, 𝑢−1]

]of 𝕌-invariant forms on

𝑋, where 𝑢 is of degree 4, the differential is 𝑑𝕌 = 𝑑 + 𝑢𝑗, and 𝑗 : Ω𝑛𝕌 → Ω𝑛+3𝕌 is again

integration along orbits of the action.

In both cases, 𝐺 = 𝕋 and 𝐺 = 𝕌, Tate cohomology �̂�*𝐺(𝑋) of a 𝐺-CW complex 𝑋has a description in terms of a nonequivariant cellular decomposition of 𝑋. Let 𝐶(𝐺) be

the standard cellular chain complex of 𝐺 ∼= 𝑆𝑟 arising from the CW decomposition withone 0-cell and one 𝑟-cell, where 𝑟 = 1 in case 𝐺 = 𝕋 and 𝑟 = 3 in case 𝐺 = 𝕌, and let𝑧 ∈ 𝐶𝑟(𝐺) correspond to the 𝑟-cell. The product 𝜋 : 𝐺×𝐺 → 𝐺 is a cellular map which,on the chain level, maps 𝑧 ⊗ 𝑧 to 0.

For every 𝐺-CW complex 𝑋 there exists a 𝐺-homotopy equivalent CW complex 𝑌

such that the action 𝜇 : 𝑌 ×𝐺 → 𝑌 is a cellular map [6, Lemma 14.1], [3, Theorem 1]. Asin [6], let

(𝐶(𝑌 ), 𝑑

)be the cellular chain complex of 𝑌 and let 𝐽 : 𝐶𝑗(𝑌 ) → 𝐶𝑗+𝑟(𝑌 ) be

given by 𝐽(𝑐) = 𝜇*(𝑧⊗𝑐). The operator 𝐽 , which corresponds to the integration operator𝑗 in the case of differential forms on 𝑋, has the properties: 𝑑𝐽 = −𝐽𝑑 and 𝐽2 = 0 and sogives rise to the bicomplex [𝑢, 𝑢−1]⊗𝐶(𝑌 ), where the powers 𝑢−𝑛 represent generatorsof the standard cellular complex of 𝐵𝐺 (that is, 𝑢 is of degree −(𝑟 + 1)) with differential

𝑑(𝑤 ⊗ 𝑐) = 𝑢𝑤 ⊗ 𝐽(𝑐) + 𝑤 ⊗ 𝑑(𝑐).

By [6, Theorem14.9], Tate cohomology �̂�*𝐺(𝑋; 𝑉 ) coincides with the cohomology

𝐻*(Hom

([𝑢, 𝑢−1]⊗ 𝐶(𝑌 ); 𝑉 ))

for any coefficient ring 𝑉 .

Using this description we can compute the Tate cohomology of the orbits 𝒪 = 𝐺/𝐻,where 𝐻 < 𝐺 is a closed subgroup.

All closed subgroups of 𝕋, different from 𝕋, are finite cyclic groups. The isomorphismclasses of closed subgroups of 𝕌 consist of two 1-dimensional subgroups: the maximaltorus 𝕋 and its normaliser, and the following 0-dimensional subgroups: the cyclic groups/𝑛, the quaternionic groups ⟨𝑥, 𝑦 | 𝑥𝑛 = 𝑦2, 𝑦−1𝑥𝑦 = 𝑥−1⟩, the special linear group𝑆𝐿2(𝔽3) (a lift of the tetrahedral subgroup of 𝑆𝑂(3)), the special linear group 𝑆𝐿2(𝔽5)(a lift of the icosahedral subgroup of 𝑆𝑂(3)), and the lift of the octahedral subgroup

of 𝑆𝑂(3) (an extension of the symmetric group 𝑆4) [12, p. 155], [11, p. 404]. Every

10 N. Mramor Kosta [4]

isomorphism class contains precisely one conjugacy class (this is proved for example in

[3]).

Proposition 1. For every finite closed subgroup 𝐻 < 𝐺, where 𝐺 is either 𝕋or 𝕌, and for every coefficient ring 𝑉 such that 𝑚 = |𝐻| is invertible in 𝑉 ,

�̂�*𝐺(𝒪; 𝑉 ) = 0.

Proof: In order to compute �̂�*𝕌(𝒪; 𝑉 ), we compute the cohomology of the bicom-plex [𝑢, 𝑢−1]⊗ 𝐶(𝒪).

Let us first consider the more complicated case 𝐺 = 𝕌. If 𝐻 < 𝕌 is finite, theorbit 𝒪 = 𝕌/𝐻 has cells in dimension 0, 1, 2, and 3. A finite part of the bicomplex[𝑢, 𝑢−1] ⊗ 𝐶(𝒪) is on figure 1, where the vertical arrows are the differential 𝑑 of 𝐶(𝒪)and the horizontal arrows are the operator 𝐽 = 𝑢⊗ 𝐽( ).

0 1⊗ C3(O) u−1 ⊗ C0(O)

0 1⊗ C2(O) 0

0 1⊗ C1(O) 0

0 u⊗ C3(O) 1⊗ C0(O) 0

0 u⊗ C2(O) 0

Figure 1: A part of the bicomplex [𝑢, 𝑢−1] ⊗ 𝐶(𝕌/𝐻), 𝐻 finite. The vertical arrowsare 𝑑 and the horizontal arrows are 𝐽 = 𝑢⊗ 𝐽( ).

It follows that on cohomology

(1) �̂�0𝕌(𝒪; 𝑉 ) ∼= 𝐻0(𝒪; 𝑉 )/Im𝐽*, �̂�3𝕌(𝒪; 𝑉 ) ∼= ker 𝐽* < 𝐻3(𝒪; 𝑉 ).

Clearly 𝐽 = 0 on elements of dimension 1, 2, and 3. Since the projection 𝑝 : 𝕌 → 𝒪= 𝕌/𝐻 is a 𝑚-fold covering projection, where 𝑚 = |𝐻|, the operator 𝐽 maps an element

[5] Generalised Tate cohomology 11

𝑎 ∈ 𝐶0(𝒪) representing the generator of 𝐻0(𝒪) to

𝐽(𝑎) = 𝜇*(𝑧 ⊗ 𝑎) = 𝑝*(𝑧) = 𝑚𝑏,

where 𝑏 ∈ 𝐶3(𝒪) represents the fundamental class [𝒪] ∈ 𝐻3(𝒪). So 𝐽* : 𝐶3(𝒪) → 𝐶0(𝒪)induces multiplication by 𝑚 on cohomology, and is an isomorphism since 𝑚 is invertible

in 𝑉 . By (1),

�̂�0𝕌(𝒪; 𝑉 ) = �̂�3𝕌(𝒪; 𝑉 ) = 0.In dimensions 1 and 2 we have

�̂�1𝕌(𝒪; 𝑉 ) = 𝐻1(𝒪; 𝑉 ), �̂�2𝕌(𝒪; 𝑉 ) = 𝐻2(𝒪; 𝑉 ).

Since 𝕌 ∼= 𝑆3 is the universal cover of 𝒪, 𝐻1(𝒪; ) = 𝐴𝑏(𝐻) is a torsion group, andthe order, which divides 𝑚, is invertible in 𝑉 so, by the universal coefficients theorem

𝐻1(𝒪; 𝑉 ) = 0 and 𝐻1(𝒪; 𝑉 ) = 0. By Poincaré duality also 𝐻2(𝒪; 𝑉 ) = 0.If 𝐺 = 𝕋, then 𝒪 ∼= 𝑆1 has a cell in dimensions 0 and 1. The bicomplex [𝑢, 𝑢−1]⊗

𝐶(𝒪) is on figure 2. By the same arguments as above, 𝐽* : 𝐶1(𝒪) → 𝐶0(𝒪) is anisomorphism, and

�̂�0(𝒪; 𝑉 ) = �̂�1(𝒪; 𝑉 ) = 0.

0 1⊗ C1(O) u−1 ⊗ C0(O) 0

0 u⊗ C1(O) 1⊗ C0(O) 0

Figure 2: A part of the bicomplex [𝑢, 𝑢−1]⊗ 𝐶(𝕋/𝐻), 𝐻 < 𝕋.

The vertical arrows are 𝑑 and the horizontal arrows are 𝐽 = 𝑢⊗ 𝐽( ).Proposition 2. For every closed subgroup 𝐻 < 𝕌 where dim 𝐻 > 0 and any

coefficient ring 𝑉

�̂�*𝕌(𝒪; ) ̸= 0.Proof: Since closed subgroups of 𝕌 are of dimension either 0 or 1, 𝐻 is 1-

dimensional, and the orbit 𝒪 has cells in dimensions 0, 1, and 2. A finite part of thebicomplex [𝑢, 𝑢−1]⊗ 𝐶(𝒪) in on figure 3.

12 N. Mramor Kosta [6]

0 u−1 ⊗ C0(O)

0 1⊗ C2(O) 0

0 1⊗ C1(O) 0

0 1⊗ C0(O) 0

u⊗ C2(O) 0

Figure 3: A part of the bicomplex [𝑢, 𝑢−1]⊗𝐶(𝕌/𝐻), dim 𝐻 > 0. The vertical arrowsare 𝑑 and the horizontal arrows are 𝐽 = 𝑢⊗ 𝐽( ).

Clearly 𝐽 = 0, so �̂�*𝕌(𝒪; 𝑘) ∼= 𝐻*(𝑋; )[𝑢, 𝑢−1] is nonzero in dimensions 0 and 2.It follows from proposition 2 that the fixed point theorem for 𝐺 = 𝕌 is not valid. An

orbit 𝒪 = 𝕌/𝐻 where dim 𝐻 = 1 is a 𝕌-CW complex with nontrivial Tate cohomologyat least in dimension 0 for any coefficient ring. The fixed point set 𝐹 of the action is

empty, though, so clearly �̂�*𝕌(𝒪; 𝑉 ) ̸= �̂�*𝕌(𝐹 ; 𝑉 ).

3. Tate cohomology of orbits of general compact Lie groups

It remains to prove the theorem for general compact Lie groups. In order to do this,

we shall show that proposition 1 is in fact valid for any compact Lie group 𝐺, that is, an

orbit of type 𝐺/𝐻 where 𝐻 is finite has trivial Tate cohomology with coefficients in any

ring 𝑉 such that the order |𝐻| is invertible in 𝑉 .For a general compact Lie group 𝐺, the ordinary reduced 𝐺-equivariant Tate coho-

mology �̂�*𝐺(𝑋; 𝑉 ) of a 𝐺 space 𝑋 with coefficients in a 𝜋0(𝐺)-module 𝑉 is obtained bysplicing the homology and cohomology of the Borel construction 𝐸𝐺×𝐺𝑋. Formally it isdefined in [6] in the following way. Let 𝑀 be a Mackey functor, that is, an additive con-

travariant functor from the full subcategory of 𝐺-spectra containing as objects suspension

spectra of orbits 𝐺/𝐻+, such that 𝑀(𝐺/𝑒) = 𝑉 , and let 𝐻𝑀 be the Eilenberg-MacLane

[7] Generalised Tate cohomology 13

𝐺-spectrum of 𝑀 , that is, the 𝐺-spectrum, unique up to homotopy, such that for each

𝐻 < 𝐺,

𝜋𝑛(𝐻𝑀𝐻) = [𝑆𝑛, 𝐻𝑀 ]𝐻 = [𝐺/𝐻+ ∧ 𝑆𝑛, 𝐻𝑀 ]𝐺 = 0, 𝑛 > 0,

𝜋0(𝐻𝑀𝐻) = [𝑆0, 𝐻𝑀 ]𝐻 = [𝐺/𝐻+ ∧ 𝑆0, 𝐻𝑀 ]𝐺 = 𝑀(𝐺/𝐻).

Let 𝐸𝐺 be a universal space for 𝐺, that is, a contractible space with a free action of 𝐺,

and 𝐸𝐺 the unreduced suspension of 𝐸𝐺. Then

�̂�𝑛𝐺(𝑋; 𝑉 ) = 𝑡(𝐻𝑀)𝑛(𝑋) =

[𝑋 ∧ 𝑆−𝑛, 𝑡(𝐻𝑀)]

𝐺,

where, for a 𝐺-spectrum 𝑘𝐺,

𝑡(𝑘𝐺) = 𝐹 (𝐸𝐺+, 𝑘𝐺) ∧ 𝐸𝐺,and 𝐹 (𝑋, 𝑘𝐺) is the 𝐺-spectrum of pointed maps with the conjugate action of 𝐺.

Proposition 3. For any compact Lie group 𝐺 and any finite closed subgroup𝐻 < 𝐺, such that 𝑚 = |𝐻| is invertible in 𝑉 , the Tate cohomology �̂�*𝐺(𝐺/𝐻+; 𝑉 ) istrivial.

Proof: Since [𝐺/𝐻+, 𝑋]𝐺 = [𝑆0, 𝑋]𝐻 , it follows directly from the definition of

Tate cohomology that

(2) �̂�𝑛𝐺(𝐺/𝐻+; 𝑉 ) = �̂�𝑛𝐻(𝑆

0; 𝑉 ).

Since 𝐸𝐺 is a model for 𝐸𝐻, the 𝐺-spectrum 𝑘𝐺, viewed as an 𝐻-spectrum is equivalent

to 𝑘𝐻 [6, Proposition 3.7]. For every orbit 𝐺/𝐻, the Borel construction (𝐺/𝐻+ ×𝐺 𝐸𝐺)is, over 𝐻, equivalent to

(𝑆0 ×𝐻 𝐸𝐻) = (𝑆0 ×𝐵𝐻),so the reduced Tate cohomology is

(3) �̂�*𝐺(𝐺/𝐻+; 𝑉 ) = �̂�*𝐻(𝑆

0; 𝑉 ).

If 𝐻 is finite, this is equal to ordinary unreduced Tate cohomology �̂�*(𝐻,𝑉 ) of the finitegroup 𝐻. It is a classical result that this is 0 if the order of 𝐻 is invertible in 𝑉 . This can

be proved using transfer in the following way. Tate cohomology of finite groups is obtained

by splicing homology and cohomology ([1, p. 134]). If 𝑖 : {𝑒} → 𝐻 is inclusion of thetrivial subgroup and 𝑡𝑟 is the transfer, then the composition (𝑡𝑟∘𝑖) induces multiplicationby |𝐻| on homology and on cohomology. Since |𝐻| is invertible in 𝑉 , these are bothisomorphisms, which factorise through 𝐻*

({𝑒}, 𝑉 ) = 0 and 𝐻*({𝑒}, 𝑉 ) = 0 respectively.It follows that �̂�*(𝐻, 𝑉 ) = 0 for all 𝑛 ̸= −1, 0. In dimensions 𝑛 = −1, 0 it follows fromthe exact sequence

0 Ĥ−1(H, V ) H0(H, V ) H0(H, V ) Ĥ0(H, V ) 0N

14 N. Mramor Kosta [8]

that �̂�0(𝐻,𝑉 ) = �̂�−1(𝐻, 𝑉 ) = 0 since the norm map

𝑁 : 𝐻0(𝐻,𝑉 ) ∼= 𝑉𝐻 → 𝐻0(𝐻, 𝑉 ) ∼= 𝑉 𝐻

is clearly an isomorphism in this case [1].

If dim 𝐻 = 𝑑 > 0, then the Tate cohomology of an orbit 𝐺/𝐻 is expressed in terms

of ordinary homology and cohomology of the classifying space 𝐵𝐻 in the following way:

�̂�*𝐺(𝐺/𝐻+; 𝑉 ) ∼= �̂�*𝐻(𝑆0; 𝑉 ) ∼=

⎧⎪⎨⎪⎩𝐻𝑛(𝐵𝐻; 𝑉 ) if 0 ⩽ 𝑛

0 if − 𝑑 ⩽ 𝑛 < 0𝐻𝑛(𝐵𝐻; 𝑉 ) if 𝑛 ⩽ −𝑑− 1

,

which is nonzero for many choices of 𝑉 .

4. Proof of the excision theorem

The proof of the excision theorem now follows directly from the following lemma

which is proved by standard homological arguments.

Lemma 1. Let 𝐺 be a compact Lie group, let 𝑋 be a 𝐺-CW complex and 𝑋 ′ asubcomplex. Assume that 𝑋 is obtained from 𝑋 ′ by attaching 𝐺-cells such that for anysubgroup 𝐻 < 𝐺 appearing as isotropy type of an attached cell, �̂�*𝐺(𝐺/𝐻𝑖; 𝑉 ) = 0. Then�̂�*𝐺(𝑋; 𝑉 ) = �̂�

*𝐺(𝑋

′; 𝑉 ).

Proof: The proof goes by induction on the dimension of the 𝐺-cells. Assume

that 𝑋 ′′ < 𝑋 is obtained by attaching all cells 𝑤𝑞𝑖 of types 𝐺/𝐻 of 𝑋 and of dimension𝑞 ⩽ (𝑛− 1) to 𝑋 ′. Let ∐ 𝑤𝑛𝛼 be the disjoint union of all cells of dimension 𝑛 in 𝑋 whichare not in 𝑋 ′. In the exact sequence of the couple

(𝑋 ′′ ∪∐ 𝑤𝑛𝛼, 𝑋 ′′)

· · · → �̂�𝑞𝐺(𝑋 ′′ ∪

∐𝑤𝑛𝛼, 𝑋

′′)→ �̂�𝑞𝐺

(𝑋 ′′ ∪

∐𝑤𝑛𝛼

)→ �̂�𝑞𝐺(𝑋 ′′) → · · ·

the left term is

�̂�𝑞𝐺

(𝑋 ′′ ∪

∐𝑤𝑛𝛼, 𝑋

′′) ∼= ⊕𝛼�̂�𝑞𝐺(𝑤𝑛𝛼, 𝜕𝑤𝑛𝛼).

For each 𝛼,

�̂�𝑞𝐺(𝑤𝑛𝛼, 𝜕𝑤

𝑛𝛼)∼= (𝒪 ×𝐷𝑚,𝒪 × 𝑆𝑚−1),

where 𝒪 is the orbit 𝐺/𝐻. Since �̂�*𝐺(𝒪) = 0, it follows by induction on 𝑚 that also�̂�*𝐺(𝒪×𝐷𝑚,𝒪×𝑆𝑚−1) = 0, and therefore �̂�*𝐺

(𝑋 ′′ ∪∐ 𝑤𝑛𝛼, 𝑋 ′′) in the exact sequence is

trivial so �̂�*𝐺(𝑋′′ ∪ 𝑤𝑛) ∼= �̂�*𝐺(𝑋 ′′).

In the special case 𝐺 = 𝕋, all proper close subgroups are finite and the excisiontheorem for 𝐺 = 𝕋 is reduced to the following reformulation of the fixed point theoremof [2].

[9] Generalised Tate cohomology 15

Corollary 1. For any 𝕋-CW complex 𝑋 and for any coefficient ring 𝑉 suchthat the order of each isotropy group 𝐻 of 𝑋 is invertible in 𝑉

�̂�*𝕋(𝑋; 𝑉 ) ∼= �̂�*𝕋(𝐹 ; 𝑉 ) ∼= 𝐻*(𝐹 ; 𝑉 )[[𝑢, 𝑢−1]

],

where 𝐹 ⊂ 𝑋 is the set of fixed points of the action, and 𝑢 is of degree 2.

References

̂̂M [1] K.S. Brown, Cohomology 𝑝𝑓 groups (Springer-Verlag, New York, 1982).

̂̂̂̂M [2] M. Cencelj, ‘Jones-Petrack cohomology’, Quart. J. Math. Oxford (2) 46 (1995), 409–415.

̂̂̂M [3] M. Cencelj and N. Mramor Kosta, ‘CW decompositions of equivariant complexes’, Bull.

Austr. Math. Soc. 65 (2002), 45–53.

̂M [4] M. Cencelj, N. Mramor Kosta and A. Vavpetič, ‘𝐺-complexes with a compatible CW

structure’, J. Math. Kyoto Univ 43 (2003), 585–597.

̂M [5] T.G. Goodwillie, ‘Cyclic homology, derivations, and the free loop space’, Topology 24

(1985), 187–215.

̂̂̂̂̂̂̂̂̂̂M [6] J.P.C. Greenlees and J.P. May, Generalized Tate cohomology, Memoirs of the American

Mathematical Society 113 (American Mathematical Society, Providence, R.I., 1995).

̂̂M [7] W.Y. Hsiang, Cohomology theory of topological transformation groups (Springer-Verlag,

Berlin, Heidelberg, New York, 1975).

̂M [8] J.D.S. Jones, ‘Cyclic homology and equivariant homology’, Invent. Math. 87 (1987),

403–423.

̂̂̂M [9] J.D.S. Jones and S.B. Petrack, ‘Le théorème des points fixes en cohomologie équivariante

en dimension infinite’, C.R. Acad. Sci. Paris Ser. I 306 (1988), 75–78.

̂M [10] J.D.S. Jones and S.B. Petrack, ‘The fixed point theorem in equivariant cohomology’,

Trans. Amer. Math. Soc. 322 (1990), 35–50.

̂M [11] I. Suzuki, Group theory I (Springer-Verlag, Berlin, Heidelberg, New York, 1982).

̂M [12] T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math-

ematics 766 (Springer-Verlag, Berlin, Heidelberg, New York, 1979).

Institute for Mathematics, Physics, Mechanics,Faculty of Computer and Information ScienceUniversity of LjubljanaSloveniae-mail: neza.mramor-kosta@fri.uni-lj.si

http://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR672956&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR1366613&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR1889377&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR2028668&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR793184&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR1230773&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR423384&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR870737&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR929113&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR1010411&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR648772&fmt=hl&l=1&r=1&dr=allhttp://www.ams.org/msnmain?fn=130&pg3=MR&s3=MR551743&fmt=hl&l=1&r=1&dr=allmailto:neza.mramor-kosta@fri.uni-lj.si

Cover PagesAustralian Mathematical SocietyThe Bulletin: EditorsCopyright StatementInformation for AuthorsEditorial Policy

Index of Volume 72 No. 1A strong excision theorem for generalised Tate cohomology1. IntroductionTheorem 1.

2. Tate cohomology of orbits of T and UProposition 1.Figure 1: A part of the bicomplex ℤ[u,u⁻1] ⊗ C(U/H), H finite.Equation (1)Figure 2: A part of the bicomplex ℤ[u,u⁻1] ⊗ C(T/H), H0.

3. Tate cohomology of orbits of general compact Lie groupsProposition 3.

4. Proof of the excision theoremLemma 1.Corollary 1.

References