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Advances
in
Mathematics
www.elsevier.com/locate/aim
Equivariant
Moore
spaces
and
the
Dade
group
Ergün Yalçın
DepartmentofMathematics,BilkentUniversity,Ankara,06800,Turkey
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received29June2016 Accepted17January2017 CommunicatedbyHenningKrause
MSC:
primary57S17 secondary20C20
Keywords:
EquivariantMoorespaces Dadegroup
Borel–Smithfunctions Orbitcategory Bisetfunctors
Let G be afinite p-group and k be afieldofcharacteristic p.
A topologicalspace X is calledan n-Moore spaceifitsreduced homologyisnonzeroonlyindimension n. Wecalla G-CW-complex X an n-Moore G-space over k if foreverysubgroup
H of G, thefixedpointset XH isan n(H)-Moore spacewith
coefficientsin k, where n(H) is afunctionof H. Weshowthat if X is afinite n-Moore G-space, thenthereducedhomology moduleof X is an endo-permutation kG-module generated byrelativesyzygies.A kG-module M is anendo-permutation moduleifEndk(M )= M⊗kM∗isapermutation kG-module.
WeconsidertheGrothendieckgroupoffiniteMoore G-spaces M(G), withadditiongivenbythejoinoperation,andrelate thisgrouptotheDadegroupgeneratedbyrelativesyzygies. ©2017ElsevierInc.Allrightsreserved.
1. Introductionandstatementof results
Let G be afinitegroupand M be aZG-module.A G-CW-complex X is calleda Moore G-space of type(M,n) if thereducedhomologygroupHi(X;Z) iszerowhenever i = n
andHn(X;Z)∼= M asZG-modules.Oneoftheclassicalproblemsinalgebraictopology,
duetoSteenrod,askswhethereveryZG-moduleisrealizableasthehomologymoduleofa Moore G-space. G. Carlsson[10]constructedcounterexamplesofnon-realizablemodules
E-mailaddress:yalcine@fen.bilkent.edu.tr. http://dx.doi.org/10.1016/j.aim.2017.01.017 0001-8708/©2017ElsevierInc.Allrightsreserved.
for finite groupsthatincludeZ/p× Z/p asasubgroupfor someprimep. Thequestion of finding a good algebraic characterization of realizable ZG-modules is still an open problem (see[21]and[4]).
InthispaperweconsiderMoore G-spaces whosefixedpointsubspacesarealsoMoore spaces. LetR be a commutativering of coefficients and let n : Sub(G) → Z denote a functionfrom subgroupsof G to integers,whichisconstantontheconjugacy classesof subgroups. Suchfunctionsareoftencalledsuperclassfunctions.
Definition 1.1. A G-CW-complex X is called an n-Moore G-space over R if for every
H ≤ G,thereducedhomologygroupHi(XH;R) is zeroforall i = n(H).
When n is theconstantfunctionwithvalue n for all H ≤ G,thehomologyat dimen-sion n can beconsideredas amoduleovertheorbitcategory Or G.If Hn(X?; R) ∼= M
as amodule overthe orbitcategory, then X is called aMooreG-space of type(M ,n).
When R =Q and XH is simply-connected forall H ≤ G, the space X is called a
ra-tional Moore G-space. Rational Moore G-spaces are studied extensivelyin equivariant homotopytheoryandmanyinterestingresultsareobtainedonhomotopytypesofthese spaces(see [16]and[11]).
Inthispaper,weallow n to beanarbitrarysuperclassfunctionandtakethecoefficient ring R as afield k of characteristic p.WedefinethegroupoffiniteMoore G-spaces over
k and relate this group to the Dade group, the group of endo-permutation modules. Since the appropriatedefinition ofaDade groupfor afinite groupis notclear yet,we restrictourselvestothesituationwhere G is a p-group, althoughtheresultshaveobvious consequencesforfinitegroupsviarestrictionto aSylow p-subgroup.
Let G be afinite group and k be afieldof characteristic p. Throughout we assume all kG-modules are finitely generated. A kG-module M is called an endo-permutation kG-module if Endk(M )= M⊗kM∗isisomorphictoapermutation kG-module when
re-gardedasa kG-module withdiagonal G-action (gf )(m)= gf (g−1m). A G-CW-complex iscalled finite if ithasfinitelymanycells.Themainresultofthepaperisthefollowing: Theorem 1.2. Let G be a finite p-group, and k be a field of characteristic p. If X is a finite n-Moore G-space over k, then the reduced homology module Hn(X, k) in dimension n = n(1) is an endo-permutation kG-module generated by relative syzygies.
ThistheoremisprovedinSections3and4.WefirstproveitfortightMoore G-spaces
(Proposition 3.8) and then extend it to thegeneral case. An n-Moore space X is said to be tight if the topological dimension of XH is equal to n(H) for every H ≤ G.
For tight Moore G-spaces, we give an explicit formula that expresses the equivalence classofthehomologygroupHn(X,k) in termsofrelativesyzygies(seeProposition 3.8).
This formula playsakey roleforrelating thegroupofMoore G-spaces tothe groupof Borel–Smithfunctionsand totheDadegroup.We nowintroducethese groupsand the homomorphismsbetweenthem.
An endo-permutation G-module is called capped if it has an indecomposable sum-mand with vertex G, or equivalently, if Endk(M ) has the trivial module k as a
summand. There is a suitable equivalence relation of endo-permutation modules, and the equivalence classes of capped endo-permutation modules form a group under the tensor product operation (see Section 2). This group is called the Dade group of the group G, denoted by Dk(G), or simply by D(G) when k is clear from the
con-text.
Fora non-empty finite G-set X, the kernel of theaugmentation map ε : kX → k is
calleda relative syzygy and denotedbyΔ(X).ItisshownbyAlperin[1]thatΔ(X) isan endo-permutation kG-module andit iscappedwhen |XG|= 1.WedefineΩ
X ∈ Dk(G) astheelement ΩX= [Δ(X)] if X = ∅ and |XG| = 1; 0 otherwise.
Thesubgroupof D(G) generated byrelativesyzygiesisdenotedby DΩ(G) anditplays
animportantroleforunderstandingtheDadegroup.
Definition1.3.WesayaMoore G-space is capped if XG hasnonzeroreducedhomology.
Theset of G-homotopy classesof capped MooreG-spaces form acommutativemonoid with addition given by [X]+ [Y ] = [X ∗ Y ], where X ∗ Y denotes the join of two
G-CW-complexes. We define thegroup ofMoore G-spaces M(G) as theGrothendieck groupof thismonoid.
The dimension function of an n-Moore G-space isdefinedasthesuperclassfunction Dim X withvalues
(Dim X)(H) = n(H) + 1
for allH ≤ G. LetC(G) denote thegroup of allsuper class functionsof G. The map Dim :M(G)→ C(G) whichtakes[X]−[Y ] toDim X−Dim Y isagrouphomomorphism sinceDim(X∗Y )= Dim X+Dim Y .InProposition 5.6,weprovethatthehomomorphism Dim is surjective. This follows from the fact that C(G) is generated by super class functionsoftheform ωX,where X denotes afinite G-set and ωX isthefunctiondefined
by
ωX(K) =
1 if XK = ∅ 0 otherwise
forall K ≤ G.Note thatifwe consider afinite G-set X as adiscrete G-CW-complex,
then X is afinite Moore G-space withdimensionfunctionDim X = ωX.
Wealsodefine agrouphomomorphism Hom :M(G) → DΩ(G) as alinearextension oftheassignment thattakestheequivalence class[X] of acapped n-Moore G-space to
the equivalence class of its reduced homology [ Hn(X; k)] in DΩ(G), where n = n(1).
ThereisalsoagrouphomomorphismΨ: C(G)→ DΩ(G) thattakes ωXtoΩXforevery G-set X (see [6,Theorem 1.7]).InProposition 5.7,weshowthat
Hom = Ψ◦ Dim .
This givesinparticularthatforan n-Moore G-space X, the equivalenceclassof its re-ducedhomology[ Hn(X; k)] in DΩ(G) isuniquelydeterminedbythefunction n. Moreover
we provethefollowingtheorem.
Theorem 1.4. Let G be a finite p-group and k a field of characteristic p. Let M0(G)
denote the kernel of the homomorphism Hom, and Cb(G) denote the group of Borel–Smith functions (see [8, Definition 3.1]). Then, there is a commuting diagram
0 M0(G) Dim0 M(G) Dim Hom DΩ(G) = 0 0 Cb(G) C(G) Ψ DΩ(G) 0
where the maps Dim and Dim0 are surjective and the horizontal sequences are exact.
In the proof of the above theorem we do not assume the exactness of the bottom sequence. It follows from the exactness of the top sequence and from the fact that the maps Dim and Dim0 are surjective (surjectivity of Dim0 follows from a theorem
of Dotzel–Hamrick [12]). Note that the exactness of the bottom sequence is the main result of Bouc–Yalçın [8] and the proof given there is completely algebraic.The proof we obtain here can be considered as a topological interpretation of this short exact sequence.
InSection6,weconsider operationsonMoore G-spaces inducedbyactions ofbisets onMoore G-spaces. Weshow thattheassignment G → M(G) overall p-groups hasan easy todescribebisetfunctor structure,where theinductionisgivenbyjoininduction, and that Hom and Dim are natural transformations of biset functors. The induction operation on M(−) is defined using join induction of G-posets JoinHKX and the key resulthereisthatthehomologyofajoininductionJoinHKX is isomorphictothetensor induction of the homology of X. Using this we obtain a topological proof for Bouc’s tensor induction formula forrelative syzygies(see Theorem 6.9). As aconsequencewe concludethatthediagraminTheorem 1.4 isadiagram ofbisetfunctors.
Thepaperisorganizedasfollows:InSection2,weintroducesomenecessarydefinitions andbackgroundonDadegroups.WeproveTheorem 1.2inSections3and4.InSection5, we introducethegroupofMoore G-spaces M(G) and proveTheorem 1.4. InSection6, wedefineabisetfunctorstructureforM(−) andshowthatthediagraminTheorem 1.4
2. PreliminariesontheDadegroup
Let p be aprimenumber, G be afinite p-group, and k be afieldofcharacteristic p.
Throughout weassumethatall kG-modules arefinitely generated.A(left) kG-module M is called an endo-permutation module if Endk(M ) = M∼ ⊗k M∗ is isomorphic to a
permutation kG-module. Herewe viewEndk(M ) asakG-module withdiagonal action
givenby(g· f)(m)= gf (g−1m). Inthissectionweintroducesomebasicdefinitionsand resultsonendo-permutation kG-modules thatwewilluseinthepaper.Formoredetails onthismaterial,wereferthereaderto[7,sect. 12.2]or[5].
Twoendo-permutationkG-modules M and N are saidto be compatible if M⊕ N is
an endo-permutation kG-module. This is equivalent to the condition that M ⊗k N∗
is a permutation kG-module (see [7, Definition 12.2.4]). When M and N are com-patible, we write M ∼ N. An endo-permutation module M is called capped if it has an indecomposable summand with vertex G, or equivalently, if Endk(M ) has the
trivial module k as a summand (see [7, Lemma 12.2.6]). The relation M ∼ N
de-fines an equivalence relation on capped endo-permutation kG-modules (see [7, Theo-rem 12.2.8]).
Everycappedendo-permutationmodule M has acappedindecomposablesummand, called the cap of M . Note that if V is a cap of M , then V ⊗k M∗ is a summand of M⊗kM∗ whichisapermutation kG-module. Thisgivesthat V⊗kM∗isapermutation kG-module, hence V ∼ M. If W is another capped indecomposable summand of M ,
then V ∼= W (see[7,Lemma 12.2.9]),sothecapof M is uniqueuptoisomorphism.Two cappedendo-permutation kG-modules areequivalentifandonlyiftheyhaveisomorphic caps(see[7,Remark 12.2.11]).
The set of equivalence classes of capped endo-permutation modules has an abelian groupstructureundertheadditiongivenby[M ]+ [N ]= [M⊗kN ]. Itiseasytoseethat
this operation is well-defined (see [7, Theorem 12.2.8]). This groupis called the Dade group of G over k and isdenotedby Dk(G),orsimplyby D(G) when thefield k is clear
fromthecontext.
Foranon-empty G-set X, thekernel ofthe augmentationmap ε : kX → k is called arelative syzygy and is denoted byΔ(X). It is shownby Alperin [1, Theorem 1]that Δ(X) isanendo-permutationmoduleand itiscappedifandonlyif|XG|= 1 (seealso
[5, Section 3.2]). For a G-set X, let ΩX denote the element inthe Dade group D(G)
definedby
ΩX=
[Δ(X)] if X = ∅ and |XG| = 1;
0 otherwise.
Notethatif XG= ∅,themoduleΔ(X) isapermutationmodule,sointhiscasewehave
ΩX = 0. Thesubgroup of the Dade group generated by the set of elements ΩX, over
allfinite G-sets X, isdenoted DΩ(G) andcalled the Dadegroupgenerated byrelative syzygies.
Remark2.1.LetO denoteacompletenoetherianlocal ringwithresiduefield k of char-acteristic p > 0. Thenotionofanendo-permutationmoduleandtheDadegroupcanbe extended to OG-modules which are O-free (called OG-lattices). In this case the Dade group is denoted by DO(G) andthere is anatural map ϕ : DO(G) → Dk(G) defined
by reduction of coefficients. An element x ∈ Dk(G) is said to have an integral lift if x = ϕ(x) forsome x∈ DO(G).Bydefinition,theelements of DΩ(G) haveintegrallifts.
This means thatwhen we are working with DΩ(G), it does notmatter if we take the coefficientsas O or k. Notealsothatarelativesyzygyover k is obtainedfrom an endo-permutation FpG-module viatensoringwith k over Fp. Inparticular,thegroup DΩ(G)
doesnotdependonthefield k as longas itisafieldwithcharacteristic p.
Now wearegoingtostatesomeresultsrelatedto relativesyzygiesthatwe aregoing to use later inthe paper. In [5, Section 3.2] these results are stated in O-coefficients, but theyalso hold ink-coefficients. So intheresults statedbelow R is acommutative coefficientringwhichiseitherafield k of characteristic p, oracompletenoetherianlocal ringO with residuefield k of characteristic p. Wereferthereaderto[5,Section3.2]for moredetails.
Definition2.2.Let G be afinitegroupand X be afinite G-set. Asequenceof RG-modules
0→ M0→ M1→ M1→ 0 iscalled X-split if thecorresponding sequence
0→ RX ⊗RM0→ RX ⊗RM1→ RX ⊗RM2→ 0,
obtainedbytensoringeverythingwith RX, splits.
There isanalternativecriterionforasequence tobe X-split.
Lemma 2.3. Let G be a finite group and X be a finite G-set. A sequence of RG-modules is X-split if and only if it splits as a sequence of RGx-modules for every stabilizer Gx in G.
Proof. See[18, Lemma2.6]. 2
Wenow statethemain technicalresultthatwewilluseinthepaper.
Lemma 2.4. Let G be a p-group and X be a finite non-empty G-set. Suppose that
0→ W → RX → V → 0
is an X-split exact sequence of RG-lattices. Then:
(1) The lattice V is an endo-permutation RG-lattice if and only if W is an endo-permutation RG-lattice.
(2) If XG=∅, then V is capped if and only if W is capped. (3) If V and W are capped endo-permutation RG-lattices, then
W = ΩX+ V in DR(G).
Proof. See[5,Lemma3.2.8]. 2 Thefollowingalsoholds:
Lemma2.5. Let G be a p-group. Suppose that X and Y are two non-empty finite G-sets such that for any subgroup H of G, the set XH is non-empty if and only if YH is non-empty. Then ΩX= ΩY in DR(G).
Proof. See[5,Lemma3.2.7]. 2 3. AlgebraicMooreG-complexes
Let G be afinitegroupandH beafamilyofsubgroupsof G closed underconjugation and taking subgroups. The orbit category OrHG over the family H is defined as the categorywhose objectsaretransitive G-sets G/H where H ∈ H,andwhose morphisms are G-maps G/H → G/K.Throughout this paper weassume thatthefamilyH is the familyofallsubgroupsof G and wedenotetheorbitcategorysimplyby ΓG:= Or G.
Let R be acommutative ring of coefficients. An RΓG-module M is a contravariant
functorfromthecategory ΓGtothecategoryof R-modules. Thevalueofan RΓG-module M at G/H is denoted M (H). ByidentifyingAutΓG(G/H) with WGH := NG(H)/H,we
canconsider M (H) as a WG(H)-module.Inparticular, M (1) is an RG-module.
Thecategoryof RΓG-modulesisanabeliancategory,sotheusualconceptsof
projec-tive module, exactsequence, and chaincomplexes are available.For more information on modulesover the orbit category, we referthe reader to Lück [17, §9, §17] and tom Dieck[22, §10–11].
Definition3.1. ForaG-set X, wedefine RΓG-module R[X?] asthe modulewithvalues
at G/H given by R[XH], with obvious induced maps. A module over the orbit
cate-gory OrHG is called free if it is isomorphic to a direct sum of modules of the form
R[(G/K)?] with K∈ H.BytheYonedalemmaeveryfree RΓ
G-moduleisprojective(see
[13,Section 2A]).
Let X be a G-CW-complex. Thereducedchaincomplexof X over theorbitcategory isthe functorC∗(X?;R) from orbitcategory Γ
G to thecategory of chaincomplexes of R-modules, where for eachH ≤ G, theobject G/H is mapped to the reducedcellular chaincomplexC∗(XH;R). Thisgivesriseto achaincomplexof RΓG-modules
C∗(X?, R) :· · · → Ci(X?; R) ∂i −→ Ci−1(X?; R)→ · · · → C0(X?; R) ε −→ R → 0
withboundarymapsgivenby RΓG-modulemapsbetweenthechainmodules Ci(X?; R),
whereforeach i ≥ 0,thechainmodule Ci(X?; R) is the RΓG-moduledefinedby G/H→ Ci(XH; R).
In the above sequence R denotes the constant functor with values R(H) = R for each H ≤ G and the identity map id : R → R as the induced map f∗: R(G/H) →
R(G/K) between R-modules for every G-map f : G/K → G/H. The augmentation map ε is definedasthe RΓG-homomorphismsuchthatforeach H ≤ G,themap ε(H) : C0(XH;R) → R isthe R-linear mapwhichtakesevery0-cell σ∈ XHto 1.Byconvention
we takeC−1(X?;R) = R and ∂ 0= ε.
Lemma 3.2. The reduced chain complex C∗(X?;R) of a G-CW-complex X is a chain complex of free RΓG-modules.
Proof. If i ≥ 0, then for each H ≤ G, the chain module Ci(XH; R) is isomorphic to
thepermutationmodule R[XH
i ],where Xiisthe G-set of i-dimensional cellsin X. This
gives an isomorphism of RΓG-modules Ci(X?; R) ∼= R[Xi?], hence Ci(X?;R) is a free RΓG-module forevery i ≥ 0 (see[17, 9.16]or [13, Ex. 2.4]).The constantfunctor R is
isomorphic to themodule R[(G/G)?] which isafree RΓ
G-module becauseweassumed H is thefamilyofallsubgroupsof G,inparticular, G ∈ H.HenceCi(X?; R) is freefor
all i ≥ −1. 2
Intherestofthesectionwestateourresultsforchaincomplexesofprojectivemodules overtheorbitcategory.Weassumethatallthechaincomplexesweconsiderare bounded from below, i.e., there is an integer s such that Ci = 0 for all i < s. We say C is finite-dimensional if there is an n such thatCi = 0 for all i ≥ n+ 1. If C = 0, then
the smallest such integeris called thedimension of C. For moreinformation on chain complexes overtheorbitcategory,wereferthereaderto[14,§2]or[13, §2,§6].
Definition3.3.LetC beachaincomplexofprojective RΓG-modulesandlet n : Sub(G) →
Z be asuper class function.We call C an n-Moore RΓG-complex if for every H ∈ H,
thehomologygroup Hi(C(H)) iszeroforevery i = n(H).WesayC is tight if forevery H ∈ H,thechaincomplexC(H) isnon-zeroandhasdimensionequalto n(H). A Moore
RΓG-complexC iscalled capped if G ∈ H and H∗(C(G)) isnon-zero.
If X is an n-Moore G-space over R as in Definition 1.1, then by Lemma 3.2, the reduced chain complex C∗(X?; R) is an n-Moore RΓG-complex. Moreover, if X is a
capped Moorespace, thenthechaincomplexC∗(X?;k) is acapped RΓG-complex.
Lemma 3.4. Let C bea chain complex of projective RΓG-modules (bounded from below). Suppose that C is a tight n-Moore RΓG-complex and H is a subgroup of G. Then, for every i ≤ n(H),the short exact sequence
0→ ker ∂i→ Ci→ im ∂i→ 0 splits as a sequence of RΓH-modules, where ΓH = Or H.
Proof. Let s be an integer such that Ci = 0 for i < s. For every K ≤ H, the chain
complexC(K) haszerohomologyexceptindimension n(K), whichisequaltothechain complex dimension of C(K). The dimension functionof aprojective chain complex is monotone(see [14,Definition2.5, Lemma2.6]).Hencewehave n(K) ≥ n(H) forevery
K≤ H.This givesthatthetruncatedcomplex
0→ ker ∂n(H) → Cn(H) → · · · → Cs→ 0 (1)
is exact when it is consideredas asequence of modules over RΓH. Note that ΓH is a
subcategory of ΓG, so there is an induced restriction map ResGH that takes projective RΓG-modulesto projective RΓH-modules(see [13, Proposition 3.7]). This impliesthat
ResGHCi isaprojective RΓH-moduleforevery i ≥ s,hencethesequence (1)splits asa
sequenceof RΓH-modules. 2
NotethatforthealgebraictheoryofMoore G-spaces, itisenoughtoconsider projec-tive RΓG-modules, butfor obtaining results related to endo-permutation modulesone
wouldneedthesechaincomplexes tobechaincomplexesoffree RΓG-modules.When G
isafinite p-group and k is afieldofcharacteristic p, thesetwoconditionsareequivalent. Lemma 3.5. Let G be a finite p-group and k be a field of characteristic p. Then every projective kΓG-module is free.
Proof. By[17, Corollary9.40],everyprojective kΓG-module P is adirectsumof
mod-ules of the form EHSHP where H ≤ G and EH and SH are functors defined in [17,
p. 170]. Since the functor SH takes projectives to projectives, SHP is a projective kNG(H)/H-module(see[17,Lemma9.31(c)]).Thegroup NG(H)/H isa p-group and k is
afieldofcharacteristic p, hence SHP is afreemodule.Thefunctor EHtakesfreemodules
tofree kΓG-modules(see[17, Lemma9.31(c)]),therefore P is afree kΓG-module. 2
Nowwe areready to provean algebraicversion ofTheorem 1.2for tightcomplexes. RecallthatachaincomplexC over RΓG is finite if itis boundedandhastheproperty
thatforeach i, thechainmoduleCi isfinitely-generatedasan RΓG-module.
Proposition 3.6. Let G be a finite p-group and k be a field of characteristic p. If C is
a finite tight Moore kΓG-complex, then Hn(C(1)) is an endo-permutation kG-module, where n = n(1).
Proof. By Lemma 3.5, we can assume that C is a finite chain complex of free
suchthatCi ∼= k[Xi?] asa kΓG-module.IfweevaluateC at1 andaugmentthecomplex
with thehomologymodule,then weobtainanexactsequenceof RG-modules
0→ Hn(C(1))→ k[Xn]→ · · · → k[Xi] ∂i
−→ k[Xi−1]→ · · · → k[Xs]→ 0
where ∂i = ∂i(1). To show that Hn(C(1)) is an endo-permutation module, we will
inductively applyLemma 2.4(i)to eachof theextensions of kG-modules
Ei : 0→ ker ∂i → k[Xi]→ im ∂i → 0
forall i such that s ≤ i≤ n.If H = Gxforsome x ∈ Xi,thenwehave i ≤ n(H) because n(H) is equalto thedimensionof thechaincomplexC(H) by thetightnesscondition. ByLemma 3.4,thesequence
0→ ker ∂i→ Ci→ im ∂i → 0
splits as a sequence of kΓH-modules, hence the sequence Ei splits as a sequence of kH-modules. Since this is true for the isotropy subgroups Gx of all the elements x ∈ Xi, by Lemma 2.3 we conclude that the sequence Ei is Xi-split. Hence by
ap-plying Lemma 2.4(i) inductively, we conclude that Hn(C(1)) is an endo-permutation kG-module. 2
Wenowgiveamoreexplicitformulafortheequivalenceclassofthereducedhomology
Hn(C(1)) intheDadegroup.
Proposition3.7. Let G be a finite p-group and k be a field of characteristic p. Let C be a finite tight n-Moore kΓG-complex such that Ci ∼= k[Xi?] for each i. If C iscapped, then Hn(C(1)) is a capped endo-permutation kG-module and the formula
[Hn(C(1))] = n
i=m+1
ΩXi
holds in DΩ(G), where n = n(1) and m = n(G).
Proof. As beforewehaveanexactsequenceof kG-modules
0→ Hn(C(1))→ k[Xn] ∂n
−→ · · · → k[Xm] ∂m
−→ · · · → k[Xs]→ 0
where m = n(G) ≤ n = n(1). We claim that Zm = ker ∂m is a capped
permuta-tion kG-module, i.e., Zmis apermutation kG-module thatincludesthetrivial module k as a summand. Once the claim is proved, by Lemma 2.4(ii) and (iii), we can con-cludethat Hn(C(1)) isacappedendo-permutationmoduleandtheformulagivenabove
Wewillshowthat Zm isapermutation kG-module suchthatthetrivialmodule k is
oneofthesummands.Toshowthis,firstnote thatbyLemma 3.4,thesequence
0→ ker ∂m→ Cm ∂m
−→ · · · → Cs→ 0
is a split exact sequence of kΓG-modules. This gives that ker ∂m is a projective kΓG-module. By Lemma 3.5, every projective kΓG-module is a free module. Hence
ker ∂m ∼= ⊕ik[G/Hi?] for some subgroups Hi ≤ G. From this we obtain that Zm ∼= ⊕ik[G/Hi],hence Zmisapermutation kG-module.
NotethatthesummandsofCithatareoftheform k[(G/G)?] formasubcomplexofC,
denotedbyCG,andwehaveC(G)= CG(G).SinceC isacappedMoore RΓ
G-complex,
thecomplexC(G) hasnontrivial homologyatdimension m. Thisimpliesthatthe sub-complex 0→ CGm ∂ G m −→ · · · → CG s → 0
also has nontrivial homology at dimension m. This gives that ker ∂m has anontrivial
summand of the form k[(G/G)?]. Hence Z
m includes the trivial module k as a
sum-mand. 2
Asacorollary oftheresultsofthis section,weobtainthefollowing result.
Proposition 3.8. Let G be a finite p-group and k be a field of characteristic p. Let X be a finite tight n-Moore G-space over k, and let Xi denote the G-set of i-dimensional cells in X for each i. Then the reduced homology group Hn(X,k), where n = n(1), is an endo-permutation kG-module. Moreover if X is also capped, then Hn(X, k) is a capped endo-permutation kG-module, and the formula
[ Hn(X; k)] = n
i=m+1
ΩXi
holds in the Dade group DΩ(G), where m = n(G).
Proof. ByLemma 3.2thereducedchaincomplexC:= C∗(X?;k) is afinite complexof
free kΓG-modules.ApplyingPropositions 3.6 and3.7tothechaincomplexC,weobtain
thedesiredconclusions. 2 4. ProofofTheorem 1.2
In Section 3 we proved that the conclusion of Theorem 1.2 holds for tight Moore
G-spaces. ToextendthisresulttoanarbitraryfiniteMoore G-space, weshowthatupto takingjoinswithrepresentationspheres,allMoore G-spaces havetightchaincomplexes up to chain homotopy. We first start with a brief discussion of the join operation on Moore G-spaces.
Let G be adiscretegroup.Given two G-CW-complexes X and Y , thejoin X ∗ Y is
definedasthequotientspace X×Y ×[0,1]/∼ withtheidentifications(x,y, 1)∼ (x, y, 1) and (x,y, 0)∼ (x, y, 0) forall x, x ∈ X and y, y ∈ Y .The G-action on X∗ Y isgiven by g(x, y, t) = (gx,gy, t) for all x ∈ X, y∈ Y ,and t ∈ [0,1].Toavoidtheusualproblems in algebraic topology with products of topological spaces, we assume the topology on productsofspacesisthecompactlygeneratedtopology.Thenthejoin X∗Y hasanatural a G-CW-complex structure.
The G-CW-complex structure on X ∗ Y can be taken as the G-CW-structure in-herited from the union (CX × Y )∪X×Y (X × CY ) where CX and CY denote the
cones of X and Y , respectively.The CW-complex structure on the products CX × Y
and X× CY are theusual G-CW-complex structures forproductsthatwe explain be-low.
Giventwo G-CW-complexes X and Y , the G-CW-structure on X×Y canbedescribed asfollows:Giventwoorbitsofcells G/H× ep and G/K× eqin X and Y , withattaching
maps ϕ and ψ, intheproductcomplexwehaveadisjointunionoforbitsofcells
HgK∈H\G/K
G/(H∩gK)× (ep× eq)
withattaching maps ϕ × ψ.Here ep× eqisconsideredasacellwithdimension p + q by
theusual homeomorphism Dp× Dq∼= Dp+q.
Remark 4.1. Note thatwhen G is acompact Lie group, this construction is nolonger possible.Inthatcaseweonlyhavea(G× G)-CW-complexstructure onthejoin X∗ Y
and ingeneralitmaynotbepossible torestrictthis to a G-CW-complex structure via diagonalmap G → G× G.ForcompactLiegroupsIllman[15,page 193]provesthatthe join X∗Y is G-homotopy equivalenttoa G-CW-complex. Alsointheaboveconstruction itispossibletotake X∗ Y withtheproducttopologyifoneofthecomplexes X or Y is
afinitecomplex(see[19,Lemma A.5]).
If X and Y are G-CW-complexes, then(X∗ Y )H= XH∗ YH forevery H ≤ G.Since
thejoinoftwoMoorespacesisaMoorespaceitiseasytoshowthatthefollowingholds. Lemma 4.2. If X is an n-Moore G-space and Y is an m-Moore G-space, then X∗ Y is an k-Moore G-space, where k = n + m + 1.
Proof. This follows from the usual calculation of homology of join of two spaces and from theaboveobservationonthefixedpointsubspacesofjoins. 2
Since it is more desirable to have a dimension function which is additive over the join operation, we define the dimension function fora Moore G-space in thefollowing way.
Definition 4.3. For an n-Moore G-space X over R, we define the dimension function Dim X : SubG→ Z tobe thesuperclassfunctionwithvalues
(Dim X)(H) = n(H) + 1 forall H≤ G.
ByLemma 4.2 wehaveDim(X∗ Y )= Dim X + Dim Y .We willtakejoinofagiven Moore G-space withahomotopy representation. A homotopyrepresentation of afinite groupG is definedas aG-CW-complex X with theproperty thatforeach H ≤ G, the fixedpointset XH isahomotopyequivalenttoan n(H)-sphere, where n(H) = dim XH.
Givenarealrepresentation V of G, theunitsphere S(V ) can betriangulatedasafinite
G-CW-complex and forevery H ≤ G,thefixed point set S(V )H = S(VH),so S(V ) is
afinitehomotopyrepresentationwithdimensionfunctionwithvalues[Dim S(V )](H)= dimRVH.If X is an n-Moore G-space, thenthejoin X∗ S(V ) isan m-Moore G-space,
where m satisfies m(H) = n(H)+ dimRVH forevery H ≤ G.
Definition4.4.Asuperclassfunction f : Sub(G) → Z iscalled monotone if f (K) ≥ f(H)
forevery K≤ H.Wesay f is strictly monotone if f (K) > f (H) for every K < H.
Weprovethefollowing.
Lemma4.5. Let X be a Moore G-space over R. Then there is a real G-representation V such that Y = X∗S(V ) is a Moore G-space with a strictly monotone dimension function and the reduced homologies of X and Y over R are isomorphic.
Proof. Let s be apositiveintegerand V be 2s copiesoftheregularrepresentationRG. Thenforeach H≤ G,wehavedimRVH= 2s|G: H|.Ifwechoose s large enough,then
the dimensionfunction of Y = X∗ S(V ) will be strictly monotone. Since the reduced homologyof S(V ) is isomorphicto R with trivial G-action, thereducedhomologyof X
and Y over R are isomorphic. 2
Proposition 4.6. Let C be a finite Moore RΓG-complex of free RΓG-modules. If the di-mension function of C is strictly monotone, then C is chain homotopy equivalent to a tight Moore RΓG-complex D such that D is a finite chain complex of free RΓG-modules.
Proof. Byapplying[13,Proposition8.7]inductively,asitisdonein[13,Corollary8.8], we obtainthat C is chainhomotopy equivalent to a tight complexD. It is clearfrom theconstructionthatD isafinitechaincomplexoffree kΓG-modules. 2
Nowwearereadyto completetheproofofTheorem 1.2.
ProofofTheorem 1.2. Let X be afinite n-Moore G-space over k. ByLemma 4.5,wecan assume thatthe function n is strictly monotone. Let C = C∗(X?;k) denote the chain
complexof X over theorbitcategory ΓG= Or G.ByProposition 4.6,C ischain
homo-topyequivalenttoatightMoore kΓG-complexD suchthatD isafinitechaincomplexof
free kΓG-modules.Applying Proposition 3.6to thechaincomplexD, weconcludethat
the n-th homology of D, and hence the n-th homology of C, is an endo-permutation
kG-module generated byrelativesyzygies. 2
Example 4.7. The conclusionof Theorem 1.2 does not hold for aMoore G-space X if
the fixed point subspace XH is not a Moore space for some H ≤ G. One can easily
construct examples of Moore G-spaces where this happens using the following general construction: Given aMoore G-space wecanassume XG = ∅ by replacing X with the suspension ΣX of X. Given two Moore G-spaces X1 and X2 with nontrivial G-fixed
points, we can take a wedge of these spaces on a fixed point. So, given two Moore
G-spaces oftypes(M1, n1) and(M2, n2),usingsuspensionsandtakingawedge,wecan
obtain a Moore G-space of type (M1⊕ M2, n), where n = max(n1, n2)+ 1. The
di-rect sum of two endo-permutation kG-modules M1⊕ M2 is not an endo-permutation
kG-module unless M1 and M2 arecompatible.Togiveanexplicitexample,wecantake
G = C3× C3 and let X1 = G/H1 and X2 = G/H2 where H1 and H2 are two
dis-tinct subgroups in G of index 3. Then X = ΣX1∨ ΣX2 is a one-dimensional Moore
G-space with reducedhomology Δ(X1)⊕ Δ(X2),where Δ(Xi)= ker{kXi → k}. This
module is not an endo-permutation module because Δ(X1)⊗k Δ(X2)∗ is not a
per-mutation kG-module. One cansee this easily by restrictingthis tensor product to H1
or H2.
Using the same idea, we can construct some other interesting examples of Moore
G-spaces.
Example 4.8. The formula in Proposition 3.8 does not hold for an arbitrary n-Moore G-space, itonlyholdsfortight n-Moore G-spaces. Toseethislet G = C3,andlet X1=
G/1 as a G-set. Let X2bea1-simplexwithatrivial G-action onit.Then X = ΣX1∨X2
isnotatightcomplexsince XG isonedimensional,butitishomotopyequivalentto S0.
We cangivea G-CW-structure to X in suchaway thatthechaincomplexfor X is of theform
0→ k[G/G] ⊕ k[G/1] → ⊕3k[G/G]→ 0.
Then forthiscomplex,thesumnm+1ΩXi iszerobutthereducedhomologymoduleis
Δ(X1),whoseequivalenceclass in DΩ(G) isΩG/1,whichisnonzero.
5. Thegroup ofMooreG-spaces
In this section, wedefine the groupof finite MooreG-spaces M(G) and relateit to thegroupofBorel–Smithfunctions Cb(G) andto theDadegroupgeneratedbyrelative
Definition5.1.WesaytwoMoore G-spaces X and Y are equivalent, denotedby X ∼ Y ,if
X and Y are G-homotopy equivalent.ByWhitehead’stheoremfor G-complexes, X and Y
are G-homotopy equivalentifandonlyifthereisa G-map f : X→ Y suchthatforevery
H ≤ G,the map onfixed point subspaces fH : XH → YH is ahomotopy equivalence
(see[9, Corollary II.5.5]or [22, Proposition II.2.7]).Wedenote theequivalence class of aMoore G-space X by [X].
It is easy to show thatif X ∼ X and Y ∼ Y, then X∗ Y ∼ X∗ Y. Hence the joinoperationdefinesanadditionoftheequivalenceclassesofMoore G-spaces givenby [X]+ [Y ]= [X∗ Y ].ThesetofequivalenceclassesofMoore G-spaces withthisaddition operationisacommutativemonoidandwecanapply theGrothendieckconstructionto thismonoidto definethegroupofMoore G-spaces.
Definition5.2. Let G be afinite p-group and k be afieldofcharacteristic p. The group of finite Moore G-spaces M(G) is defined as the Grothendieck group of G-homotopy
classesoffinite,cappedMoore G-spaces withadditiongivenby[X]+ [Y ]= [X∗ Y ]. Since we are only interested in the group of finite Moore spaces, from now on we willassumeallMoore G-spaces arefinite G-CW-complexes. Notethateveryelement of
M(G) isavirtualMooreG-space [X]− [Y ],and thattwo suchvirtualMoore G-spaces
[X1]− [Y1] and[X2]− [Y2] areequalinM(G) ifthere isaMoore G-space Z such that
X1∗Y2∗Z is G-homotopy equivalentto X2∗Y1∗Z.Inparticular,forallMoore G-spaces
X and Y and everyrealrepresentation V , wehave[X]− [Y ]= [X∗ S(V )]− [Y ∗ S(V )]. Usingthiswecanprovethefollowing:
Lemma5.3. Every element in M(G) can be expressed as [X]− [Y ] where X and Y are Moore G-spaces over k with strictly monotone dimension functions.
Proof. Let[X]− [Y ]∈ M(G).UsingtheargumentintheproofofLemma 4.5,itiseasy toseethatthere isarealrepresentation V such thatboth X∗ S(V ) and Y ∗ S(V ) have
strictly monotone dimension functions.Since [X]− [Y ]= [X∗ S(V )]− [Y ∗ S(V )], we obtainthedesiredconclusion. 2
RecallthatinDefinition 4.3,wedefinedthedimensionfunctionofan n-Moore G-space X as thesuperclass functionDim X : Sub(G)→ Z satisfying(Dim X)(H)= n(H)+ 1 forevery H ≤ G.Note thatif X is atight n-Moore G-space, then n coincides withthe geometric dimension function, but in general n is actually the homological dimension function, giving the homological dimension of fixed point subspaces. Hence Dim X is well-definedontheequivalenceclassesofMoore G-spaces.
Let C(G) denote the groupof all super class functionsf : Sub(G) → Z. Note that
C(G) is a free abelian group with rank equal to the number of conjugacy classes of subgroupsin G.
Definition5.4.Theassignment[X]→ Dim X canbeextendedlinearlytoobtainagroup homomorphism
Dim :M(G) → C(G).
Wecall thehomomorphismDim the dimension homomorphism.
If X is a finite G-set such that XG = pt, then X is a capped Moore G-space as a discrete G-CW-complex. In this case Dim X = ωX where ωX is the element of C(G)
definedby
ωX(K) =
1 if XK = ∅
0 otherwise
forevery K≤ G.Let{eH} denotetheidempotentbasisfor C(G) defined by eH(K)= 1
if H and K are conjugatein G and zerootherwise.Notethatforevery H≤ G,wehave
ωG/H =
K≤GHeK, where thesum is overall K such thatK
g ≤ H for some g ∈ G.
Since the transition matrix between {ωG/H} and {eK} is an upper triangular matrix
with1’sonthediagonal,theset{ωG/H},overthesetofconjugacyclassesofsubgroups H ≤ G,isabasisfor C(G). Weconcludethefollowing.
Lemma 5.5. The set of super class functions {ωG/H} over all transitive G-sets G/H is a basis for C(G). Moreover, for every H ≤ G, we have eH =K≤GHμG(K,H)ωG/K where μG(K,H) denotes the Möbius function of the poset of conjugacy classes of sub-groups of G.
Proof. See[6,Lemma2.2]. 2
Thefollowing isimmediatefromthis lemma.
Proposition 5.6. The dimension homomorphism Dim :M(G)→ C(G) that takes [X] to
its dimension function Dim X is surjective.
Proof. For H ≤ G, letX = G/H if H = G, and X = G/G G/G if H = G. Then,
X is a capped Moore G-space and dim X = ωG/H. Hence by Lemma 5.5, the map
Dim :M(G)→ C(G) issurjective. 2
NowconsiderthegrouphomomorphismHom :M(G)→ DΩ(G),definedasthelinear extensionoftheassignmentwhichtakesanisomorphismclass[X] ofan n-Moore G-space X to the equivalence class of the n-th reduced homology [ Hn(X,k)] in DΩ(G), where n = n(1).Thisextensionispossiblebecausetheassignment[X]→ Hom([X]) isadditive. Note thatif[X1]− [Y1]= [X2]− [Y2], thenthereisaMoore G-space Z such that
ThisgivesthatHom([X1])− Hom([Y1])= Hom([X2])− Hom([Y2]) in DΩ(G).So,Hom is
awell-definedhomomorphism.
There isathird homomorphismΨ: C(G) → DΩ(G) which canbe uniquelydefined asthegrouphomomorphismwhich takes ωG/H to ΩG/H.In[6,Theorem 1.7],itis also
provedthatΨ takes ωX toΩXforevery G-set X, byshowingthattherelationssatisfied
by ωX alsohold forΩX.Nowwestatethemain theoremofthissection:
Proposition 5.7. Let Ψ : C(G) → DΩ(G) denote the homomorphism defined by
Ψ(ωG/H)= ΩG/H, and let Hom :M(G)→ DΩ(G) be the homomorphism which takes the equivalence class of a finite, capped Moore G-space [X] to the equivalence class [Hn(X;k)] in DΩ(G). Then
Hom = Ψ◦ Dim .
In particular, if X is a finite, capped n-Moore G-space, then the equivalence class
[Hn(X; k)] in DΩ(G) depends only on the function n.
To prove Proposition 5.7, we need to introduce a property that is found in chain complexes of G-simplicial complexes. Let X be a finite G-simplicial complex and let C:= C∗(X?; k) denote thereducedchaincomplexof X over theorbitcategory Γ
G.The
complex C isa finite chain complex of free kΓG-modules. LetXi denote the G-set of i-dimensional simplices inX for every i. Note thatsince X is a G-simplicial complex, thecollection{Xi} satisfiesthefollowingproperty:
(∗∗) Foreverysubgroup H ≤ G,if XH
i = ∅ for some i, then XjH = ∅ forevery j≤ i.
Note thatif X is aG-CW-complex and Xi denotes the G-set of i-dimensional cells
in X, thenthecollection{Xi} doesnotsatisfythispropertyingeneral.Ifthisproperty
holdsfora G-CW-complex X, thenwe sayX is a full G-CW-complex. Moregenerally, wedefinethefollowing:
Definition5.8. LetC beafinitefreechaincomplexof RΓG-modules.Foreach i, let Xi
denotethe G-set suchthatCi∼= R[Xi?].WesayC isa full RΓG-complex if thecollection
of G-sets {Xi} satisfiestheproperty(∗∗).
Forchaincomplexes thatarefull, wehavethefollowing observation:
Lemma 5.9. Let C be a finite chain complex kΓG of dimension n. Suppose that C is a full complex, and Xi denotes the G-set such that Ci ∼= R[Xi?] for each i. Let f be the super class function defined by f (H) = dim C(H)+ 1 for all H ≤ G. Then
f = n
i=0 ωXi.
Proof. Let H ≤ G.ThesumiωXi(H) isequaltothenumberof i such that X
H i = ∅.
Since XH
i = ∅ if andonly if i satisfies 0≤ i ≤ dim XH,we obtain that
iωXi(H) =
dim C(H)+ 1. 2
Now, wearereadyto proveProposition 5.7.
Proof of Proposition 5.7. Let [X]− [Y ] be an element in M(G). By Lemma 5.3, we canassume X and Y are MooreG-spaces with strictly monotonedimension functions. Moreover we can assume that both X and Y are G-simplicial complexes. This is be-cause every G-CW-complex is G-homotopy equivalent to a G-simplicial complex (see
[19,Proposition A.4]).LetC:= C∗(X?; k) denote thereducedchaincomplexfor X over
theorbitcategory.
Since the dimension function of X is strictly monotone, by Proposition 4.6 the complex C is chain homotopy equivalent to a tight complex D, which is by con-struction a finite chain complex of free kΓG-modules. Moreover, we can take D
to be a full complex. To see this, observe that since X is a simplicial complex, C is a full complex. The construction of D involves erasing chain modules of C above the homological dimension, hence we can assume that D is also a full com-plex.
For each i, let Xi denote the finite G-set such thatDi ∼= k[Xi?]. ByLemma 5.9,for
each H ≤ G,wehavedim D(H)+ 1=ni=0ωXi(H),wheredim D(H) denotesthechain
complexdimensionofD. SinceD isatight complex,we have n(H) = dim D(H) forall
H ≤ G,so wehaveDim D=i=0n ωXi. ByProposition 3.7,theequation
[ Hn(X; k)] = n
i=m+1
ΩXi
holds intheDadegroup DΩ(G),where m = n(G).Note thatsinceD is afullcomplex,
XG
i = ∅ for all i ≤ m.ThismeansthatΩXi = 0 forall i ≤ m, henceweconclude
[ Hn(X; k)] = n i=0 ΩXi = Ψ( n i=0 ωXi) = Ψ(Dim D).
Thesameequalityholdsfor[Y ],henceHom = Ψ◦ Dim. 2
The rest of the section is devoted to the proof of Theorem 1.4 stated in the intro-duction. LetM0(G) denotethe kernel of thehomomorphism Hom :M(G) → DΩ(G).
Note thatHom issurjectivebecauseHom([X])= ΩX when X is afinite G-set suchthat |XG|= 1.ByProposition 5.7,wehaveHom = Ψ◦ Dim,sothemap Ψ isalsosurjective.
0 M0(G) Dim0 M(G) Dim Hom DΩ(G) = 0 0 ker Ψ C(G) Ψ DΩ(G) 0
where thehorizontalsequencesare exact.ByProposition 5.6, thehomomorphism Dim issurjective,hencebytheSnakeLemmaDim0 isalsosurjective.Tocomplete theproof
of Theorem 1.4, it remains to show that ker Ψ is equal to the group of Borel–Smith functions Cb(G).RecallthatBorel–Smithfunctionsaresuperclass functionssatisfying
certainconditionscalledBorel–Smithconditions.Alistoftheseconditionscanbefound in[8,Definition 3.1]or [22,Definition 5.1].
Proposition5.10. Let G be a p-group. Then, the kernel of the homomorphism Ψ: C(G)→
DΩ(G) is equal to the group of Borel–Smith functions C
b(G).
Proof. Aproofofthis statementcanbe foundin[8,Theorem 1.2],buttheproof given there usesthe biset functorstructure ofthe morphism Ψ: C(G)→ DΩ(G),hence the tensorinductionformula ofBouc.Here we giveanargumentindependent ofthetensor inductionformula.
Let f ∈ Cb(G) be a Borel–Smith function. Then by Dotzel–Hamrick [12] there is
a virtual real representation U − V such that Dim U − Dim V = f. Since the unit spheres S(U ) and S(V ) are orientable homology spheresover k, both S(U ) and S(V )
are Moore G-spaces and the element [S(U )]− [S(V )] is in M0(G). This proves that
f ∈ im(Dim0)= ker Ψ.
For the converse, let f = Dim0([X]− [Y ]) for some [X]− [Y ] ∈ M0(G). Then,
Hom([X]) = Hom([Y ]). We want to show thatf satisfies the Borel–Smith conditions. Since theBorel–Smithconditions aregiven as conditions oncertain subquotients, first notethatfor anysubquotient H/L, wecanlook at(H/L)-spaces XL and YL,and the dimensionfunctionforthevirtualMoore G-space [XL]−[YL] wouldsatisfyBorel–Smith
conditionsifandonlyifthefunction f satisfies theBorel–Smithconditioncorresponding tothesubquotient H/L. Forthesesubquotientsitiseasytocheckthateverysuperclass function f in ker Ψ satisfiestheBorel–Smithconditions(see[8,page 12]). 2
Thiscompletestheproof ofTheorem 1.4. Wecanviewthistheoremas atopological interpretation of the exact sequence given in [8, Theorem 1.2]. There is an interest-ing corollary of Theorem 1.4 that gives a slight generalization of the Dotzel–Hamrick Theorem(see[12]).
Proposition5.11. Let G be a finite p-group and let k be a field of characteristic p. Sup-pose that X is a finite Moore G-space of dimension n such that Hn(X;k) is a capped permutation kG-module. Then the super class function Dim X satisfies the Borel–Smith conditions.
Proof. Bytheassumptiononhomology,[Hn(X;k)] = 0 in DΩ(G),hence[X]∈ M0(G).
Now theresultfollows fromTheorem 1.4. 2 6. OperationsonMooreG-spaces
The main aim of this section is to show that the assignment G → M(G) defined
over acollectionof p-groups G has aneasyto describebisetfunctorstructure andthat the maps Hom and Dim are both natural transformations of biset functors. We also give a topological proof of Bouc’s tensor induction formula for relative syzygies (see
Theorem 6.9belowforastatement).
LetC denoteacollectionof p-groups closedundertakingsubgroupsandsubquotients, and let R be acommutativering with unity.An (H, K)-biset is aset U together with aleft H-action and aright K-action such that(hu)k = h(uk) foreveryh ∈ H, u ∈ U,
and k ∈ K. The C-biset category over R is the category whose objects are H ∈ C
and whose morphisms Hom(K,H) for H, K∈ C are givenby R-linear combinations of (H, K)-bisets, wherethecompositionoftwomorphismsisdefinedbythelinearextension oftheassignment(U, V ) → U ×KV for U an (H,K)-biset and V a (K,L)-biset. A biset
functor F on C over R is afunctor F from theC-bisetcategoryover R to thecategoryof
R-modules. Wereferthereaderto[7]formoredetailsonbisetfunctorsforfinitegroups. To define a biset functor structure on M(−), we need to define the action of an (H, K)-biset U on an isomorphism class [X] in M(H) andextend it linearly. To sim-plify the notation we will define these actions on arepresentative of each equivalence class and show that the definition is independent of the choice of the representative. Every (H, K)-biset canbe expressed asa compositionof 5typesofbasic bisets,called restriction, induction, isolation, inflation, and deflation bisets (see [7, Lemma 2.3.26]). Exceptfortheinductionbiset,theactionofabisetonaMoore G-space iseasytodefine. For example, ifϕ : H → K isagrouphomomorphism and U = (H× K)/Δ(ϕ) where
Δ(ϕ) ={(h, ϕ(h)) | h ∈ H}, then for aMoore K-space X, we define M(U)(X) as the Moore space X together with the H-action given by hx = ϕ(h)x. This gives us the action ofrestriction,isolation, andinflation bisetsonaMoorespace.Theactionof the deflationbisetcanalsobedefinedeasilybytakingfixedpointsubspaces:Givenanormal subgroup N in G and aMoore G-space X, wedefineDefGG/NX as the G/N -space XN.
Theactionoftheinductionbiset U =H(H)K,where K≤ H ishardertodefine.This
operationiscalled join induction and thedifficultycomesfromdescribingtheequivariant cellstructureoftheresultingspace.Tomakethistaskeasierwewilldefinejoininduction onaMoore G-space whose G-CW-structure comesfrom arealization ofa G-poset.
Recallthata G-poset isapartiallyorderedset X together witha G-action suchthat
x ≤ y implies gx ≤ gy for all g ∈ G.Associated to aG-poset X, there is a simplicial
G-complex whose n-simplices are given by chains of the form x0 < x1 < · · · < xn
where xi∈ X.Thissimplicialcomplexiscalledtheassociatedflagcomplexof X (or the
ordercomplexof X) andisdenotedbyFlag(X).Wedenotethegeometricrealizationof Flag(X) by |X|.The complexFlag(X) isan admissiblesimplicial G-complex, i.e., ifit
fixesasimplex, itfixesall itsvertices. Since Flag(X) isan admissibleG-CW-complex,
therealization|X| hasa G-CW-complex structure.Formoredetailson G-posets werefer thereaderto[7,Definition 11.2.7].
By an equivariant version of the simplicial approximation theorem, every
G-CW-complexis G-homotopy equivalenttoasimplicial G-complex (see[19,Proposition A.4]). Given a G-simplicial complex X, by takingthebarycentric subdivision,we canfurther assume that X is the flag complex of the poset of simplices in X. Therefore, up to
G-homotopy we can alwaysassume thata given MooreG-space is therealization of a
G-poset X.
Let X and Y be two G-posets. Theproductof X and Y is definedtobe the G-poset X× Y where the G-action is given by the diagonal action g(x, y) = (gx,gy), and the order relationis given by(x,y) ≤ (x, y) if and onlyif x ≤ x and y ≤ y. Thejoin of two G-posets X and Y is defined to be the disjoint union XY together with extra order relations x ≤ y for all x ∈ X and y ∈ Y . However this description of the join isnot suitablefor definingjoininductionsinceit isnot symmetric.Instead weuse the followingdefinition:
Definition 6.1. For aG-poset X, let cX denote the cone of X where cX = {0X}X
withtrivial G-action on0X.Theorderrelationson cX are thesameastheorderrelations
on X together with anextrarelation0X ≤ x forall x ∈ X.Wedefine the(symmetric)
joinof two G-posets X and Y as theposetdefinedby
X∗ Y := (cX × cY ) − {(0X, 0Y)}.
Throughoutthissection,whenwerefertothejoinoftwoposets,wealwaysmeanthe symmetric join defined above. The realization |X ∗ Y | of the (symmetric) join of two
G-posets X and Y is G-homeomorphic to thejoin|X|∗ |Y | ofrealizationsof X and Y .
This is proved in[20, Proposition1.9] but belowwe prove this moregenerally for the joinoffinitelymany G-posets. Wedefinethe(symmetric)join X1∗ · · · ∗ Xnof G-posets X1, X2, . . . , Xn as the G-poset
icXi− {(0X1, . . . , 0Xn)} with the diagonal G-action.
Note that the geometric join |X1|∗ · · · ∗ |Xn| can be identified with the subspace of
ic|Xi| formedbyelements t1x1+· · · + tnxn suchthat ti∈ [0,1] and
iti = 1.Here c|Xi| denotestheidentificationspace X×[0,1]/∼ where(x,0)∼ (x, 0) forall x, x ∈ X.
Alternativelywecanconsiderelements of c|Xi| asexpressions tixi where ti∈ [0,1] and xi ∈ |Xi|. Wehavethefollowingobservation.
Proposition 6.2. Let {Xi} be a finite set of G-posets. Then |∗iXi| isG-homeomorphic to the (topological) join ∗i|Xi| of the realizations of the Xi.
Proof. Therealization |∗ Xi|=|icXi− {(0X1, . . . , 0Xn)}| can be identifiedwith the
unionofsubspaces
i
intheproduct ic|Xi|. Usingtheradialprojectionfrom thepoint{(0X1, . . . , 0Xn)} in
ic|Xi|, we can write a G-homeomorphism between this subspace and the geometric
join|X1|∗ · · · ∗ |Xn|. Thishomeomorphismtakes thepoint (t1x1, . . . , tnxn) inic|Xi|
to (t1x1, . . . , tnxn) where ti= ti/(iti) forall i. Notethatthisargumentonlyworksif
we takethecompactopen topologyontheproduct,nottheproduct topology(see[23, Theorem 3.1]). 2
Let U be afinite(H, K)-biset and X be a K-poset. Define tU(X) astheset tU(X) := MapK(Uop, X)
of all functionsf : U → X suchthat f (uk) = k−1f (u). The posetstructure on tU(X)
is defined by declaring f1 ≤ f2 if and only if for every u ∈ U, f1(u) ≤ f2(u). There
is an H-action on tU(X) given by (hf )(u) = f (h−1u) for all h ∈ H, u ∈ U. The set tU(X) isan H-poset withrespect to this action.The assignment X → tU(X) is called
thegeneralizedtensorinductionofposetsassociated to U (see [7,11.2.14]).
Definition 6.3. Let K and H be finite groups and U be a finite (H,K)-biset. For a
K-poset X, wedefinethe join induction induced by U on X as the H-poset
JoinUX := tU(cX)− {f0}
where f0 istheconstantfunctiondefinedby f0(u)= 0X forall u ∈ U.When U =HHK
istheinductionbiset,thenwedenotethejoininductionoperationJoinU byJoinHK,and
we callit join induction from K to H.
Thefollowing resultjustifiesthisdefinition.
Proposition6.4. Let R be a coefficient ring and let X be a K-poset such that the realiza-tion |X| is a Moore K-space over R. Then for every (H, K)-biset U , the realization of the H-poset JoinUX is a Moore H-space over R.
Proof. We need toshow thatforevery L ≤ H,the fixedpoint subspace|JoinUX|L = |(JoinUX)L| isaMoorespaceover R. Wehave
(JoinUX)L= HomH(H/L, tU(cX))− {f0} = HomK(Uop×H(H/L), cX)− {f0}.
By[7,Lemma11.2.26], wehave
Uop×H(H/L) ∼=
u∈L\U/K K/Lu
where Luisthesubgroupof K defined by Lu={k ∈ K : uk = lufor some l∈ L}.Using
(JoinUX)L= HomK( u∈L\U/K K/Lu, cX)− {f0} = ( u∈L\U/K (cX)Lu)− {(0X, . . . , 0X)}
ApplyingProposition 6.2,weconclude
| JoinUX|L=∗u∈L\U/K|X|L
u
. (2)
Sincethe joinofacollectionof Moorespacesis aMoorespace, |JoinUX|L isaMoore
spaceover R. 2
Let U be an(H,K)-biset. If f : X→ Xisa K-poset mapthatinducesa K-homotopy
equivalence|f|:|X|→ |X|,then|JoinUf|:|JoinUX|→ |JoinUX| isan H-homotopy
equivalence.Toseethis, observethatforevery L ≤ H theinducedmap
| JoinUf|L:| JoinUX|L→ | JoinUX|L
is a homotopy equivalence by the fixed point formula given in Equation (2). Hence byWhitehead’stheorem for G-complexes, |JoinUf| is a G-homotopy equivalence. This
proves that JoinUX defines a well-defined map on the equivalence classes of Moore K-spaces.
Giventwo K-posets X and Y , wehave
JoinU(X∗ Y ) = tU(c(X∗ Y )) − {f0} = tU(cX× cY ) − {f0} = JoinU(X)∗ JoinU(Y ).
Hencewe concludethatJoinU inducesawell-defined grouphomomorphism M(U): M(K) → M(H)
for every (H, K)-set U . We show below that this operation satisfies all the necessary conditionsfordefiningabisetfunctor.
Proposition6.5. There exists a biset functor M overp-groups such that for any p-group G, M(G) isthe group of finite Moore G-spaces, and for any (H,K)-biset U ,
M(U) : M(K) → M(H)
is the group homomorphism induced by the generalized join induction JoinU.
Proof. Let U be a(H,K)-biset and V be an(L, H)-biset. Forevery K-poset X,
Since tV(tU(cX))= tV×HU(cX) (see[7,Proposition 11.2.20]),weconcludethat
JoinV(JoinUX) = JoinV×HUX.
Byasimilarargument,weseethatif U and Uaretwo(H, K)-bisets, thenJoinUUX =
JoinUX∗ JoinUX for everyMoore K-space X. HenceM(U):M(K)→ M(H) defines
abisetfunctorM(−) onanycollectionC of p-subgroups thatisclosedunderconjugations and takingsubquotients. 2
Now we will show that the dimension homomorphism Dim : M(G) → C(G) is a natural transformation of biset functors. Note that the biset action on the group of superclassfunctionsisdefinedas thedual ofthebisetactionontheBurnsidering.Let
B(G) denote the Burnside ring of the groupG and B∗(G) = Hom(B(G),Z). We can identify C(G) with B∗(G) by sending f to a homomorphism thattakes the transitive
G-set G/H to f (H). Underthisidentificationweoftenwrite f (G/H) for f (H) when we wantto thinkof f as anelement of B∗(G).
Fora(H,K)-biset U , theactionof U on B∗(G) isdefinedasthedualofthe U -action
on B(G). Inparticular, for f ∈ B∗(K), we define thegeneralized inductionJndUf as
thesuperclassfunctionthatsatisfies
(JndUf )(H/L) = f (Uop×H(H/L)) =
u∈L\U/K
f (K/Lu) (3)
forevery L ≤ H (see[8,page 7]formoredetails).
Proposition 6.6. Let U be a (H,K)-biset and let X be a Moore K-space. Then JoinUX is a Moore H-space with dimension function Dim(JoinUX) = JndU(Dim X). Hence the dimension homomorphism Dim : M(−) → C(−) is a natural transformation of biset functors.
Proof. Let L ≤ H.UsingtheformulasinEquations(2)and(3), weobtain [Dim(JoinUX)](H/L) =
u∈L\U/K
(Dim X)(K/Lu) = [JndU(Dim X)](H/L). 2
Now we consider the homomorphism Hom : M(G) → DΩ(G).This homomorphism alsodefinesanaturaltransformationofbisetfunctors,butforthiswefirstneedtoexplain the bisetfunctorstructure onDΩ(−).Againit isrelatively easytodefine theactionof
an(H, K)-biset onanendo-permutation kK-module M when thebiset U is arestriction, inflation,isogation,ordeflationbiset.Theactionoftheinductionbisetishardertodefine anditisdoneusingtensorinductionof kG-modules. Todefinetensorinductionoperation on DΩ(G) oneneedsto provethattensorinductionofarelative syzygyisgenerated by
relative syzygies. To prove this, Bouc [5] proved the tensor induction formula stated belowasTheorem 6.9.Wewillfirstgiveatopologicalproofforthisformula.
Let K be asubgroupof H, andlet{h1K, . . . , hsK} beasetof cosetrepresentatives
forcosetsof K in H. Foreach h ∈ H,thereisapermutation π of theset{1,. . . , s} such
thatforevery i ∈ {1,. . . , s},
hhi= hπ(i)ki
for some ki ∈ K. The tensor induction TenHKM is defined as the kH-module which
is equal to the tensor product M ⊗k · · · ⊗k M (s times) as a k-vector space, and the H-action isdefinedby
h(m1⊗ · · · ⊗ ms) = kπ−1(1)mπ−1(1)⊗ · · · ⊗ kπ−1(s)mπ−1(s)
forevery h ∈ H.Thetensorinductiondistributesovertensor products TenHK(M1⊗ M2) = TenHKM1⊗ TenHKM2
and there is a Mackeyformula for tensor inductionsimilar to the Mackeyformula for additiveinduction(see[2,Proposition3.15.2]).
Proposition 6.7. Let K ≤ H be p-groups, and let X be a Moore K-space over k with nonzero reduced homology at dimension n. Then the reduced homology of JoinHKX is isomorphic to TenHK( Hn(X; k)).
Proof. As before we canassume thattheMoore space X is arealization of a G-poset X and that join induction is defined by JoinKHX = HomK(H, cX) − {f0}. Let
{h1K, . . . , hsK} be aset of left coset representatives of K in H. We canconsider the H-poset HomK(H,cX) as aproduct of K-posets
s
i=1cX with the H-action givenby h(x1, . . . , xs) = (kπ−1(1)xπ−1(1), . . . , kπ−1(s)xπ−1(s))
forevery h ∈ H.Thepermutation π and theelements k1, . . . , ks∈ K aredefinedas in
the definition of tensor induction. As inthe proof of Proposition 6.2, we can identify therealization ofthis posetwith thegeometric joinof s copies of therealization of X. Thisgives asimplicialcomplex, alsodenotedJoinKHX, where thesimplices ofJoinHKX
are unions of simplices of the flag complex Flag(X). Hence a simplex of JoinHKX is
of the form σ = ∪si=1σi where σi is a simplex of Flag(X) or an empty set, for every i ∈ {1,. . . , s}.The H-action on σ is givenby
hσ = s
i=1
kπ−1(i)σπ−1(i)
forevery h ∈ H. Notethatthe H-action onJoinHKX is notadmissible,i.e.,anelement
h ∈ H cantakeasimplextoitselfwithoutfixingitpointwise.Wecantakethebarycentric subdivisionofJoinHKX to obtainanadmissiblecomplex.Sinceanadmissible H-complex
