On smooth manifolds with the homotopy type of a homology sphere

Contents lists available atScienceDirect

Topology

and

its

Applications

www.elsevier.com/locate/topol

On

smooth

manifolds

with

the

homotopy

type

of

a

homology

sphere

Mehmet Akif Erdal

DepartmentofMathematics,BilkentUniversity,Ankara,06800,Turkey

a r t i c l e i n f o a b s t r a c t

Article history:

Received 4 July 2016

Received in revised form 30 October 2017

Accepted 17 November 2017 Available online 22 November 2017

MSC: 57Q20 55Q45 Keywords: Homology sphere Poincaré duality K-theory Cobordism Spectral sequence

InthispaperwestudyM(X),thesetofdiffeomorphismclassesofsmoothmanifolds withthesimplehomotopytype ofX,viaa mapΨ fromM(X) intothequotient ofK(X)= [X,BSO] bytheactionofthegroupofhomotopyclassesofsimpleself equivalencesofX.ThemapΨ describeswhichbundlesoverX canoccurasnormal bundlesofmanifoldsinM(X).WedeterminetheimageofΨ whenX belongstoa certainclassofhomologyspheres.Inparticular,wefindconditionsonelementsof K(X) thatguaranteetheyarepullbacksofnormalbundlesofmanifoldsinM(X). ©2017ElsevierB.V.Allrightsreserved.

1. Introduction

Unless otherwise stated, by a manifold we mean a smooth, oriented, closed manifold with dimension greater than or equal to 5. Given a simple Poincaré complex X with formal dimension m, a classical problem intopologyis to understandtheset ofdiffeomorphismclasses ofsmooth manifoldsinthesimple homotopy type of X.For suchan aim, afundamental object to study is thesmooth simple structure set

Ss_{(X) (see[1]page125–126fornotationanddetails).Elementsof S}s_{(X) areequivalenceclassesofsimple} homotopy equivalencesω : M → X fromanm-dimensionalmanifold M .Twosuchhomotopyequivalences

ω1: M1 → X andω2: M2 → X aresaid tobe equivalent ifthere is adiffeomorphism g : M1→ M2 such

that ω1 is homotopic to thecomposition ω2◦ g.An element ofSs(X) is called asimplesmooth manifold

structure onX. Note thatcomposition of an element inSs_{(X) with asimple self equivalence of X gives} another element inSs_{(X), althoughthemanifold isstill thesame.Hence,we needto quotientoutsimple} self equivalences of X inorder toget theset of diffeomorphismclassesof smoothmanifolds inthesimple

E-mailaddress:merdal@fen.bilkent.edu.tr.

https://doi.org/10.1016/j.topol.2017.11.006

homotopy typeofX. Denote byAuts(X) thegroup ofhomotopy classesof simpleself equivalencesof X. ThenAuts(X) actsonSs(X) bycomposition.Thesetofdiffeomorphismclassesofsmoothmanifoldsinthe simplehomotopy typeofX,M(X), is definedasthe setof orbitsofSs(X) under theactionof Auts(X), i.e.M(X):=Ss_(X)/Aut

s(X).

Let K(X) denote the group of homotopy classes of maps [X,BSO] (here,we abandonthe traditional notation KSO(X) forsimplicity). Every simplehomotopy equivalenceX → X induces anautomorphism onK(X). LetAuts(K(X)) denote thesubgroupof Aut(K(X)) thatconsist ofautomorphisms inducedby the simple self equivalences of X. There is a canonical action of Auts(K(X)) on K(X) again given by composition.WedenotebyK(X) theset oforbitsofK(X) undertheactionofAuts(K(X)).

Aspointedoutin[1]computationsofAuts(X) andSs(X) areingeneraldifficult,sodoesthecomputation ofM(X).Ontheotherhand,computationsofK(X) andAuts(K(X)) areeasierinmostcasesasK(−) is a(generalized)cohomologygroup(see[2]).Inthispaper, wecompareK(X) with M(X) whereX belongs

toacertainclassofPoincarécomplexes.

There isamap Ψ:M(X)→ K(X) definedby[ω : M→ X]→ (ω−1)∗(ν) whereν : M → BSO denotes

the normal bundle of M (see Proposition 2.2). For a prime q, by a Z/q-homology m-sphere we mean a simple Poincaré complex X of formal dimension m such that H_∗(X;Z/q) ∼= H∗(Sm;Z/q). For a general reference to Poincaré complexes we refer to [3] and [4]. Our purpose is to determine the image of Ψ for certain such homology spheres. Here we also assume that such a homology sphere admits a degree one normalmap (equivalently theSpivak normalfibrationhasavector bundlereduction), sinceotherwisethe problemis trivial.

Let m be an oddnumber and S be a subset of the set of primes between (m+ 4)/4 and (m+ 2)/2. DenotebyK(X)(S,q1)⊂ K(X) thesetoforbitsinK(X) thatcanberepresentedbyelementsξ suchthatfor

eachp in S the firstmod p Wu classof ξ satisfytheidentity q₁p(ξ)+ qp₁(X) = 0 (see[5]or [6]).Note that

K(Sm₎∼_{= 0 form oddandm= 1(mod 8).Themain objectofthispaperis toprovethefollowing:}

Theorem1.1. Letm beanoddnumberandS beasubsetofthesetofprimesbetween(m+4)/4 and(m+2)/2.

LetX withagivenmapf : Sm_{→ X beaZ/q-homologym-sphere,sothatf isaZ/q-homologyisomorphism}

foreveryprimeq < (m+ 2)/2 withq /∈ S.Assumefurtherthatπ1(X) isofoddorder.Thentheimage ofΨ

consists of orbits inK(X)(S,q1) that are represented by elements in thekernel of f

∗ _{: K(X)→ K(S}m_{). In}

particular,ifm= 1(mod 8), thentheimage ofΨ isK(X)(S,q1).Furthermore,ifS =∅,thenΨ issurjective.

ObservethatasS getslarger,theimageofΨ getssmaller.Inparticular,ifT ={q prime : q < m+2₂ }\S,

thenwedonotneedtomaketheassumptionofTheorem 1.1onthemod q Wuclassesfortheprimesq∈ T .

On the other hand, for an oddprime q, X being a Z/q-homologysphere implies there is no q-torsion in

K(X).Hence,primes inT alsoaffecttheimage ofΨ.

It is well-known, due to [7], that a degree one normal map can be surgered to a simple homotopy equivalence if and only if the associated surgery obstruction vanishes. The main result in [8] states that ifπ1(X) is ofoddorder, then theodddimensionalsurgery obstruction groups,Lsm(Z[π1(X)]), vanish,i.e.

everydegreeonenormal map canbesurgered to asimplehomotopy equivalence. Anessentialstepinthe proofofTheorem 1.1isthat,underthestatedconditions,anelementinK(X) admitsadegreeonenormal mapifanonlyifitisinthekerneloff∗andithasthesamemod p WuclassesastheSpivaknormalfibration of X foreach p∈ S. In particular, ifm= 1(mod 8), thenbundles admitting degreeonenormal mapsare completely determined by theirmod p Wu classes for p∈ S. In the casewhen S = ∅ andm = 1(mod 8)

everystablevectorbundleover X admitsadegreeonenormalmap.Therestistodeterminetheactionof Auts(K(X)) onK(X),whichisgivenbytherestrictionofthecanonicalactionofAut(K(X)).

SomeexamplesofsuchX comefromthesmoothsphericalspaceforms.Someapplicationsofourtheorem arediscussedinSection4.Smiththeorymayalso provideexamples,althoughwedonotmentionanysuch exampleinthis note.

2. Notationandpreliminaries

Let X be asimplePoincaré complexwith formal dimension m. Given astable vector bundle ξ : X → BSO, a ξ-manifold is a manifold whose stable normal bundles lifts to X through ξ, we refer [9] for more details (in [9] such objects are called (B,f )-manifolds). We denote by Ωk(ξ) the cobordism group of k-dimensional ξ-manifolds. An element of Ωk(ξ) is often denoted by [ρ : M → X], where M is a

k-dimensional manifold and ρ is a lifting of its stable normal bundle to X through ξ, and brackets de-note thehomotopy class ofsuch liftings(see [10] Proposition 2forthis notation). Such amap ρ is called a normal map, and if degree of ρ is equal to 1,i.e. ρ_∗[M ]= [X] ∈ Hm(X;Z), then it is called a degree

one normal map (see[11] Definition3.46). Due to thePontrjagin–Thom construction,the groupΩk(ξ) is isomorphicto k-thhomotopygroupofM ξ,theThom spectraassociatedto ξ[12].

Our primary tool is theJames spectral sequence, which is avariantof the Atiyah–Hirzebruchspectral sequence(see[10],Section II).Leth beageneralizedhomologytheoryrepresentedbyaconnectivespectrum,

F → X → B bef anh-orientable fibration withfiber F and ξ : X→ BSO be astable vector bundle. The James spectral sequence forh, f andξ has E2-pageEs,t2 = Hs(B;ht(M ξ|F)) and converges to hs+t(M ξ). Inthecasewhenh isthestablehomotopy,theedgehomomorphismofthisspectral sequencecomingfrom thebase lineisas follows:

Proposition 2.1(see[10]Proposition2). The edgehomomorphismof theJamesspectralsequenceforstable homotopy, f : X → B andξ : X → BSO is ahomomorphismed : Ωn(ξ)→ Hn(B,Z) given by

ed[ρ : M→ X] = f∗◦ ρ∗[M ]

forevery element [ρ: M → X]∈ Ωn(ξ).

TheAtiyah–HirzebruchspectralsequencesforM ξ isisomorphictotheJamesspectralsequenceforstable homotopy,id: X → X andξ : X → BSO.ThisfollowsfromthefactthatM ξ|_∗isthespherespectrum.This isomorphismis givenbytheThom isomorphism(seeproofof Proposition1in[10]). Inthispaper,we will onlyusethisedgehomomorphismoftheJamesspectralsequenceforthestablehomotopy,theidentitymap

id: X→ X andagivenstable vectorbundleξ : X→ BSO.Inthiscaseed : Ωn(ξ)→ Hn(X,Z) isthemap givenby[ρ: M → X]→ ρ_∗[M ] foreveryelement[ρ: M → X]∈ Ωn(ξ).Theotheredgehomomorphismfor this spectralsequencewill bedenotedbyed : π¯ ∗(S)→ π∗(M ξ),whereS denotesthespherespectrum.

RecallthatK(X) denotesthequotientK(X)/Auts(K(X)) (seeSection1).LetΦ: K(X)→ K(X) bethe quotient map.Wedefine amap Ψ fromM(X) toK(X) asfollows:LetM beasmooth manifoldequipped withasimplehomotopyequivalenceω : M → X andletν bethestablenormalbundleofM .Letg : X→ M

be thehomotopy inverseof ω.Then the pullbackbundle g∗(ν) definesan element inK(X). If[M ] is the diffeomorphismclassofM inM(X),wedefine Ψ[M ]:= Φ(g∗([ν])).

Proposition 2.2. Ψ iswelldefined.

Proof. LetK beanothermanifoldintheorbit[M ] withnormalbundleκ,withadiffeomorphismt: K → M and withasimplehomotopyequivalenceh: X→ K.Sincet∗([ν])= [κ],wehaveh∗t∗([ν])= h∗([κ]).Since

ω : M → X isthehomotopyinverseof g,wehaveh∗([κ])= h∗t∗([ν]) = h∗t∗ω∗g∗([ν]).Hence,h∗([κ]) and

g∗([ν]) differbyanautomorphisminAuts(K(X)) asthecompositionω◦ t◦ h ishomotopictoasimpleself homotopy equivalenceofX.Bydefinition,inK(X) theyarethesame. 2

Letp= 2b+ 1 beanoddprime.Foravectorbundle,or ingeneralasphericalfibration,ξ overX,there exist cohomologyclassesqp_k(ξ) inH4bk_{(X;Z/p),knownasmod p Wuclasses,introducedin[5].Wewriteq}

insteadof qp_k if theprimewe consider is clear from thecontext.These classesare definedby theidentity qk(ξ)= θ−1Pkθ(1).Here,Pn denotestheSteenrod’s reducedp-thpoweroperationand θ : H∗(X;Z/p)→

H∗(T ξ;Z/p) denotestheThomisomorphism.Formoredetailsonmod p Wuclasseswereferto[6],Ch.19. Foreachprimep,letqp₁(X) denotethenegativeofthemod p WuclassofνX,theSpivaknormalfibration ofX (which existssinceX isafinitecomplex,see [4]).GivenS aset ofprimes,letK(X)(S,q1)denote the

subsetof K(X) thatconsist of elementsξ such thatforeach p inS the firstmod p Wuclass of ξ satisfies

theidentityqp₁(ξ)+ qp₁(X)= 0 (orequivalently qp₁(ξ)= qp₁(νX)).Sincetheclassqp1(X) isahomotopytype

invariant of X (see [6] Ch. 19), the subset K(X)(S,q1) is invariant under the action of Auts(K(X)). We

denotethequotientof thisactionbyK(X)(S,q1).Inparticular, ifS =∅,thenK(X)(S,q1)= K(X).

Notation 2.3.E_∗,∗∗ (ξ) will denote the James spectral sequence for thestable homotopy as the generalized homology theory, identity fibrationid : X → X andthe stable bundle ξ : X→ BSO. Theabbreviations JSS and AHSS willbe usedforthe JamesandAtiyah–HirzebruchSpectralsequencesrespectively.Forany finite spectrumE,Eq∧ will denote theq-nilpotent completionof E attheprimeq (also called localization atZ/q,corresponding tolocalizationattheMoorespectrumMZq ofZ/q),see[13].

3. Mainresults

LetX beasimplePoincarécomplexwithformal dimensionm.Weimpose thefollowingconditionona stablevector bundleξ : X→ BSO:

Condition3.1. Foreachr≤ m the differential dr_{: E}r

m,0(ξ)→ Em−r,r−1r (ξ) intheJames spectralsequence

iszero.

Observe that the image of the edge homomorphism of E_∗,∗∗ (ξ) in Hm(X,Z) is the intersection of the kernelsofallofthedifferentialswithsourceE_m,0∗ (ξ),i.e. im(ed)=_rker(dr_{).Thus,Condition3.1implies} thatthegroupE2

m,0(ξ)= Hm(X;Z) isequaltoEm,0∞ (ξ),i.e. edgehomomorphismissurjective.Foragiven class [ρ: M → X] in Ωm(ξ) we haveed[ρ: M → X]= ρ∗[M ].Therefore, wecanfindaclass [ρ: M → X] inΩm(ξ) suchthated[ρ: M → X]= ρ∗[M ] isagenerator ofHm(X;Z) withthepreferred orientation.As aresult,we getadegreeonenormalmap ρ: M → X,i.e.wehaveasurgeryproblem.

IfCondition3.1doesnotholdforξ,i.e.wehaveanon-trivialdifferentialdr: E_m,0r (ξ)→ E_m−r,r−1r (ξ) for somer, then theedge homomorphismcannotbe surjective. This meansρ_∗[M ] cannotbe ageneratorof

Hm(X;Z),i.e.ρ cannotbeadegreeonemap.Hence,thereisnotadegreeonenormalmapthatrepresents aclassinΩm(ξ).Asaresult,thereisnotamanifoldsimplehomotopyequivalenttoX whosestablenormal bundleliftstoX through ξ.Hence,wehavethefollowing lemma:

Lemma 3.2. A stable vector bundle ξ admits a degree one normal map if and only if Condition 3.1 holds for ξ.

For the JSS for ξ, E_∗,∗∗ (ξ), there is a corresponding (isomorphic) AHSS for the Thom spectrum M ξ,

i.e. the AHSS whose E2-page is H_∗(M ξ,π_∗S(∗)) which converges to π_∗(M ξ), with the isomorphism given by the Thom isomorphism. For a given prime q, it is well known that the q-primary part of πS

k is zero

whenever 0 < k < 2q− 3 (see [14]). We use finiteness of πS

k [15]. On each mod q torsion part, the first non-trivial differentialsof the AHSS are given bythe duals of thestable primary cohomologyoperations. DuetoWuformulas,whenwepasstothe JSS weneedtoknowtheactionofSteenrodalgebraontheThom class. Forp= 2 theactionofSteenrod squaresontheThom classU ∈ H∗(M ξ;Z/2) isdetermined bythe Stiefel–Whitneyclasses.Infactthe(mod 2) WuformulaassertsthatSqi_{(U )= U∪ w}

i (see[6],p. 91). Let S/q denote the homology theory given by S_q∧. Let E be a spectrum. Consider the AHSS for the homology theory S/q, i.e. the coefficient groups will be π_∗(S∧_q). Due to naturality of the AHSS, the first

non-zero differentials have to be stable primary cohomology operations independent of the generalized cohomologytheory,seepp.208[2].Foreachi with0< i< 2q−3 wehaveπi(S∧q)= 0 andπ2q−3(S∧q)=Z/q. Thus, thefirstnon-trivialdifferentialinthis AHSS appearsatthe(2q− 2)-thpage.Thisdifferentialhasto be astable primary cohomologyoperation. Theonlymod q operationsinthis rangeare 0 anddual ofthe mod q Steenrod operationP1_{.Asintheproof ofLemmain[10,pp.751],lettingE = Σ}2q−2_{HZ/p asatest}

case, one cansee that d2q−2 is not always zero.The d2 differential inE_∗,∗∗ (ξ) isgiven by the dual of the map x→ Sq2_{(x)+ w}

2(ξ)∪ x, see [10] Proposition1.Letus writeq1 for qq1, where q isafixedoddprime.

Weobtainasimilarformulaforthefirstnon-zerodifferentialsinE_∗,∗∗ (ξ) actingonmod q torsionpart.

Lemma 3.3. Foreach n≥ 2q − 2 thedifferential on themod q part d2q−2 _{: E}2q−2

n,0 (ξ)→ E

2q−2

n−2q+2,2q−3(ξ) is

equalto thedual of themap

δ : Hn−2q+2(X;Z/q) → Hn(X;Z/q)

defined asx→ P1_{(x)+ q}

1(ξ)∪ x, composedwithmod q reduction.

Proof. Consider the AHSS forM ξ and S/q.Inthiscasethe coefficientgroupsofthe AHSS willbe π_∗(S∧_q) and itwillconvergeto π_∗(M ξ_q∧).Fromaboveremarks,thedifferentiald2q−2 _{inthe AHSS forM ξ andS/q,} is thedual ofthemod q Steenrodoperation

P1_{: H}n−2q+2_{(M ξ,Z/q) → H}n_{(M ξ,Z/q).}

BytheThomisomorphismtheoremanelementofH∗(M ξ,Z/q) isoftheformU∪ σ whereσ∈ H∗(X;Z/q) and U istheThom class.Onthepassagetothe JSS,Cartan’sformulaimplies

P1_{(U∪ σ) = U ∪ P}1_{(σ) +P}1_{(U )∪ σ = U ∪ P}1_{(σ) + U∪ q}

1(ξ)∪ σ

hence inthe Jamesspectral sequence thesedifferentials becomeduals ofthemaps σ→ P1_{(σ)+ q}

1(ξ)∪ σ

composedwith mod q reduction. 2

Wehavethefollowinglemmaforthedifferentialdm_{with sourceE}m m,0(ξ):

Lemma 3.4.Let q be a prime and m be an odd number. Let X be a Z/q-homology sphere with a given

Z/q-homology isomorphism f : Sm _{→ X.Then for any stable vector bundleξ : X → BSO thatis in the}

kernel of f∗: K(X)→ K(Sm),theimage of thedifferentialdm inE_0,mm ₋₁(ξ) hastrivialq-torsion.

Proof. Let: Sm_{→ BSO bethestablevectorbundlegivenbythecompositionξ◦f.Themapf : S}m_{→ X} induces amap of spectra M f : M → Mξ. Since f is a Z/q-homologyisomorphism, the induced map is also Z/q-homologyisomorphism,due toThomisomorphism.BothM ξ and M  areconnective spectraand of finite type. The space X is q-good (see Definition I.5.1 in [16]) due to 5.5 of [17]. Thus, the induced map M f_q∧ : M ∧_q → Mξ∧_q is ahomotopy equivalence(see [13]Proposition2.5 andTheorem 3.1).Wehave

f∗(ξ)= 0 inK(Sm).Hence, isatrivialbundle. TheThomspectrumM  is thenhomotopyequivalent to the wedgeofspectra S∨ Σm_{S,as itisthesuspension spectrumofS}m_{∨ S}0_{. RecallthatE}∗

∗,∗() isthe JSS for theidentity fibration Sm _{→ S}m _{and the trivialstable vector bundle. Hence, E}∗

∗,∗() collapses on the second page.Asaresult,q-torsioninE0,mm −1() survivestotheE0,m∞ −1().Theresultfollowsbycomparing theq-torsion inE_0,mm ₋₁(),viaM f ,withtheq-torsion inE_0,mm ₋₁(ξ). 2

Lemma3.5. Assumeq = 2 andm= 1(mod 8) inLemma 3.4.Then,if ξ /∈ ker(f∗), thenCondition3.1does not holdforE_∗,∗∗ (ξ).

Proof. Suppose thatξ /∈ ker(f∗) and Condition3.1 holds forE_∗,∗∗ (ξ). Letξ◦ f = μ : Sm _{→ BSO be the} nontrivial element in K(Sm). Then, by acomparison as in the proof of Lemma 3.4, Condition 3.1 holds for E_∗,∗∗ (μ). Thus, there is a degree one normal map ρ : M → Sm _{representing a class in Ω}

m(μ). Since

Ls

m(Z) = 0, see [18], every such map can be surgered to asimple homotopy equivalence, ρ : ˜˜ M → Sm. However, it is well knownthat everyhomotopy sphere is stably parallelizable(see [18]). Hence,we get a contradiction,asμ◦ ˜ρ isnontrivialinK(Sm). 2

Proof ofTheorem 1.1. Sinceboth|π1(X)| andm areodd,duetoTheorem1in[8],thesurgeryobstruction

groupsvanish. Hence,every degreeonenormal map canbe surgered to ahomotopy equivalence. We will showthatelementsinK(X)(S,q1)aretheonesthatadmitadegreeonenormalmap.

Let [ξ] beanorbit inK(X)(S,q1) represented byξ : X → BSO.Consider theJames spectral sequence,

E_∗,∗∗ (ξ).Letq beaprimewithq < (m+ 2)/2 andq /∈ S.ThenX isaZ/q-homologysphere.ByLemma 3.4

theimageofdm_{hastrivialq-torsion.SinceE}r

m−r,r−1(ξ)= Hm−r(X;πSr−1) doesnotcontainq-torsionwhen

r < m,imageofthedifferentialsdr_{: E}r

m,0(ξ)→ Em−r,r−1r (ξ) havetrivialq-torsionaswell.Hence,allofthe differentialsbasedatEr

m,0(ξ) havetrivialq-torsion intheirimages.

Now, let p∈ S. Then 2p− 2 < m < 4p− 4. It is well known thatfor t < 4p− 5 the p-torsion in π_tS

vanishesexcept when t= 2p− 3 (seefor example[14], III Theorem 3.13,B).Since 2p− 2< m< 4p− 4, Em

0,m−1 hastrivialp-torsion, wehavedm= 0. Hence,theonlydifferentialwhose imagecancontainmod-p torsionappearsat degree2p− 2.ByLemma 3.3,thedifferentiald2p−2 onthep-torsionpartisequaltothe dualofthemap

δ : Hm−2p+2(X;Z/p) → Hm(X;Z/p)

definedasx→ P1(x)+q1(ξ)∪x,composedwith(mod p) reduction.Letx beanelementinHm−2p+2(X;Z/p).

ByPoincaréduality,thereexistsanelements inH2p−2_{(X;Z/p) suchthatP}1_{(x)= s∪x (see[19]Section 2).}

Bydefinition s = q1(X). Then d2p−2 is trivialon mod-p torsion as ξ is an element in K(X)(S,q1), i.e. as

qp₁(X)+ qp₁(ξ)= 0.

Now, assume that qp₁(X) + qp₁(ξ) = 0. Then by Poincaré duality there exist an element a ∈ Hm−2p+2(X;Z/p) suchthata∪ (qp₁(X)+ q₁p(ξ)) isnonzeroinHm_{(X;Z/p),i.e.d}2p−2 _{= 0.}

As a result, Condition 3.1 holds for ξ if and only if ξ ∈ K(X)(S,q1) ∩ ker(f

∗_{). By Lemma 3.2 ξ ∈}

K(X)(S,q1)∩ ker(f∗) ifandonlyifξ admitsadegreeonenormalmap.Hence,wecandosurgeryandobtain

a smooth manifold M with asimple homotopy equivalence ω : M → X that represent aclass inΩm(ξ) if and only if ξ ∈ K(X)(S,q1)∩ ker(f

∗_{). By Lemma 3.4, the image of Ψ consists of orbits in K(X)}

(S,q1)

representedbyelementsinthekernel off∗: K(X)→ K(Sm_).

Inthecasewhenm= 1(mod 8) wehaveK(Sm_{)= 0 byBottperiodicity,i.e.imageofΨ isK(X)}

(S,q1).If

S =∅,thenK(X)(S,q1)= K(X),i.e.Ψ isasurjection. Thiscompletestheproof. 2

Thefollowingcorollaryisessentiallystrongerthanthemainresult.

Corollary 3.6.Let m and S be as in Theorem 1.1. Suppose that X and X are Z/q-homology m-spheres fitting intoazigzag Sm f→ X g← X,sothat both f andg are Z/q-homology isomorphisms forevery prime q < (m+ 2)/2 with q /∈ S. If π1(X) isafree product of finitely manyodd order groups,then theimage of

Ψ contains allorbits inK(X)(S,q1) whichare represented by elements ing

∗_(ker(f∗_{)), where f}∗ _andg∗ _are

theinducedmapsK(Sm)f

∗

← K(X₎g∗

Proof. Assumeq /∈ S withq < (m+ 2)/2.Forabundleξ overX,Condition3.1holds forξ ifandonlyif

ξ∈ K(X)(S,q1)∩ ker(f∗) foreachq. Onecancomparespectralsequencesforξ andg∗(ξ) asintheproof

of Lemma 3.4, andshowthatdmdifferentialistrivialonEm,0r (g∗(ξ)) wheneverξ∈ ker(f∗) foreachsuch primeq.Thus,Condition3.1holdsforg∗(ξ),wheneverξ∈ K(X)(S,q1)∩ker(f∗).Repeatingthearguments

of Lemma 3.5, onecanshow thatif ξ ∈ ker(f/ ∗), thenCondition 3.1 does nothold for g∗(ξ).The result follows fromTheorem 5in[20],togetherwithLemma 3.2above. 2

Corollary 3.6,forexample,allowsustodosimilarestimationsforconnectedsumsofmanifoldssatisfying theassumptions ofTheorem 1.1.

Remark3.7. ItcanbeseenfromtheproofofTheorem 1.1(resp.Corollary 3.6)thatwedonotneedasingle map f (or g) which is simultaneously aZ/q-homology isomorphism for everyprime q < (m+ 2)/2 with

q /∈ S. Itisenoughthatfor everyprimeq < (m+ 2)/2 withq /∈ S there exist mapsfqand gq (depending on q) whichare Z/q-homologyisomorphisms. Inthis case,we need to replaceker(f∗) (org∗(ker(f∗)))by intersectionoverq ofallker(f_q∗) (org_q∗(ker(f_q∗))).

Let Θm denote thegroup ofhomotopy m-spheres.For anysmooth m-manifold M , thereis asubgroup

I(M ) of Θm called the inertia group of M , defined as {Σ ∈ Θm : Σ#M ∼= M}, where ∼= here means diffeomorphic(see[21]).TwomanifoldsM1andM2aresaidtobealmostdiffeomorphicifthereisaΣ∈ Θm such thatM1#Σm∼= M2.It isknownthatalmost diffeomorphicmanifoldshaveisomorphicstable normal

bundles, ashomotopyspheresarestablyparallelizable(see[18]).Thus,theirimagesarethesameunderΨ. InordertodeterminethesetofmanifoldsthatarealmostdiffeomorphictoM ,oneneedstocomputeI(M ).

Hence,todetermineM(X),itisnecessarytoknowI(M ) forevery[M ] inM(X).ItisknownthatI(M ) is

notahomotopytypeinvariant, infactitisnoteven aPL-homeomorphismtypeinvariant,see forexample

[22].Asaresult,completedeterminationofM(X) maynotbe possibleinthisgenerality.

Thefollowing corollarysaysthatframed manifoldsdonotboundinΩm(ξ) forsomeξ : X → BSO. Corollary3.8. UndertheassumptionsofTheorem 1.1togetherwithm= 1(mod 8) andS =∅,theedgemap

¯

ed : π_∗(S)→ π_∗(M ξ) isan inclusionforanystable vector bundleξ : X→ BSO.

Proof. As intheproof ofTheorem 1.1 foraprimeq thefirst differentialinE_∗,∗∗ (ξ) thatactsnon-trivially onq-torsion appearsindimension2q− 2.SinceS =∅,X isaZ/q-homologysphereforeveryprimeq with

2q− 2< m.Hence,dr= 0 for everyr < m.Asintheproof ofLemma 3.4, bycomparing withE_∗,∗∗ () we get dm= 0 (asE_∗,∗∗ () collapsesat thesecond page,due to degreereasons).Therefore,thefirst nontrivial differential appearswhenr≥ m+ 1.Butthen thetargetof dr _{shouldbe zero.Hence,E}∗

∗,∗(ξ) collapses at thesecond page,andwegetthated : π¯ _∗(S)→ π_∗(M ξ) isaninclusion. 2

Observe that the degree of f (as in Theorem 1.1) plays the important role here, as it is co-prime to smallerprimes.Onecanaskwhatthenecessaryandsufficientconditionsareonthepair(X,ξ) so thatthe naturalmap ed : π¯ s_∗→ π∗(M ξ) induced bytheinclusionof pointisinjective. Itiswell knownthatsuchis notthecaseforclassicalThomspectra likeM O orM SO (see[12]).Inthecasewhenξ isatrivialbundle, there are examplesfor which this is true. Another possible question is: For which spaces X,this natural mapπs

∗→ π∗(M ξ) isinjectiveforeverystablevectorbundleξ : X→ BSO.Corollary 3.8providesjustone suchexample.

Supposethatq = 2 andm= 1(mod 8) inLemma 3.4.Inthiscasef inducestheidentityoncohomology withcoefficientsZ/2.BottperiodicitytheoremassertsthatK(Sm_{)= K}−m_(S0_{)=Z/2.Themapf induces}

a map on the Atiyah–Hirzebruch spectral sequences. At the second page we have f∗ : Hm_{(X;Z/2) →}

Hm_(Sm_{;Z/2),whichisan isomorphism.Themod-2 classinH}m_(Sm_{;Z/2) survives totheinfinitypage of} the Atiyah–Hirzebruchspectralsequence forK(Sm_{). Hence,f}∗ _{is asurjectionontheinfinitypage bythe}

naturalityofthe AHSS.Itfollowsthatf∗: K(X)→ K(Sm_{) issurjective(forthecasewhenX isaspherical} spaceform,thisfollowsfrom [23],Theorem 1-(b)).Hence,wehavethefollowing remark:

Remark3.9.Inthecasewhenq = 2 andm= 1(mod 8) inLemma 3.4,wehave[K(X): ker(f∗)]= 2.

Since2∈ S in/ Theorem 1.1,wehave[K(X): ker(f∗)]= 2 aswell.Thus,wecandeterminetheimagein thecasewhenm= 1(mod 8) aswell.

4. Examples

LetLk(n) denotethequotientspace S2k+1/Z/n ofthefreelinearactionofZ/n on S2k+1.If (q,n)= 1, then Lk(n) is aZ/q-homologysphere with thecovering projection being theZ/q-homology isomorphism. LetLk_{(n,μ) denotetheorbitspace ofafreeactionμ of Z/n onS}2k+1_{where μ acts byhomeomorphisms.}

SuchLk_{(n,μ) areoftencalledfakelensspaces(see[24]and[25],[26]formoredetailsontopologicaland[27],}

[28]for smoothfakelensspaces).Forany suchactionμ,onecanalwaysfindalensspaceLk_{(n) homotopy} equivalent to Lk(n,μ) (see[26] P. 456). If p is aprimeand k is anintegerwith k ≤ 2p− 3, Theorem 1.1

applies toany fake lensspace Lk_{(p,μ). Ifk < p− 1, thenS =∅ andif p− 1≤ k ≤ 2p− 3, thenS ={p}} (notethatdimensionofLk_{(p,μ) is2k + 1).Inthiscase,ifT ={q prime : q ≤ k +1}\{p},thenforanyprime}

q∈ T , Lk_{(p,μ) isaZ/q homologysphereandK(L}k_{(p,μ)) doesnothaveanyelement oforderq forq odd.} Ingeneral,ifn isanaturalnumbernotdivisiblebyprimes lessthanorequalto k+2

2 ,μ isanactionofZ/n

onS2k+1,S ={pprime : p|n} andT ={q prime : q ≤ k +1}\S,thenTheorem 1.1appliestoLk(n,μ) where

theimage ofΨ consists oforbitsinK(Lk(n,μ))(S,q1) thatarerepresentedbyelements inker(f∗), wheref

isthecoveringprojection.Again,forq∈ T oddprime,thegroupK(Lk_{(n,μ)) does nothaveanyq-torsion.} Werefer to Theorem 2in[29]for theK-theory of alensspace and to [30]for calculationof thegroupof homotopy classes of self homotopy equivalences of a lens space. For the particular cases when k = p− 3

andk = 2p− 4,wecangetfromTheorem 2.Ain[30] thatAuts(K(Lk(p,μ))) hasonly2-elements,namely theidentityandtheautomorphismmappinganelement toitsalgebraicinverse.Hence,inthesecaseseach orbit(excepttheorbitof0) inK(Lk(p,μ)) hasexactlytwoelements.

Another class of examples can be obtained from spherical space forms. There is a vast literature on classificationof spherical spaceforms, see forexample [31], [32] and [33]. Let Σ bea homotopym-sphere

withm≥ 5.Letπ beagroupthatcanactfreelyandsmoothlyonΣ andletX = Σ/π,sothatf : Σ→ X isa principalπ-bundle.Letp≥ 3 bethesmallestprimedividingtheorderofπ.Thenthereisamapϕ: X → Bπ thatclassifies f .ThegroupofselfequivalencesAut(X) ofX containsanormalsubgroupisomorphicto all innerautomorphismInn(π) ofπ (see[30]Corollary1andTheorem1.4).Notethat,aninnerautomorphism inducestheidentityonall(generalized)cohomologygroupsofBπ,duetothecommutativityincohomology. Letα : X→ X inAut(X) beaself equivalence.Considerthediagram

X Bπ

X ϕ Bπ

α∗ α

ϕ

sothatα andα∗ inducethesamemaponπ1(X)= π.Byanargumentasin[34]Theorem7.26,thediagram

commutes up to homotopy. It is well-known that ϕ induces a surjection on K (see for example [23]). Thereby,innerautomorphismsofπ induceidentityonK(X) aswell.DenotebyOut(X)= Aut(X)/Inn(π)

thegroupofouterself-equivalencesofX.Then,inordertodetermineAuts(K(X)),weonlyneedtoconsider automorphismsinducedbyselfequivalencesbelongingtoafixedsetofrepresentativesofcosetsinOut(X).

For m= 1(mod 8) and m< 2p− 2,Theorem 1.1applies toX,so thatΨ:M(X)→ K(X) issurjective. In this case S =∅. Ifm < 4p− 4 and noother prime betweenp and 2p divides theorder of π, then the image ofΨ is determinedbythefirstmod -p WuclassoftheSpivaknormalbundleofX.Ingeneral,ifS is

thesetofprimesbetweenp and2p whichdividetheorderofπ,thentheimageofΨ isK(X)(S,q1).Werefer

to [23] for the computationofK(X) (for X = Σ/π as above) and the resultsgiven in[35] (andalthough indirectly,in[36])forthecomputationofAut(X).Ofcourseweonlyneedthesecomputationswhenπ1(X)

isofoddorder.TheactionofAuts(K(X)) onK(X) isgivenbytherestrictionoftheusualcanonicalaction of theautomorphism groupAut(K(X)) onK(X),whichcanbeunderstoodonce K(X) isknown.

Let m andS be as inTheorem 1.1. LetX0 and X1 with given maps fi : Sm → Xi fori = 0,1 be two Z/q-homology m-spheres,so thatbothf0 andf1 areZ/q-homologyisomorphismsfor q < (m+ 2)/2 with

q /∈ S,i.e.Theorem 1.1appliestobothX0andX1.Then,X0#X1isalsoaZ/q-homologysphereforprimes

q < (m+ 2)/2 with q /∈ S (whichfollowseasilyfrom Mayer–Vietorissequence)andthefundamentalgroup of X0#X1 isthe freeproductπ1(X0)∗ π1(X1) (which followsfrom asimpleapplicationofVan Kampen’s

Theorem).Thus,Corollary 3.6appliestotheconnectedsumX0#X1.Ifthereexistsamapf : Sm→ X0#X1

satisfying the conditions of Theorem 1.1, then we can choose g as the identity map.In this case, im(Ψ) consists of allorbits in K(X0#X1)(S,q1) which are represented by elements inker(f∗). If we cannot find

such amap f , thenwe canapply Corollary 3.6by using thezigzagsSm fi

→ Xi gi

← X0#X1, where gi’s are theobviouscollapsemaps. Inthiscase,im(Ψ) containsallorbitsinK(X0#X1)(S,q1)whicharerepresented

byelements ing∗(ker(f∗)).

For a given finite CW -complex X, denote by Aut(Ki_{(X)) the subgroup of Aut(K}i_{(X)) that consists} of automorphisms induced by elements inAut(X) (wesimply write Aut(K(X)) when i= 0). Due to the naturalityofsuspensionisomorphismandBottperiodicity,wecanidentifyAut(K(X)) withAut(K(Σ8iX)).

Hence,thenaturalmapfromAut(X) toAut(K(X)) factorsthroughthegroupofstableselfequivalencesof

X,whichisequaltocolimiAut(Σ8iX) (see forexample[37],[38]and[39]formoredetailsaboutthegroup of stable self equivalences).IfX is aZ/q-homologysphereforaprimeq, then allBettinumbersof X are

lessthan or equalto1.Thus, dueto Theorem 1.1-(a) in[39],thegroupof stable selfequivalences ofX is

finite,whichimpliesthatAuts(K(X)) (whichisasubgroupofAut(K(X)))isfinite. Acknowledgements

IwishtothankÖzgünÜnlüandMatthewGelvinfortheirvaluableadvices.Ialsothanktheanonymous referee for the valuable comments. This research was partially supported by TÜBİTAK-BİDEB-2214/A Programme.

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