6 pagesIn a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of coﬁbrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra. Content

Hai, Nguyen D. H.

Schwartz, Lionel

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Realizing a complex of unstable modules

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Realizing a complex of unstable modulesNguyen D. H. Hai, Lionel SchwartzTo cite this version:Nguyen D. H. Hai, Lionel Schwartz. Realizing a complex of unstable modules. 6 pages. 2009.<hal-00420917>HAL Id: hal-00420917https://hal.archives-ouvertes.fr/hal-00420917Submitted on 30 Sep 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Realizing a complex of unstable modulesNguyen Dang Ho Hai, Lionel SchwartzSeptember 30, 2009AbstractIn a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolutionof the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum wasproved. In this paper one shows that the above mentioned resolutions can be realized topologically.In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. Theresolutions above are obtained by splicing together these short exact sequences. Thus the injectiveresolutions are realizable in the best possible sense. In fact our construction appears to be in somesense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically(not only homotopically) certain dual Brown-Gitler spectra.Contents1 Introduction 12 Spectra satisfying the Brown-Gitler property and related spectra 32.1 The Mitchell-Priddy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The Brown-Gitler spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The Lannes-Zarati spectra [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Proof of theorem 1.3 41 IntroductionBelow H∗X , resp. H∗X , will denote the singular mod 2 cohomology, resp. homology, of a space or aspectra X . All spaces or spectra will be supposed to be 2-completed and to have finite mod 2 cohomologyin each degree. Let A be the mod 2 Steenrod algebra and let U be the category of unstable A-modules.Let Ln be the Steinberg summand in the mod 2 cohomology H∗B(Z/2)n, as defined by Mitchell andPriddy [6]. Let also J(ℓ) be the ℓ-th Brown-Gitler module. These are both injective unstable A-modulesas well as their tensor product Lk ⊗ J(ℓ) following Lannes and Zarati [5].The Brown-Gitler modules are the mod 2 cohomology of spectra, even of spaces if ℓ is odd. We willdenote these spectra by T(ℓ). If ℓ is odd this is a suspension spectrum.Let Vn = (Z/2)n. Let us denote by L′n the unstable module which is obtained as follows. Let en be theSteinberg idempotent in F2[GLn(F2)]. Let r˜egn : Vn → O(2n − 1) be the regular reduced representation ofVn over the field R.Let r˜eg⊕kn = r˜egn + · · · + r˜egn, be the direct sum of k copies of this representation. Let BVgreg⊕knn bethe Thom space associated to r˜eg⊕kn , i.e. the Thom space of the vector bundle EVn ×Vn Rk(2n−1) → BVn.The Steinberg idempotent determines a map of these Thom spaces after one suspension. Let us denote [9]by M(n)k the Thom spectrum en ·BVgreg⊕knn . In particular if k = 1 it is the spectrum L(n) of Mitchell and1Priddy, and we will simply denote it L′(n) if k = 2. We will denote by Ln (as usual) and by L′n their mod2 cohomology.Let us now recall:Theorem 1.1 ([2, 3]) For all n ≥ 1 there exists an acyclic complex of unstable injective A-modules0→ L′n → Ln → Ln−1 ⊗ J(1)→ Ln−2 ⊗ J(3)→ · · · → L(1)⊗ J(2n−1 − 1)→ J(2n − 1)→ 0.This complex is a minimal injective resolution of L′n.Here is the main result of this article.Theorem 1.2 The above complexes are realizable in the sense there exists a complex of spectraT(2n − 1) −→ L(1) ∧T(2n−1 − 1) −→ . . . −→ L(i) ∧T(2n−i − 1) −→ . . . −→ L(n) −→ L′(n)so that when applying mod 2 singular cohomolgy one gets the algebraic complexes. The composite of twosuccessive maps is homotopically trivial, moreover all Toda brackets vanish.The result follows from the following stronger result:Theorem 1.3 There are cofibrations of spectra, 1 ≤ k ≤ n,C(k − 1) −→ L(k) ∧T(2n−k − 1) −→ C(k)so that• The induced sequences in mod 2 cohomology are short exact;• C(n) = L′(n);• C(0) = T(2n − 1).Because of [2, 3], and further applications we are focused on the case of 1.2. However this theorem isquite general. The proof of the general case proceeds exactly as in the particular case. Thus, we will onlystate:Theorem 1.4 Let M be the mod 2 cohomology of a spectrum X. Let us assume that M has an injectiveresolution of finite type in the category of unstable modules U . By finite type one means that any term inthe injective resolution is a finite direct sum of indecomposable modules. Then this resolution can be realizedtopologically by spectra, in the sense described above as soon as the algebraic map from M to its injectivehull in U can be realized.The above theorems are in the category of spectra, however the spectra L(i)∧T(2n−i−1) are suspensionspectra if n > i. In Theorem 1.2 only L(n) and L′(n) are not. They are only direct summand of a suspensionspectrum (after one suspension). It is thus reasonable to ask whether or not the complex of spectra canbe realized as a complex of spaces (in an appropriate sense) after one suspension. The technology to dothat is provided by the work of P. Goerss, J. Lannes and F. Morel [1]. However even if some preliminarycomputations are positive they are very technical in nature and we have not been able up to now to decidethis question. Thus we only propose:Conjecture 1.5 The preceding cofibrations in 1.2 can be realized as cofibrations of spaces, except for thelast one that can be realized after one suspension and which is a retract of the suspension of a cofibrationof spaces.In the general case it does not look plausible to hope for such a result, at least it is necessary to requiresomething on the map realizing the embedding in the injective hull. Next we mentioned applications of1.2. It is clear that 1.3 leads to the construction of spectral sequences that we will consider either in a laterversion of this paper or in another one.22 Spectra satisfying the Brown-Gitler property and related spec-traIn this section we recollect some facts from [4]. Most (but not all) of the material in this paper comesfrom earlier work, nicely collected here. Moreover we will need an original result of this paper. We shallsay that a spectrum is space-like if it is a wedge summand of a suspension spectrum. For such a spectrumS the evaluation map Σ∞Ω∞S −→ S has a section and induces a monomorphism in cohomology.Let now S be a spectrum whose mod 2 singular cohomology is an injective unstable A-module. FollowingN. Kuhn we shall say that S has the Brown-Gitler property if the natural mapf 7→ f∗, [S,X ] −→ HomU (H∗X,H∗S)from homotopy classes from S to X to the set of maps of unstable modules HomU (H∗X,H∗S) is onto forall space-like spectrum X .Theorem 2.1 ([4]) Let S be a spectrum whose mod 2 cohomology is an injective unstable A-module. ThenS satisfies the Brown-Gitler property if and only if either of the following condition holds:• the evaluation map induces a monomorphism H∗S −→ H∗Σ∞Ω∞S in cohomology,• the spectrum is space-like,• there exists a space-like spectrum Z and a map f : S −→ Z which induces an epimorphism incohomology.It is clear that a spectrum satisfying the Brown-Gitler property is uniquely defined by its cohomology.Recall that an unstable module M is reduced, if either it does not contain a non-trivial suspension, orif the map from M into itself, x ∈Mn, Sq0 : x 7→ Sqnx is injective.Below are the main examples of interest for us of such spectra.2.1 The Mitchell-Priddy spectraThe standard GLn(F2) action on Vn induces one on BVn, on the direct sum of all non-trivial line bundlesover BVn, and on the direct sum of k copies of this vector bundle. Thus it acts also on their Thom spacesBVgreg⊕knn . As a consequence the Steinberg idempotent en [8] induces a stable map on BVgreg⊕knn , this map isdefined after one suspension. Consider the telescope spectrum of this map [9], this is a stable summand ofBVgreg⊕knn , denoted by en · BVgreg⊕knn . Following Takayasu we will shorten en · BVgreg⊕knn to M(n)k.In particular M(n)0 = M(n), M(n)1 = L(n) are the spectra of Michell and Priddy and by definitionM(n)2 = L′(n). The spectra M(n) and L(n) satisfy the Brown-Gitler property.The cohomology of these spectra follows from the Thom isomorphism theorem:H∗M(n)k = ωknenF2[x1, · · · , xn] = ωk−1n Lnwhere ωn is the Euler class of the reduced regular representation r˜egn, or the top Dickson invariant ofGLn(F2): the product of all non-trivial linear forms in F2[x1, · · · , xn].Theorem 2.2 (Takayasu [9]) For k ≥ 0, there exist a cofibration sequenceΣkM(n− 1)2k+1in,k−−→ M(n)kjn,k−−→ M(n)k+1.This induces the following short exact sequence in cohomology:0→ ωknLn → ωk−1n Ln → Σkω2kn−1Ln−1 → 0.Splicing the cofibration sequences together yields:Σ2n−1M(0)2n → · · · → Σ2k−1M(n− k)2k → Σ2k−1−1M(n− k + 1)2k−1 → · · · → ΣM(n− 1)2 → M(n)1 → M(n)2.32.2 The Brown-Gitler spectraLet J(ℓ) the ℓ-th Brown-Gitler module (cf. [7, Chapter 2]). It is characterised by the fact that there isa natural equivalenceHomU (M,J(ℓ)) ∼=Mℓ∗.The unstable module J(ℓ) is the injective hull of ΣℓF2 in the category U . It is known that there existspectra T(ℓ) with mod 2 singular cohomology J(ℓ) [1]. This was originally proved by E. Brown et S. Gitler.If ℓ is odd T(ℓ) is a suspension spectrum. More precisely we haveTheorem 2.3 Let n > 2. There exists a 1-connected pointed space T1(n) (that we are going to call then-th Brown-Gitler space), unique up to homotopy, such that the following holds:• There is an isomorphism of unstable modules H∗T1(n) ∼= ΣJ(n).• The induced map in cohomology by the evaluation ΣΩT1(n) −→ T1(n) is a monomorphism.• T1(n) is 2-complete.This space has moreover the following properties:• It is retract of the suspension of a pointed space.• For all pointed spaces Y the natural map[T1(n),ΣY ]∗ −→ HomU (H˜∗(ΣY ),ΣJ(n))is surjective.Thus the suspension spectrum of T1(n) is ΣT(n), and if n is odd T(n) is the suspension spectrum ofthe space T1(n− 1).These were the first known spectra to have the two first conditions of Theorem 2.1.2.3 The Lannes-Zarati spectra [5]Theorem 2.4 The unstable module Ln ⊗ J(k) is an injective object in U .This is just a particular case of the Lannes-Zarati theorem. This module is the reduced mod 2 coho-mology of L(n) ∧ T(k). If k is odd this is the suspension spectrum of L(n) ∧ T1(k − 1). This is becauseT1(k − 1) has a co-H-space structure.3 Proof of theorem 1.3Let us recall some notations. The morphismsfn−k,n : Lk+1 ⊗ J(2n−k−1 − 1) −→ Lk ⊗ J(2n−k − 1)of the complex of Theorem 1.1 are defined in [2, 3]. Let us denote by Ck the image of the map fn−k,n,which is also the kernel of fn−k+1,n. If the spectra C(k) exist one has H∗C(k) ∼= Ck.We are now going to proceed by descending induction. The first step of the proof is done by observingthat the cofibrationC(n− 1) −→ L(n) −→ C(n)is nothing else than Takayasu’s cofibrationΣM(n− 1)3 −→ M(n)1 → M(n)2.4However the right map is just the zero section, and the left hand spectrum could be defined as above.We will assume that we have got a spectrum C(k − 1), 2 ≤ k ≤ n, which is the fibre of a map fromL(k) ∧T(2n−k − 1) to C(k) and short exact sequences in cohomologyH∗(C(k)) −→ H∗(L(k) ∧T(2n−k − 1)) −→ H∗(C(k − 1)).We will make use of the following:Proposition 3.1 ([4]) Let X be a 0-connected spectrum such that H∗X is of finite type and let Z be aspace. Suppose there exists a map f : X −→ Σ∞Z which is an epimorphism in cohomology. Then theevaluation map Σ∞Ω∞X −→ X induces a monomorphism in cohomology.One can apply this proposition to the spectrum C(k − 1) and the suspension spectrum Σ∞Z of whichL(k) ∧T(2n−k − 1) is a wedge summand. Thus the evaluation map Σ∞Ω∞C(k− 1) −→ C(k− 1) induces amonomorphism in cohomology:Ck−1 ∼= H∗C(k − 1) −→ H∗Ω∞C(k − 1).As a consequence, as Lk⊗J(2n−k−1) is injective in U , the canonical map Ck−1 −→ Lk−1⊗J(2n−k+1−1)extends to a map of unstable modulesH∗Ω∞C(k − 1) −→ Lk−1 ⊗ J(2n−k+1 − 1).It follows from 2.1 that this map is induced by a map of spectraγ : L(k − 1) ∧T(2n−k+1 − 1) −→ Σ∞Ω∞C(k − 1).Composing γ with the evaluation ǫ : Σ∞Ω∞C(k − 1) −→ C(k − 1) one gets a mapǫγ : L(k − 1) ∧T(2n−k+1 − 1)→ C(k − 1)inducing in cohomology the canonical map Ck−1 −→ Lk−1 ⊗ J(2n−k+1 − 1). One takes the fibre C(k − 2)of the map ǫγ to obtain a cofibration of spectra which induces a short exact sequence in cohomology.The final step of the induction deserves care. One needs to show that the spectrum C(0) is indeedequivalent to T(2n − 1), a priori it has the right cohomology. By construction there is a map C(0) −→L(1) ∧T(2n−1 − 1) which induces an epimorphism in cohomology. As L(1) ∧T(2n−1 − 1) is space-like, itfollows from 2.1 that C(0) satisfies the Brown-Gitler property, thus C(0) ≃ T(2n − 1).Remark 3.21. One has a fibration of spaces:Ω∞C(k − 1) −→ Ω∞L(k − 1) ∧T(2n−k+1 − 1) −→ Ω∞C(k).This is even a fibration of H-spaces which allows us to compute H∗Ω∞C(k−1). In fact the Eilenberg-Moore spectral sequence shows that:H∗Ω∞C(k − 1) ∼= H∗(Ω∞L(k − 1) ∧T(2n−k+1 − 1))⊗H∗Ω∞C(k) F2.However we don’t need this computation in the above proof, thus we don’t describe (in terms ofDyer-Lashof operations) the result further.2. The final step of the proof suggests it is possible to describe the space T1(2n − 2) as a sub-complexof (BZ/2)∧n. We hope to be able to say more about that later.5References[1] Paul Goerss, Jean Lannes, and Fabien Morel, Vecteurs de Witt non commutatifs et repre´sentabilite´ del’homologie modulo p, Invent. Math. 108 (1992), no. 1, 163–227. MR MR1156389 (93e:55014)[2] Nguyen Dang Ho Hai, Lionel Schwartz, and Tran Ngoc Nam, La fonction ge´ne´ratrice de Minc et une“conjecture de Segal” pour certains spectres de Thom, (preprint).[3] , Re´solution de certains modules instables et fonction de partition de Minc, C. R. Math. Acad.Sci. Paris 347 (2009), no. 2, 599–602.[4] David J. Hunter and Nicholas J. Kuhn, Characterizations of spectra with U-injective cohomologywhich satisfy the Brown-Gitler property, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1171–1190. MRMR1621749 (2000j:55016)[5] Jean Lannes and Sa¨ıd Zarati, Sur les U-injectifs, Ann. Sci. E´cole Norm. Sup. (4) 19 (1986), no. 2,303–333. MR MR868302 (89a:55032)[6] Stephen A. Mitchell and Stewart B. Priddy, Stable splittings derived from the Steinberg module, Topology22 (1983), no. 3, 285–298. MR MR710102 (85f:55005)[7] Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture,Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR MR1282727(95d:55017)[8] Robert Steinberg, Prime power representations of finite linear groups, Canad. J. Math. 8 (1956), 580–591. MR MR0080669 (18,281d)[9] Shin-ichiro Takayasu, On stable summands of Thom spectra of B(Z/2)n associated to Steinberg modules,J. Math. Kyoto Univ. 39 (1999), no. 2, 377–398. MR MR1709300 (2000g:55011)LAGA, Universite´ de Paris XIII, 93430 Villetaneuse, France6

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