CONTENTS. 1. Introduction Organization Acknowledgements Model Structures Homotopy Sheaves 9

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1 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY KIRSTEN WICKELGREN AND BEN WILLIAMS ABSTRACT. We gve a tool for understandng smplcal desuspenson n A 1 -algebrac topology: we show that X Ω(S 1 X) Ω(S 1 X X) s a fber sequence up to homotopy n 2-localzed A 1 algebrac topology for X = (S 1 ) m G q m wth m > 1. It follows that there s an EHP sequence spectral sequence Z (2) π A1 n+1+ (S2n+2m+1 (G m) 2q ) Z (2) π A1,s (S m (G m) q ). CONTENTS 1. Introducton Organzaton Acknowledgements 7 2. Overvew of A 1 homotopy theory Model Structures Homotopy Sheaves Compact Objects and Flasque Model Structures Spectra Long Exact Sequences of Homotopy Sheaves A 1 Unstable and Stable Ponts Localzaton P and A 1 Localzaton P Localzaton of Spectra P and A 1 Localzaton of Spectra The Grothendeck Wtt Group The homotopy of spheres 25 Date: Wednesday, March 15, Mathematcs Subject Classfcaton. Prmary 55Q40, 55Q25, 14F42 Secondary 55S35, 19D45. The frst author was supported by an Amercan Insttute of Mathematcs fve year fellowshp and NSF grants DMS and DMS Some of ths work was done whle the frst author was n resdence at MSRI durng the Sprng 2014 Algebrac Topology semester, supported by NSF grant Further work was done whle both authors were n resdence at the Insttut Mttag- Leffler durng the specal program Algebro-geometrc and homotopcal methods. We thank MSRI and Insttut Mttag-Leffler for the pleasant and productve vsts we enjoyed wth them. 1

2 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY Twst classes The Hlton-Mlnor splttng The low-dmensonal smplces of the J constructon The Stable Isomorphsm The dagonal Combnatorcs Decomposng the second James Hopf map The Stable Weak Equvalence Fber of the James-Hopf map A spectral sequence A functoralty property A cancelaton property A 1 smplcal EHP fber sequence 53 References INTRODUCTION Let Σ denote the suspenson functor from ponted smplcal sets (or topologcal spaces) to tself, defned ΣX := S 1 X. For some maps f : ΣY ΣX, there s a g : Y X such that f = Σg. In ths case, f s sad to desuspend and g s called a desuspenson of f. Under certan condtons, the obstructon to desuspendng f s a generalzed Hopf nvarant, as s proven by the exstence of the EHP sequence (1) X ΩΣX ΩΣ 2 X of James [Jam55] [Jam56a] [Jam56b] [Jam57] and Toda [Tod56] [Tod62] whch nduces a long exact sequence n homotopy groups n a range, see for example [Tod52] or [Wh12, XII Theorem 2.2]. Namely, f desuspends f and only f the generalzed Hopf nvarant H(f) : Y ΩΣY Ωf ΩΣX ΩΣ 2 X s null. Because calculatons can become easer after applyng suspenson, t s useful to have such a systematc tool for studyng desuspenson. By work of James [Jam56a] [Jam56b], t s known that when X s an odd dmensonal sphere, (1) s a fber sequence, and when X s an even dmensonal sphere, (1) s a fber sequence after localzng at 2. In partcular, for any sphere, (1) s a 2-local fber sequence. Snce the suspenson of a sphere s agan a sphere, the correspondng fber sequences for all spheres form an exact couple, thereby defnng the EHP spectral sequence [Mah82]. The EHP spectral sequence s a tool for calculatng unstable homotopy groups of spheres. See for example, the extensve calculatons of Toda n [Tod62].

3 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 3 We provde the analogous tools for ΣX = S 1 X n A 1 -algebrac topology, dentfyng the obstructon to S 1 - desuspenson of a map whose codoman s any sphere wth a generalzed Hopf nvarant, and relatng S 1 - stable homotopy groups of spheres to unstable homotopy groups, after 2-localzaton, by the correspondng EHP specal sequence. We leave the p-localzed sequence for future work. Place ourselves n the settng of A 1 -algebrac topology over a feld [MV99] [Mor12]; let Sm k denote the category of smooth schemes over a perfect feld k, and consder the smplcal model category spre(sm k ) of smplcal presheaves on Sm k wth the A 1 njectve local model structure, whch wll be recalled n Secton 2. Ths model structure can be localzed at a set of prmes P (see [Hor06] and Secton 3) gvng rse to the notaton of a P-local fber sequence up to homotopy. See Defnton 8.1. Defne the notaton S n+qα = (S 1 ) n (G m ) q. Let Ω( ) denote the ponted A 1 -mappng space Map spre(smk ) (S 1, L A 1 ), where L A 1 denotes A 1 fbrant replacement. Theorem 1.1. Let X = S n+qα wth n > 1. There s a 2-local A 1 -fber sequence up to homotopy X ΩΣX ΩΣX 2. Let π A1 denote the th A 1 homotopy sheaf, and more generally defne π A1 +vα (X) to be the sheaf assocated to the presheaf takng a smooth k-scheme U to the A 1 -homotopy classes of maps from S +jα U + to X. The stable A 1 homotopy groups are defned as the colmt π s,a1 +vα (X) = colm r π A1 +r+vα (Σr X). Theorem 1.2. (Smplcal EHP sequence) Choose n, q and v n Z 0 wth n 2. There s a spectral sequence (E r,j, d r : E r,j Er 1,j r ) Z (2) π s,a1 n+(v q)α = Z (2) π A1,s +vα Sn+qα wth E 1,j = Z (2) π A1 j+1++vα (S2j+2n+1+2qα ) f 2n 1 + j and otherwse E 1,j = 0. Choose n > n. There s a spectral sequence (E r,j, d r : E r,j Er 1,j r ) Z (2) π A1 +vα Sn +qα wth E 1,j = Z (2) π A1 j+1++vα (S2j+2n+1+2qα ) f 2n 1 + j and j < n n, and E 1,j = 0 otherwse. Theorem 1.2 follows drectly from Theorem 1.1. Theorem 1.1 s a summary of a more refned theorem, gvng condtons under whch (1) s a fber sequence wthout 2-localzaton. To state ths theorem, let GW(k) denote the Grothendeck-Wtt group of k, and consder the element of GW(k) gven by 1 = (1 + ρη), where η s the motvc Hopf map and ρ = [ 1] n the notaton of [Mor12, Defnton 3.1]. Let K MW denote Mlnor-Wtt K-theory defned [Mor12, Defnton 3.1]. For a set of prmes P, wrte Z P for the rng Z wth formal multplcatve nverses adjoned for all prmes not n P. Theorem 1.3. Let X = S n+qα wth n > 1, and let e = ( 1) n+q 1 q. Let P be a set of prmes. The sequence X ΩΣX ΩΣX 2 s a P-local A 1 -fber sequence up to homotopy f 1 + m(1 + e) are unts n GW(k) Z P for all postve ntegers m. Corollary 1.4. In the settng of Theorem 1.3, the sequence s always a 2-local A 1 -fber sequence up to homotopy. s an A 1 -fber sequence up to homotopy when e = 1 or when n + q s odd and the feld k s not formally real. In partcular, the sequence s an A 1 -fber sequence up to homotopy when n s odd and q s even. when n + q s odd and k = C, or more generally, when n + q s odd and k s any feld such that 2η = 0 n K MW.

4 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 4 Although the statement of Theorem 1.3 s a drect analogue of the correspondng theorem n algebrac topology, the proof gven here s not a straghtforward generalzaton of a proof n algebrac topology. The dffculty s that A 1 -fber sequences are problematc and A 1 -homotopy groups are not necessarly fntely generated. Standard tools lke the Serre spectral sequence are not currently avalable. If a theorem holds for every smplcal set n a functoral manner, t may globalze n the followng sense. Frst, one may be able to obtan n spre a naïve analogue by startng wth smplcal presheaves nstead of smplcal sets, performng correspondng operatons, producng correspondng maps n spre. If the theorem n algebrac topology says that some map s always a weak equvalence (respectvely weak equvalence through a range), t may be mmedate that the correspondng map s a global weak equvalences (respectvely global weak equvalence through a range). If the A 1 -nvarant analogues of the operatons consdered n the theorem are obtaned by applyng L A 1 to the naïve analogue (defned by applyng the operaton n smplcal set to the sectons over each U Sm k ), then the theorem holds n A 1 -algebrac topology. Ths s the case of the Hlton-Mlnor splttng shown below: Theorem 1.5. There s a natural somorphsm n the A 1 -homotopy category. ΣΩΣX Σ n=1 X n Ths s also the case for the statement that for any smplcal presheaf X, the sequence (1) s a fber sequence up to homotopy n the range 3n 2, meanng π A1 3n 2X π A1 3n 1ΣX π A1 3n 1Σ(X 2 )... π A1 X π A1 +1ΣX π A1 +1Σ(X 2 ) π A1 1X... s exact. Ths fact s shown n jont work wth A. Asok and J. Fasel [AFWB14]. Ths s not the case for Theorem 1.1 and Theorem 1.3,.e. these theorems are not proven by globalzng a correspondng result n algebrac topology, where the sequence (1) fals to be exact for X = S n S n. See Example Here s a sketch of the proof of Theorem 1.1; ts purpose s to help the reader understand the proof gven n ths paper, and also to explan the smlartes wth, and dfferences from the stuaton n classcal algebrac topology. Let J(X) denote the free monod on a ponted object X n smplcal presheaves on Sm k, where Sm k denotes smooth schemes over a perfect feld k. In algebrac topology, the free monod on a ponted object s canoncally homotopy equvalent to the loops of the suspenson. It was understood by Faben Morel that the same result holds n A 1 -algebrac topology. Indeed, a result of Morel mples that L A 1J(X) s smplcally equvalent to ΩL A 1ΣX, for X ponted, fbrant and connected. (The phrase smplcally equvalent means weakly equvalent n the njectve Nsnevch local model structure. Here, fbrant means wth respect to ths model structure as well.) We show the versons of ths result that we need n Secton 5. By globalzng a constructon from algebrac topology [Wh12, VII 2], there s a sequence X J(X) J(X 2 ), where X J(X) s the canoncal map nduced from the adjuncton between Σ and Ω, and J(X) J(X 2 ) s the James-Hopf map.e., the above maps exst n A 1 -algebrac topology and the composte map X J(X 2 ) s nullhomotopc (smplcally). Thus there s an nduced map n the homotopy category from X to the P- localzed A 1 -homotopy fber of J(X) J(X 2 ), where P s a set of prmes. Use the notaton h : X F for ths map. Theorems 1.1 and 1.3 say that for X a sphere, h s a P-localzed A 1 -homotopy equvalence for approprate P, and t s proved as follows.

5 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 5 By Theorem 1.5, there s a map of S 1 -spectra b : Σ J(X) Σ X. Usng the tensor structure of spectra over spaces, t follows that there s a map of S 1 -spectra whch fts nto the commutatve dagram c : Σ J(X) Σ (X J(X 2 )) Σ J(X) Σ X J(X 2 ). Σ J(X 2 ) (See Secton 6.4, and, for general X, see Secton 7.2, n partcular, the dscusson followng Constructon 7.11.) The two spaces J(X) and X J(X 2 ) are the same sze n the sense that stably they are both weakly equvalent to Σ n=1 X n. To see ths, note that Σ J(X) = Σ n=1 X n by Theorem 1.5; Σ (X J(X 2 )) = Σ X Σ J(X 2 ) Σ (X J(X 2 )) because the product of two spaces s stably equvalent to the wedge of ther smash wth ther wedge,.e. Σ (X Y) = Σ (X Y X Y). By Theorem 1.5, we have stable weak equvalences J(X 2 ) = n=1 X 2n and X J(X 2 ) = n=1 X 2n+1. These equvalences, when combned wth the prevous, show that stably X J(X 2 ) = n=1 X n. It s not always the case, however, that the stable map c : Σ J(X) Σ (X J(X 2 )) constructed above s a weak equvalence, see Example In algebrac topology, ths map s a weak equvalence for X an odd sphere, and an equvalence after nvertng 2 for X an even sphere. We show an analogous fact n A 1 -algebrac topology, n the followng way. By the Hlton-Mlnor theorem, the map c can be vewed as a matrx, whch tself s the product of matrces correspondng to the dagonal of J(X) and a combnaton of b wth the James-Hopf map J(X) J(X 2 ). Nck Kuhn s calculatons of the stable decomposton of the dagonal of J(X) (see [Kuh01]) and the stable decomposton of the James-Hopf map (see [Kuh87, 6]) n algebrac topology globalze to gve the matrx entres of c n terms of sums of permutatons of smash powers of X. Morel computes that the swap map X X X X s e, and more mportantly, any permutaton σ on X m s equvalent to e sgn(σ) n the homotopy category (see [Mor12, Lemma 3.43]). Snce X s a co-h space, N. Kuhn s results mply that the matrx entres of c are dagonal, and when combned wth Morel s result, we calculate the nth such entry to be ( (2n)! 2 n n! 1)(e + 1) for n even. ( ( (2(n 1))! n+1 1)(e + 1))( 2 n 1 (n 1)! 2 + n 1 2 e) for n odd. Note that (2n)!/(2 n n!) = 1(3)(5) (2n 1) s an odd nteger, so that the nth dagonal term of ths matrx s of the form 1 + m(e + 1), wth m an nteger, for n even, and a product of two such terms for n odd. Note that e 2 = 1 n the homotopy category, because e s the class of the swap. It follows that the product of two terms of the form (1 + m(e + 1)) s also of ths form because (e + 1) 2 = 2(1 + e). Also note that for any postve nteger m, we have that ((m + 1) + me)((m + 1) me) = 2m + 1, whence (m+1)+me s a unt after localzng at 2. It follows that c s a weak equvalence after 2-localzaton. More generally, c s a weak equvalence after P-localzaton whenever all the terms (m + 1) + me are unts n GW(k) Z P. See Proposton Ths produces the correspondng hypothess n Theorem 1.3. We can furthermore characterze exactly when (m + 1) + me s a unt n GW(k) for all m: ether e = 1 or the feld s not formally real and e = 1. See Corollary 4.8.

6 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 6 We are then n the stuaton where we have two P-localzed A 1 -fber sequences (2) F J(X) J(X 2 ) (3) X X J(X 2 ) J(X 2 ), and a stable equvalence between the total spaces whch respects the maps to the base. We would lke to cancel off the base J(X 2 ) to conclude that there s an equvalence between the fbers. Ths s ndeed what we do, however, there are two major obstacles to overcome wth ths approach. The frst s that the standard tool to measure the sze of a fber of a fbraton n terms of the base and total space s the Serre spectral sequence, and at present there no Serre spectral sequence for A 1 -fber sequences. The desred such sequence would use a homology theory lke H A1 (see [Mor12, Defnton 6.29]) because of the need for analogues of the Hurewcz theorem as n [Mor12, Chapter 6.3] to conclude a weak equvalence between the fbers. We use S 1 -stable A 1 homotopy groups on the obvous analogue of the Serre spectral sequence defned by lftng the skeletal fltraton on the base to express the total space as a fltered lmt of cofbratons, and then makng an exact couple by applyng π s,p,a1. Ths gves a spectral sequence even for a global fbraton, but t s not clear that t can be controlled. We provde some of the desred control n Secton 7.2. Assume for smplcty that the base s reduced n the sense that ts 0-skeleton s a sngle pont, as s the case for J(X 2 ). The E 1 -page can be then dentfed wth π s,p,a1 appled to a wedge ndexed by the non-degenerate smplces of the base of the fbraton. Ths wedge constructon takes P-local A 1 -weak equvalences of the fber (respectvely P-local A 1 -weak equvalences n a range) to P-A 1 weak equvalences (respectvely n a range). See Lemmas 7.13, 7.16, and We then show that ths dentfcaton of the E 1 -page s natural wth respect to maps, and even natural wth respect to the stable map c dscussed above. See Lemma Ths dentfcaton of the E 1 -page does not behave well wth respect to weak equvalences of the base, as t nvolves the specfc smplces of the base. It s suffcent here because the map on the base s the dentty. We do not understand the E 2 page. We then have a map of spectral sequences from the spectral sequence assocated to (2) to the spectral sequence assocated to (3). We wsh to use ths map of spectral sequences to show that the stable weak equvalences of the base and total space mply a stable A 1 equvalence of the fbers, after approprately localzng. Then comes the second dffculty. There are nfntely many non-vanshng stable homotopy groups of the fbers n queston, and these groups themselves are not necessarly fntely generated abelan groups. We need to show that there s an somorphsm of these E 1 -pages, but to do ths, we need to allow for the possblty that all terms of both spectral sequences are non-zero non-fntely generated groups. We gve an nductve argument to do ths n Proposton 7.20, and mmedately followng the proposton there s a verbal descrpton of what happened. The strategy of ths proof of the motvc EHP sequence s modeled on the proof of the EHP sequence gven n Mchael Hopkns s stable homotopy course at Harvard Unversty n the fall of Hopkns credts ths proof to James [Jam55] [Jam56b] together wth some deas of Ganea [Gan68]. In ths argument, the orgnal Serre spectral sequence s used; there s no need to work n spectra, as calculatons n (co)homology suffce. Snce the (co)homology of spheres n algebrac topology s concentrated n two degrees, there s no analogue of Proposton It s also possble to compute the frst dfferental n the EHP sequence of Theorem 1.2, and ths computaton wll be made avalable n a jont paper wth Asok, Fasel and the present authors [AFWB14]. Computatons of unstable motvc homotopy groups of spheres can be appled to classcal problems n the theory of projectve modules, for example to the problem of determnng when algebrac vector bundles

7 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 7 decompose as a drect sum of algebrac vector bundles of smaller rank. See [Mor12, Chapter 8], [AF14b], and [AF14a]. In a dfferent drecton, t can be shown that there s an A 1 weak equvalence Σ(P 1 {0, 1, }) = Σ(G m G m ) between the S 1 suspensons of P 1 {0, 1, } and G m G m. By comparng the actons of the absolute Galos group on geometrc étale fundamental groups, t can be shown that ths weak equvalence does not desuspend [Wc15]. Because the acton of the absolute Galos group on πét 1 (P1 {0, 1, }) s both ted Q to nterestng mathematcs [Iha91] and obstructs desuspenson, t s potentally also of nterest to have systematc tools lke those provded by the EHP sequence to study the obstructons to desuspenson Organzaton. The organzaton of ths paper s as follows: Theorem 1.2 s proven n Secton 8 as Theorems 8.5 and 8.6. Theorem 1.1 s proven n Secton 8 as Theorem 8.3. The core of these arguments s the cancelaton property of Secton 7.3. The substtute for the Serre spectral sequence s developed n Secton 7. In Secton 6, the motvc James-Hopf map and the dagonal of the James constructon are computed stably as matrces wth entres n GW(k). Secton 5 proves the Hlton-Mlnor splttng. Secton 4 gves results on the Grothendeck Wtt group that are needed to understand when the matrces computated n Secton 6 are nvertble. Secton 3 provdes needed results on localzatons of spre(sm k ) and Spt(Sm k ), and Secton 2 ntroduces the needed notaton and background on A 1 -homotopy theory Acknowledgements. We wsh to strongly thank Aravnd Asok and Jean Fasel for ther generosty n sharng wth us unpublshed notes about the James constructon n A 1 algebrac topology. The frst author wshes to thank Mchael Hopkns for hs stable homotopy course n the Fall of 2012, and the entre homotopy theory communty n Cambrdge Massachusetts for ther energy, enthusasm, and perspcacty. We are also pleased to thank Emly Rehl for help usng smplcal model categores, and Aravnd Asok, Jean Fasel, Davd Gepner, Danel Isaksen, and Kyle Ormbsy for useful dscussons. We also thank an anonymous referee for useful comments, n partcular for pontng out a gap n a prevous verson of the proof of Proposton OVERVIEW OF A 1 HOMOTOPY THEORY In the sequel, we wll have to draw on many results regardng A 1 homotopy. We collect those results n ths secton for ease of reference. We make no clam that any of these results are orgnal. Let k be a feld such that the results of [Mor12] hold over Spec k. It suffces that k be perfect, but we hope that n future ths requrement may be weakened. Let Sm k denote a small category equvalent to the category of smooth, fnte type k-schemes. The category spre(sm k ) s the category of smplcal presheaves on Sm k, and spre(sm k ) the category of ponted smplcal presheaves. The category Sm k s consdered embedded n spre(sm k ) va the Yoneda embeddng. The termnal object of Sm k and spre(sm k ) s therefore Spec k, whch s also denoted by k and dependng on the context. The notaton Map(X, Y) denotes the nternal mappng object where t appears, generally n spre(sm k ). Many categores appearng n the sequel are smplcally enrched, and n them SMap(X, Y) wll denote a smplcal mappng object. Where there s a model structure, a, present we wll use the notaton [X, Y] a to denote the set of maps n the homotopy category from X to Y. The notaton [X, Y] wll be used when a s clear from the context. If K s a smplcal set, then we wrte K for the set of smplces n K.

8 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY Model Structures. Ths paper makes use of two famles of model structure on the category spre(sm k ) and ts descendants. In the frst place, the local njectve model structure of [Jar87] ntroduced there as the global model structure and the local flasque model structure of [Isa05]. Our use of these terms follows [Isa05]. These model structures are Qullen equvalent. Each gves rse to descendent model structures by A 1 or P localzaton or by stablzaton. The flasque model structures are employed only to prove techncal results regardng spectra; when flasque s not specfed, t s to be understood that the njectve structures are meant. The weak equvalences n the njectve local and the flasque local model structures are the local weak equvalences those maps that nduce somorphsms on homotopy sheaves, properly defned: [Jar87]. In the semnal work [MV99], these maps are called smplcal weak equvalences n order to emphasse ther non-algebrac character. Both the njectve and the flasque local model structures are left Bousfeld localzatons of global model structures on spre(sm k ); a global model structure beng one where the weak equvalences are those maps φ : X Y that nduce weak equvalences φ(u) : X(U) Y(U) for all objects U of Sm k. Both the global njectve and the global flasque model structures are left proper, smplcal, cellular, see [Hor06], and combnatoral so that left Bousfeld localzatons of ether at any set of morphsms exst and are agan left proper, smplcal and cellular. In the njectve model structures all objects are cofbrant, and therefore these model structures are tractable n the sense of [Bar10]. The njectve global model structure s symmetrc monodal, the structure beng gven by and the usual nternal mappng object, ths beng the evdent extenson of the symmetrc monodal structure on sset. By [Bar10, 4.46], any left Bousfeld localzaton, a, of the njectve local model structure wll nhert the structure of a module over the njectve local model structure. In partcular, any object X of spre(sm k ) gves rse to a left Qullen functor X : spre(sm k ) spre(sm k ) where spre(sm k ) s endowed wth the structure a. Snce X preserves trval cofbratons and all objects are cofbrant, by Ken Brown s lemma, [Hov99, Lemma ], t preserves weak equvalences. Lemma 2.1. Suppose a s a left Bousfeld localzaton of the njectve global model structure on spre(sm k ), and suppose X s an object of spre(sm k ). The functor X preserves a weak equvalences. These model structures all have ponted analogs, and a standard argument allows us to deduce: Corollary 2.2. Suppose a s a left Bousfeld localzaton of the njectve global model structure on spre(sm k ) and a s the assocated model structure on spre(sm k ). Suppose X s an object of spre(sm k ). The functor X preserves a weak equvalences. Proof. Let f : Z Y be a a weak equvalence. Because a s a smplcal model category n whch all objects are cofbrant and monomorphsms are cofbratons, t follows from [Re14, Corollary ] that d X f : X Z X Y s a a weak equvalence.

9 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 9 By Lemma 2.1, X f : X Z X Y s a a weak equvalence. Note that d X f, d, X f, and X f determne a map of push-out squares as n the commutatve dagram X Z X Z X Z X Y X Y X Y Furthermore, X Z X Z and X Y X Y are cofbratons because they are monomorphsms. It now follows from [Re14, Corollary ] that X f : X Z X Y s a a weak equvalence as clamed Homotopy Sheaves. If X s an object of spre(sm k ) or spre(sm k ), we wrte L Ns X for a functoral fbrant replacement n the local njectve model structure, and L fl Ns n the local flasque model structure. We wrte L A 1 or L fl A for a functoral fbrant replacement n the approprate A 1 model structures. 1 Snce the purpose of ths paper s to establsh some denttes regardng A 1 homotopy sheaves, t behoves us to defne what a homotopy sheaf means n the sequel. The followng defntons date at least to [Jar87]. Defnton 2.3. If X s an object of spre(sm k ), then we defne π pre 0 (X) as the presheaf U π 0 ( X(U) ) where U s an object of Sm k, and where X(U) ndcates a geometrc realzaton of X(U). We defne π 0 (X) as the assocated Nsnevch sheaf to π pre 0 (X). Proposton 2.4. If X s an object of spre(sm k ) such that X(U) s a fbrant smplcal set for all objects U of Sm k, then π 0 (X) s the sheaf assocated to the presheaf coequalzer: ( ) d 1 U coeq X(U) 1 X(U) 0. d 0 The condton that X(U) be a fbrant smplcal set, to wt a Kan complex, for all U wll be satsfed f X s a fbrant object n a model structure for whch Λ n d U n U s a trval cofbraton. Ths ncludes all local and flasque model structures. Defnton 2.5. If X s an object of spre(sm k ), wth basepont x 0 X, then we defne π pre (X, x 0 ) for 1 as the presheaf U π ( X(U), x 0 ) where U s an object of Sm k, and where x 0, n an abuse of notaton, ndcates the basepont of X(U) nduced by x 0 X. We defne π (X, x 0 ) as the assocated Nsnevch sheaf. The basepont x 0 wll generally be understood and omtted. The reader s remnded that X(U) may have connected components that do not appear n the global sectons, X( ). In ths case, the sheaves of groups π (X, x 0 ) as defned above are nsuffcent to descrbe the homotopy theory of X.

10 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 10 It s the case that the functor π ( ) takes smplcal weak equvalences to somorphsms, and π A1 0 weak equvalences to somorphsms. takes A1 If X s an object of spre(sm k ), we reserve the notaton Ω X for the derved loop space Map (S n, L Ns X). In partcular, Ω 0 X = L Ns X. We rely on the followng result throughout. Proposton 2.6. Equp spre(sm k ) wth the Nsnevch local model structure. If X s an object of spre(sm k ), and f 0, then π (X) s the sheaf assocated to the presheaf U [Σ (U + ), X]. Proof. By applyng a functoral fbrant replacement functor f necessary, we may assume that X(U) s a fbrant smplcal set for all objects U of Sm k. In the followng sequence of somorphsms, all homotopy groups consdered are smplcal homotopy groups of fbrant smplcal sets [S U +, X] = π0 (SMap (S U +, X)) = π 0 (SMap (S, Map (U +, X))) = π 0 (SMap (S, Map(U, X))) = π 0 (sset (S, Map(U, X)( ))) = π 0 (sset (S, X(U))) = π (X(U)), as requred. Corollary 2.7. If X s an object of spre(sm k ), and f 0, then π (X) = π 0 (Ω X). Proof. The result follows from the proposton and the adjuncton [Σ (U + ), X] = [U+, Ω X]. Snce takng global sectons, X X( ), s takng a stalk, we also have the followng corollary. Corollary 2.8. If X s an object of spre(sm k ), and f 0, then π (X)( ) = [S, X]. If, j 0 we defne S +jα = S G j m, where G m s ponted at the ratonal pont 1. If j 0 and X s an object of spre(sm k ) +, we defne π A1 +jα (X) as π (Map (Gm j, L A 1X)); t s somorphc to the sheaf assocated to the presheaf U [S +jα U +, X] A 1. Takng global sectons, we have π A1 +jα(x)(k) = [S +jα, X] A Compact Objects and Flasque Model Structures. We say that an object X of spre(sm k ) s compact f colm Map (X, F ) = Map (X, colm F ) whenever F s a fltered system n spre(sm k ), and smlarly for spre(sm k ). An argument smlar to that of [DI05, Lemma 9.13] shows that ponted smooth schemes are compact, and t s easy to see that fnte

11 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 11 constant smplcal presheaves are compact. If A, B are ponted compact objects, then A B s compact, and all fnte colmts of compact objects are agan compact. We shall frequently make use of the followng. Proposton 2.9. Let a be a left Bousfeld localzaton of the global njectve model structure on spre(sm k ) (resp. spre(sm k ) ). Let I be a fltered small category and X I an I-shaped dagram n spre(sm k ) (resp. spre(sm k ) ). Then the natural map hocolm X I colm X I s a global weak equvalence. Proof. We descrbe the case of spre(sm k ), that of spre(sm k ) s smlar. By constructon, [Hr03, Chapter 18], (hocolm X I )(U) = hocolm(x I (U)) for all U Sm k, and smlarly for colm. The result then follows from the classcal fact that the natural map hocolm X I (U) colm X I (U) s a weak equvalence, [BK72, XII.3.5]. Proposton If X 0 X 1... s a sequental dagram n spre(sm k ), then the natural map of sheaves s an somorphsm. colm π 0 (X ) π 0 (colm X ) Proof. By Proposton 2.9, we may replace {X } by a naturally weakly equvalent dagram wthout changng the homotopy type of colm X. The group colm π 0 (X ) s also unchanged by such a procedure. We can therefore assume that X (U) s a fbrant smplcal set for all objects U of Sm k. Snce, accordng to Proposton 2.4, π 0 (Y) s the sheaf assocated to a coequalzer of presheaves, provded Y takes values n fbrant smplcal sets, the result follows by commutng colmts. The njectve local model structure on spre(sm k ) suffers from a techncal drawback when one wshes to calculate wth fltered colmts, whch s that fltered colmts of fbrant objects are not necessarly fbrant themselves. Ths s the problem that motvates the constructon of the flasque model structures of [Isa05], and one can see the presence of flasque or flasque-lke condtons appearng often throughout the lterature when calculatons wth fltered colmts are beng carred out, see [Jar00], [DI05], [Mor05]. We therefore consder two flasque model structures on spre(sm k ): the local flasque structure n whch the weak equvalences are the smplcal weak equvalences, and the A 1 flasque structure n whch the weak equvalences are the A 1 weak equvalences. These model structures apply also to spre(sm k ). These model structures are smplcal, proper and cellular, and the A 1 structures are left Bousfeld localzatons of the local model structure. There s a square of Qullen adjunctons (4) Injectve Local Flasque Local Injectve A 1 Flasque A 1 where the arrows ndcate the left adjonts, and each arrow s the dentty functor on spre(sm k ). The horzontal arrows represent Qullen equvalences. A smlar dagram obtans n spre(sm k ). Not all objects are cofbrant n spre(sm k ) or spre(sm k ) n the flasque model structures, n contrast to the case of the njectve structures. Snce the A 1 flasque structures are left Bousfeld localzatons of the local flasque structures, the cofbrant objects n one model structure agree wth the cofbrant objects n the other. The results of [Isa05], specfcally Lemmas 3.13, 6.2, show that all ponted smplcal sets and all quotents X/Y of monomorphsms Y X n Sm k are flasque cofbrant n spre(sm k ). Ths ncludes all smooth

12 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 12 schemes ponted at a ratonal pont. Lemma 3.14 of [Isa05] shows that fnte smash products of flasque cofbrant objects are agan flasque cofbrant n spre(sm k ). Proposton If F s a fltered dagram of objects of spre(sm k ), and f X s a compact and flasque cofbrant object of spre(sm k ), then there s a zgzag of local (resp. A 1 ) weak equvalences: (5) colm Map (X, RF ) Map (X, R colm RF ) Map (X, R colm F ), where R denotes an njectve local (resp. njectve A 1 ) functoral fbrant replacement. Proof. By Proposton 2.9, the local (resp. A 1 ) homotopy type of a fltered colmt s nvarant under termwse replacement by locally (resp. A 1 ) equvalent objects. Fltered colmts of flasque fbrant objects are agan flasque fbrant, see [Isa05]. The objects RF are flasque fbrant, so the colmt colm RF s flasque fbrant, as s R colm RF. There s a global weak equvalence colm RF R colm RF. Snce R preserves weak equvalences, we also have R colm RF R colm F. Snce the object X s flasque cofbrant, the functor Map (X, ) preserves trval flasque fbratons, and by Ken Brown s lemma, weak equvalences between flasque fbrant objects. The map Map (X, colm RF ) Map (X, R colm F ) s therefore a weak equvalence. The result now follows from the compactness of X. Corollary If F r s a fltered system of objects of spre(sm k ), and f, j 0 are ntegers, then there are natural somorphsms of sheaves π (colm F r ) = colm π (F r ) r r and π A1 +jα(colm F r ) = colm π A1 r r +jα(f r ). Proof. Combne Corollary 2.7 and Propostons 2.10, 2.11, notng that the objects S +jα are compact and flasque cofbrant. We warn the reader that π A1 +jα (Ωr X) dffers from π A1 +r+jα (X) n general, [Mor12, Theorem 6.46] Spectra. We take [Hov01] as our man reference for the theory of spectra n model structures such as those we consder here. We shall requre only naïve spectra, rather than symmetrc spectra. For us a spectrum, E, shall be an S 1 spectrum, consstng of a sequence {E } =0 of objects of spre(sm k), equpped wth bondng maps σ : ΣE E +1. The maps of spectra E E beng defned as levelwse maps E E whch furthermore commute wth the bondng maps, we have a category of presheaves of spectra, whch we denote by Spt(Sm k ). Just as we have two notons of weak equvalence on spre(sm k ), the local and the A 1, we shall have two knds of weak equvalence between objects of Spt(Sm k ), the stable and the A 1. There s a set, I n the notaton of [Isa05], of generatng cofbratons for whch the domans and codomans all posses the property that we call compact, whch [Isa05] calls ω small and whch s stronger than the property that [Hov01] calls fntely presented. Moreover, both model structures are localzatons of an objectwse flasque model structure havng a set, J n the notaton of [Isa05], whch agan conssts of maps havng fntely-presented domans and codomans. By the arguments of [Hov01, Secton 4], these model structures are almost fntely generated. The theory of [Hov01, Secton 3] establshes a stable model structure on Spt(Sm k ) based on any cellular, left proper model structure, a, on spre(sm k ). In partcular, ths apples when a s a left Bousfeld localzaton

13 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 13 of the global njectve or global flasque model structure, and therefore when t s one of the four structures of (4). The results of [Hov01, Secton 5] ensure that we have Qullen adjunctons and equvalences between these model structures: (6) Stable Injectve Local Stable Flasque Local Stable Injectve A 1 Stable Flasque A 1. Snce the functors of (4) are the dentty functors, the same s true of the functors of (6); only the model structure vares. We wrte stable weak equvalence for the weak equvalences of the stable njectve local and stable flasque local model structures, and stable A 1 equvalence for the weak equvalences of the stable njectve A 1 and the stable flasque A 1 model structures. In keepng wth our conventon, we wrte L A 1E to denote a fbrant replacement of E n the stable A 1 model structures. Snce the underlyng unstable model structures are proper, we may apply fbrant-replacement functors levelwse to objects n Spt(Sm k ) to obtan maps of spectra: E RE gven by E RE, the fbrant replacement n any one of the four unstable model structures under consderaton. There s also a spectrum-level nfnte loop space functor, Θ that takes a spectrum E to the spectrum havng -th space (Θ E) = colm k Map (Sk, E +k ). Proposton A map f : E E of Spt(Sm k ) s a stable weak equvalences (resp. a stable A 1 equvalence) f and only f Θ (Rf) : (Θ RE) (Θ RE ) s a weak equvalence for all, where R represents the flasque local fbrant replacement functor (resp. flasque A 1 fbrant replacement functor). Proof. Ths s a specal case of [Hov01, Theorem 4.12]. The ancllary hypotheses gven there, that sequental colmts n commute wth fnte products and that Map (S 1, ) commutes wth sequental lmts, are satsfed n spre(sm k ). One can verfy that a spectrum E s weakly equvalent to the spectrum one obtans from E by replacng each space E by the connected component of the basepont n E. We may therefore assume that E s connected, meanng we do not have to worry about the problem of non-globally-defned components. For any nteger, there s an adjuncton of categores (7) Σ : spre(sm k ) Spt(Sm k ) : Ev where the spectrum Σ X s the spectrum the j-th space of whch s Σ j X f j, and otherwse, and where the bondng maps are the evdent ones. The rght adjont Ev takes E to E. Proposton Suppose a s a left Bousfeld localzaton of ether the global njectve or the global flasque model structure on spre(sm k ). Then the adjont functors of (7) form a Qullen par between the ponted model structure on spre(sm k ) and the stable model structure on Spt(Sm k ) nduced by a. Proof. Ths follows from Defnton 1.2, Proposton 1.15 and Defnton 3.3 of [Hov01]. The left-derved functor of Ev 0 n the flasque model structures are the functors Ω : {E n } n colm Ω k RE n+k k

14 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 14 where R s ether L fl Ns or Lfl A 1, as approprate. The smash product on spre(sm k ) extends to an acton of spre(sm k ) on Spt(Sm k ), and the functors Σ preserve ths structure, and n partcular are smplcal functors, [Hov01, Secton 6]. For an object E of Spt(Sm k ), we defne the stable homotopy sheaves π s (E) as the colmt π s (E) = colm r π +r(e r ). Snce π +r (E r ) s the sheaf assocated to the presheaf U π +r (E r (U)), and sheaffcaton commutes wth colmts, t follows that π s (E) may also be descrbed as the sheaf assocated to the presheaf U π s (E(U)) where the stable homotopy group s the ordnary stable homotopy group of smplcal spectra. Ths defnton of the sheaf π s s used n [Mor05]. We smlarly defne the stable A 1 homotopy sheaves π s,a1 +jα (E) as the colmt π s,a1 +jα(e) = colm r πa1 +r+jα(e r ). Proposton Let f : E E be a map n Spt(Sm k ). Then (1) f s a stable weak equvalence f and only f π s (f) s an somorphsm for all ; (2) f s an A 1 stable weak equvalence f and only f π s,a1 (f) s an somorphsm for all. Proof. The map f : E E s a stable weak equvalence f and only f the maps (Θ L fl Ns E) (Θ L fl Ns E ) are smplcal weak equvalences for all. The space (Θ L fl Ns E) j s and ts + j-th homotopy sheaf s, by Corollary 2.12, The result for π s follows. colm r Ωr (L fl NsE j+r ) π +j (colm r Ωr (L fl NsE j+r )) = colm j+r π +j+r(e j+r ) = π s (E). For A 1 equvalence, the same argument apples mutats mutands. Wrtng L fl A for the flasque A 1 fbrant 1 replacement functor, we see that the + j-th homotopy sheaf of the j-th level of the A 1 stable fbrant replacement Θ (L fl A E) s π 1 +j (colm r Ω r (L fl A E 1 j+r )) whch smplfes to π s (Lfl A E) = π s,a1 1 (E) Ths proposton says that the defnton of stable weak equvalence used n ths paper agrees wth that of [Mor05]. Proposton For any object E of Spt(Sm k ) and any nonnegatve ntegers, and j, (1) The sheaf assocated to the presheaf s π s (E) (2) The sheaf assocated to the presheaf s π s,a1 +jα (E). U [Σ (S U + ), E] U [Σ (S +jα U + ), E] A 1 Proof. We prove the frst statement.

15 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 15 The gven presheaf, by adjuncton, s U [S U +, Ev E], where the functor Ev s a derved functor n the flasque stable model structure. By reference to [Hov01], we wrte ths presheaf more explctly as whch s assocated to the sheaf as asserted. U [S U +, colm r Ωr L fl NsE +r] π (colm r Ωr L fl NsE +r) = colm r π +( +r)(e +r) = π s (E), The proof of the second statement s smlar, wth the provso that L fl Ns s replaced by Lfl A 1, and one concludes that the sheaf beng represented t π s (Map (G j m, L fl A 1 E)) = π s,a1 +jα (E). Corollary For any, j 0, and any object E of Spt(Sm k ), takng global sectons gves and π s (E)( ) = [Σ S, E] π s,a1 +jα(e)( ) = [Σ S +jα, E] A 1. Proposton Suppose {E n } s a fltered system of objects n Spt(Sm k ). Then the natural maps and are somorphsms. colm π s (E ) π s (colm E ) n n colm π s,a1 n +jα(e ) π s,a1 +jα(colm E ) n Proof. These follow from Corollary 2.12 and the observatons that takng colmts commute and that colmts of spectra are calculated termwse Long Exact Sequences of Homotopy Sheaves. We wll use the term cofber sequence only n a lmted sense: a cofber sequence n a ponted model category M s a sequence of maps X Y Z such that X Y s a cofbraton of cofbrant spaces and Z s a categorcal pushout of X Y. A fber sequence s dual. The mage of a cofber sequence n ho M may also be called a cofber sequence, as n [Hov99, Chapter 6]. The noton of fber sequence s dual. The derved functors of left-qullen functors preserve cofber sequences, and dually the derved functors of rght-qullen functors preserve fber sequences. Suppose a s a model structure on spre(sm k ), obtaned as a left Bousfeld localzaton of the flasqueor njectve-local model structure. Consder Spt(Sm k ), endowed wth the stable model structure derved from a, [Hov01, Secton 3]. By [Hov01, Theorem 3.9], the homotopy category ho a Spt(Sm k ) s a trangulated category n the sense of [Hov99, Chapter 7.1]. Wrte π s,a +jα (E) for the sheaf assocated to the presheaf U [S G j m U +, E] sa, where the set of maps s calculated n ho a (Spt(Sm k )). Then we have the followng result as an mmedate corollary of sheaffyng Lemma of [Hov99]. Proposton If X Y Z s a cofber sequence n ho a (Spt(Sm k )), then the nduced sequence of homotopy sheaves π s,a +jα(x) π s,a +jα(y) π s,a +jα(z) π s,a 1+jα(X) s an exact sequence of sheaves of abelan groups.

16 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 16 Examples nclude π s, πs,a1 +jα as well as πs,p and π s,p,a1 +jα of Secton A 1 Unstable and Stable. We say that a spectrum E s A 1 n connected f π s,a1 (E) = 0 for all n. From the above defnton of π s,a1, combned wth [Mor12, Theorem 6.38] sayng that L A 1 does not decrease the connectvty of connected objects, and that L A 1 commutes wth Ω for smply-connected objects, we deduce the followng lemma: Lemma If X s an A 1 n connected object of spre(sm k ), then Σ X s A 1 n connected. Recall that a map f : X Y of connected objects of spre(sm k ) s sad to be n connected f the homotopy fber s (n 1)-connected, and A 1 n-connected f the A 1 -homotopy fber s A 1 (n 1)-connected. By use of [Mor12, Theorem 6.53, Lemma 6.54] and the A 1 connectvty theorem, we deduce that f X Y s n-connected wth n 1 and f moreover π 1 (Y) s strongly A 1 nvarant, then X Y s A 1 n-connected. These condtons hold when X s smply connected, or when n 2 and X s A 1 local. The followng result s due to Asok Fasel, [AF13] Proposton 2.21 (The Blakers Massey Theorem of Asok Fasel). Suppose f : X Y s an A 1 n-connected map of connected objects n spre(sm k ) and X s A 1 m-connected, wth n, m 1, then the morphsm hofb A 1 f ΩL A 1 hocofb f s m + n-connected. Proof. We rely on a homotopy excson result, a consequence of the Blakers Massey theorem, that says that the result of ths proposton holds n the settng of classcal topology, [Wh12, VII Theorem 7.12]. We may replace f : X Y by an equvalent A 1 -fbraton of A 1 -fbrant objects wthout changng the A 1 homotopy type of hofb A 1 f or of hocofb f. The A 1 homotopy fber of f therefore agrees wth the ordnary fber and therefore also wth the smplcal homotopy fber. The classcal homotopy excson result, appled at ponts, now says that the map hofb f Ω hocofb f s smplcally (m + n)-connected. Snce m + n 2, and hofb f = hofb A 1 f s A 1 -local, t follows that π 1 (Ω hocofb f) s strongly A 1 -nvarant and then by [Mor12, Theorem 6.56] t follows that s (m + n)-connected. hofb A 1 f L A 1 hofb A 1 f L A 1Ω hocofb f The connectvty hypotheses mply that π 1 (Y) = π A1 1 (Y) s trval, and therefore by the van Kampen theorem, that hocofb f s smply connected. Ths mples by [Mor12, Theorem 6.46] that L A 1Ω hocofb f ΩL A 1 hocofb. Ths completes the proof. Corollary Suppose f : X Y s a map of A 1 smply connected objects n spre(sm k ) such that the homotopy cofber hocofb f s A 1 contractble. Then f s an A 1 weak equvalence. Proof. We show by nducton that hofb A 1 f s arbtrarly hghly connected. Snce X and Y are smply connected, hofb A 1 f s 0-connected, so f s 1-connected. Suppose we know that hofb A 1 f s d-connected, then applyng Proposton 2.21 wth n = d + 1 and m = 1, we deduce that hofb A 1 f Ω hocofb f s A 1 (d + 2)-connected, so that π d+1 (hofb A 1 f) s trval. Corollary Suppose f : X Y s a map of A 1 smply connected objects n spre(sm k ) such that Σ f : Σ X Σ Y s an A 1 -weak equvalence, then f s an A 1 weak equvalence.

17 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 17 Proof. We may replace f by a fbraton of A 1 -fbrant objects. The map f s necessarly 1-connected, and from the proposton we deduce that π 1 (hocofb f) π 0 (hofb f), whch s trval. Snce Σ s a left Qullen functor, t preserves cofber sequences n the derved category, and we deduce that Σ hocofb f s A 1 contractble. Snce hocofb f s smply connected, the A 1 Hurewcz theorem mples that hocofb f s A 1 contractble. An appeal to Corollary 2.22 now completes the argument Ponts. The ste Sh Ns (Sm k ) s well known to have enough ponts. Let Q be a conservatve set of ponts of Sh Ns (Sm k ). For each element q Q, there s an adjuncton of categores q : Sh Ns (Sm k ) Set : q, where q, as well as preservng all colmts, preserves fnte lmts. There s a Qullen adjuncton q : spre(sm k ) sset : q from the njectve local model structure on spre(sm k ) to the usual model structure on sset. Ths extends n the obvous way to the ponted model categores, and to the categores of spectra q : Spt(Sm k ) Spt : q. For an object X of spre(sm k ), there s, by reference to 2.4, an somorphsm q π 0 (X) = π 0 (q X). It s also the case that p Ω (L Ns X) Ω (Ex p X). Ths gves us the followng proposton Proposton If X s an object of spre(sm k ) and s a postve nteger and q a pont of Sh Ns (Sm k ), then there s an somorphsm of groups π ( q X ) = q π (X). Corollary If X s an object of Spt(Sm k ), and f s an nteger, then there s an somorphsm of abelan groups π s (q X) = q π s (X). These facts are specal cases of results concernng -topo, [Lur09, ]. They are well-known, see for nstance [Mor05, 2.2 p14], but seldom stated. 3. LOCALIZATION Let P denote a nonempty set of prme deals of Z, and P = (p) P (Z \ (p)) the set of ntegers not lyng n any of these deals. We wrte Z P for the localzaton (P ) 1 Z, and Z (p) n the case where P = {(p)} conssts of a sngle deal. Followng [CP93], where the followng s carred out n the category of CW complexes, we defne S 1 τ = S 1, a Kan complex equvalent to 1 / 1, and S n τ = S 1 τ ( ) k 1 / k 1 +. For any nteger n, defne ρ 1 n : S 1 τ S 1 τ to be the usual degree-n self-map of S 1, and extend ths to maps ρ k n : S k τ S k τ by ρ k n = ρ 1 n d. Defne T P to be the set of maps T P = {ρ k n : k 1, n P }. For each map ρ k n : S k τ S k τ n T P and each object U of Sm k, we may defne a self-map ρ k n d U of S k τ U. Denote the set of such maps by T P. The local njectve and flasque model structures on spre(sm k ) are cellular n the sense of Hrschhorn, [Hr03]; a proof for the njectve case appears n [Hor06, Lemma 1.5] and the flasque case s treated n [Isa05]. We may therefore apply the general machnery of [Hr03] and left-bousfeld-localze spre(sm k ) at the set T P. We call the resultng model structures P local, and f P = {(p)} we call the resultng model structures

18 THE SIMPLICIAL EHP SEQUENCE IN A 1 ALGEBRAIC TOPOLOGY 18 p local. Wrte L P for the functoral fbrant replacement functor n each model category. In the case where P = {(p)}, we may wrte L (p). The localzaton of the usual model structure on sset wth respect to the set T P of maps s a form of P local model structure on sset, we refer the reader to [CP93], especally [CP93, Secton 8], for the comparsons between dfferent P localzatons n classcal topology and for a dscusson of non-nlpotent objects. For nlpotent objects, the varous P localzaton functors agree up to weak equvalence. Lemma 3.1. Wth notaton as above, f s s a pont of Sh Ns (Sm k ), the adjunctons and s : spre(sm k ) s : spre(sm k ) sset : s sset : s are monodal Qullen adjunctons between the P local model categores, where spre(sm k ) and spre(sm k ) may be gven ether the flasque or the njectve model structure. Proof. It s suffcent to prove the unponted cases, the ponted follow mmedately. The proofs n the flasque and njectve cases are the same. Followng [Hr03, Theorem ], the adjont par s : spre(sm k ) sset : s s a Qullen adjuncton between the P local model structure on the left and the model structure on sset obtaned by localzaton at the set of maps s (ρ k n d U ) : s (S k τ U) s (S k τ U) where ρ k n T P. Denote ths set of maps by s T P. It wll suffce to show that localzaton of sset at s T P agrees wth localzaton of sset at T P. Snce evaluaton at s commutes wth fber products, the maps of s T P maps are of the form ρk n d s U, and settng U =, we see that T P s T P. The maps of s T P are, moreover, weak equvalences n the localzaton of sset at T P. It follows that the localzaton of sset at s T P s smply the ordnary P localzaton of sset. We note n addton that the model categores appearng above are smplcal model categores, and the adjunctons appearng are adjunctons of smplcal model categores n the sense of [Hov99, Chapter 4.2]. We contnue to work prncpally n the njectve local not-localzed-at-p model structures, but wrte A P B to ndcate that A s weakly equvalent to B n the P local structure, or equvalently that L P A L P B. The notaton A (p) B wll be used where approprate. We wll use the flasque model structures only when dealng wth spectra. In ths secton we wll occasonally wrte groups π (X) n multplcatve notaton even when the groups are abelan. The n th power map of a group G wll be the map x x n, whch s necessarly a homomorphsm f G s abelan, and s preserved by group homomorphsms n any case. If P s a set of prmes, then a group G s sad to be P local f the n th power map s a bjecton on G whenever n s not dvsble by any of the prmes n P. We wll say that a presheaf of groups s P local f all groups of sectons are P local, and a sheaf of groups s P local f the approprate n th power maps are somorphsms of sheaves of sets. Proposton 3.2. If X s a connected object of spre(sm k ), and P s a set of prmes, then the sheaves π (L P X) are P local sheaves of groups.