Classification of Rational Homotopy Type for 8-Cohomological Dimension Elliptic Spaces

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1 Aves Pure Mthemts Publshe Ole Jury 0 ( Clssfto of Rtol Homotopy Type for -Cohomologl Dmeso Ellpt Spes Mohme Rh Hll Hss Lme My Isml Mmou Fulté es Sees Aï Cho Csbl Moroo Cetre Pégogque Régol Rbt Moroo Eml: {rhll hlmee}@hotmlom mmoumysml@gmlom Reeve September 0; revse November 0; epte November 5 0 ABSTRACT The fferet methos use to lssfy rtol homotopy types of mfols re geerl fstg vrous (see []) I ths pper we re tereste to prtulr se tht of smply oete ellpt spes eote X by s- m H X; X 0 ussg ts ohomologl meso Here we wll the suss the se whe Keywors: Rtol Homotopy Theory; Ellpt Spes; Clssfto; Rtol Homotopy Type; Mml Moel of Sullv Itrouto Let us frst rell some bs eftos of rtol homotopy theory A smply oete spe X s lle ellpt f both of H X; π X re fte meso tht ts ohomologl Euler-Por hr terst s gve s X: m H X; 0 We wll fx ths throughout ths pper The spe s lle rtol f π X s -vetor spe If t s ot by [] we ssote rtol smply oete spe eote verfyg X ; H X X H X X slgebrs π svetorspes The rtol homotopy type of X s efe s the homotopy type of ts rtolzto X Our purpose ths pper to gve omplete lssfto ths rtol m H X; X 0 homotopy type whe Prelmres The rtol homotopy theory ws foue the the e of the sxtes by Del Qulle Des Sullv Oe of the tehl gget of ths theory s the mml moel of Sullv t s free -ommuttve fferetl gre lgebr V ssote to y smply oete CW omplex X of fte type [] Here V s -gre vetor spe wth m V V < eomposble fferetl; tht mes V V ( oes ot hve ler prt) tht Correspog uthor 0 It s well ow tht the mml moel V etermes the rtol homotopy type of X the sese tht X V H X; H V slgebrs π svetorspes For exmple the mml moel of eve sphere s of the form x y wth x y x 0 y x H ; x x whle the mml moel of sphere s of the form x y wth y y 0 It wll be utle for our proofs to rell the reer ths smple propertes For homogeeous elemet x of V x eotes ts egree whh verfes the followg: xy x y y x; xy x y x xy (Lebz formul) I prtulr x 0 whe x s xy yx whe x s eve X : mv s lle the homotop 0 Euler-Por hrterst of X I [5] S Hlper hve show the followg: π H V Oe other oto tht we wll use throughout ths pper s the forml meso of X gve s f X : mx H X ; 0 We ow from [5] tht whe re the elemets of homogee- π () Copyrght 0 SRes

2 M R HILAL ET AL ous bss of V f X () eve Our proofs re essetly bse o ths equlty ombe wth other equlty estblshe by J Freler S Hlper [] tht eve f X f X Flly let us rell tht H X; stsfes the Por ulty tht mes tht the multplto H X; H X; H X; s o egeerte bler form (here f X eotes the so lle fumetl lss of H X; ) For the reer tereste by more etls bout the rtol homotopy theory we reomme the bs referee [] The M Theorem I ll the remer of ths pper X eotes smply oete ellpt spe wth m H X; 0 V wll eotes t mml moel Put bss for H X; wth the oto tht tht V wth The followg tble summrzes the lssfto of ts rtol homotopy type Rtol homotopy type of X f X fx # f X Lege s s p fx p p f X p p f X p p Y E E : the totl spe of the fber bule wth p q s bse spe () Lege: ) I [] I M Jmes hs troue the oept of reue prout whe X s bse spe He put X X X : X X x x x x : p p p ) From ths ostruto pple to eve sphere rses the Jmes sphere p stsfyg p H ; p The use of the eotto p mes mpltly tht s suppose to be eve As the most of our proofs wll be by otrto we wll mr suh proofs by (by otrto) ts begg by (QED) whe ts e I the sprt esre to smplfy the leture of ths pper we wll subve t o my propostos lemms theorems The frst oe s tht: Lemm There exsts 5 suh tht < Proof Suppose tht f X 5 0 The Cse Where 5 Proposto If 5 X hs the rtol homotopy type (rht) of wth f X Proof Se V We stgush two ses: ) s The 0 Let E be the vetor spe spe by If m E we te s bss of E Let b b homogeeous bss of omplemet of V wth b b b therefore b 0 b H V wht m- ples tht m H V 9 of X s wth 0 So the mml moel Ths s extly the mml moel of If me the there exst suh tht 0 the Aorg to the Por ulty we hve so 0 Ths s mpossble ) s eve The f X re bef X use tht 0 Therefore 0 there exst tree geertors b of V wth eve egrees suh tht b The eve b f X Copyrght 0 SRes

3 M R HILAL ET AL Ths s mpossble The Cse Where 5 Lemm If 5 f X re eve s 0 Proof Frst beuse of the Por ulty we hve f X s eve hve 0 the sme prty Hee s s eve (By otrto) Suppose ow tht 0 se 0 the V Otherwse the Por ulty let us to suppose tht 5 to olue tht 5 tht re lso geertors of V So 5 5 f X f X Ths s mpossble (QED) Lemm If 5 there exsts homogeeous geertor b of V stsfyg b Proof Se 0 we ssume tht tht V Ot herwse there exsts homogeeous geertor b of V suh b Lemm 5 If 5 V x x x y D wth: Dx Dx Dy 0 Dy y xxy x x et y Proof We hve H V H W D W x x x y We efe the lgebr homomorph- sm : W D V s x x x b y s to beuse t trsforms the bss x x x y of W o the lerly epeet fmly b Let V0 W V V0 V se H V H V0 the V V V \ 0 0 Assume th t V 0 otht m x xv ser V suh x 0 b As 0 b 0 We hve to suss two ses: ; I th s se therefore eve b 9 > f X Ths s mpossble m m m 0 >5 > fx so 0 Let V suh I ths se V m s prtulr But 0 beuse f ot we wll hve m > fx Ths s mpossble Proposto If 5 the X hve oe of the followg rht: s fx # s X Proof Let us rell tht fx re eve tht re Frst se: 0 Se 0 the 0 0 Hee s bss for H X; therefore H X; b b e X hs the rht of Seo se: 0 Here re both o ull beuse the opposte se we wll hve b or b b s geertor of V ths ses 5 > f X Ths s mpossble Rep ths les us to olue tht X hve the rht of # The Cse Where 5 Proposto If 5 f f X s eve X hve the rht of p wth f X p Proof Beuse of the prty of fx the ulty of Por the ft tht 0 re respetve ly eve so 0 Assume tht 0 tht 5 there exst P P suh tht P P for 5 Ths mp les the mpossble stuto tht 0 but lso tht our seo ssumpto s flse Thus eessrly 5 re both geertors of V tht 5 > f X f X Ths other mpossble stuto mples tht our frst ssumpto s lso flse Put 0 ths se prtulr re geertors of V for The Por ulty let us to wrte to olue tht 0 tht 0 flly to wrte 5 Rell tht 0 beuse of the prty of the egree V b x wth b 0 x Ths s the mof p ml moel Copyrght 0 SRes

4 M R HILAL ET AL Proposto If 5 f fx s X hs oe of followg rht: p wth fx p p wth f X p p wth f X p Proof We wll suss three ses: Frst se: s 5 s eve Suppose eve tht 5 V eve 5 5 fx Ths s mpossble So wth 0 s egree geertor of V for or A sme ustfto s the lst proof let us to olue tht 5 p tht X hve the rht of Seo se: s eve 0 Se 5 s m Assume for exmple tht 0 wth or s Therefore 0 Let suppose tht re oller wrte 0 so 5 V Se tht 0 tht there exst two egree geertors of V b suh tht b We olue tht 5 b 5 f X > f X (mpossble) Put The mml moel of X wll be of the form V b b b wth 0 b b b 0 b p p e X Thr se: s eve 0 As the frst se we wrte 5 S e m Suppose tht 0 wrte ( 0 ) b e 0 Tht otrts the m hypothess our thr se Hee the mml moel of X wll be of the form V b b b wth b 0 b b e p X The Cse Whe re 5 Proposto 9 If 5 f X s eve x hve oe the rht of p Proof As f X s eve 0 re respetvely eve Suppose (by otrto) tht or s ull (for exmple 0) The ulty of Por sures tht 5 V 5 > f X Ths s mpossble (QED) Put 5 0 beuse tht Th s les us to te to olue tht 0 Hee V x y y y wth x y y 0 y x x y y p e p X Lemm 0 If 5 f X s 5 V Let us suppose V (for exmple) suss two ses: s eve 0 th ere exsts ge- ertor of V suh tht If s the >fx mpossble The s eve eessry m e there exsts geertor of V verfyg wth 0 or 0 Cosequetly fx wht s oe g mpossble stuto s beuse of the Por ulty we must hve be eve m Let 9 V suh tht 9 9 eve f X Ths s mpossble Lemm If 5 f X s 0 0 Proof (By otrto) Assume for exmple tht 0 By the preeet lemm the ulty of Por we hve Therefore 5 but f X s Ths s otrto (QED) Proposto If 5 f X s x h ve oe of the followg rht: wth Y Y hve mml Copyrght 0 SRes

5 M R HILAL ET AL 9 moel of the form b uv wth b 0 u b v b Proof By the two lst lemms we hve s V 0 V wth V But m H V (se lssfe by the frst uthor hs thess) X Y Y or Y Y V b u v b 0 u b v b 5 Cse Where 5 Lemm If 5 or Proof Suppose tht V there exst two geertors of V stsfyg 5 wth V We stgush two ses: Frst se: s eve s As f X s eve 0 s eve osequetly > fx eve Seo se: s As f X s eve 0 s eve fx eve The two ses re both mpossble Lemm If 5 : ) 0 ) ) V ) 5 Pro of ) suppose tht 0 s eve Se f X s eve 0 5 re both Put V 5 > f X (o trto) ) We hve If s eve s If s the result s evet beuse tht 0 ) It s mmete osequee of ) Hee we te 5 V Se ) 0 there exsts suh tht So Proposto 5 If 5 the X hve the hrt of or tht of Proof Put 5 the mml moel of X hve oe of the followg forms: V x y y y wth x y y 0 y x x y y e X V x y y y wth x y y 0 y x yy x y y e X Cse Where 5 Lemm If 5 5 Proof Let 5 V suss my ses: eve ) 5 V there exst two geertors x y of V suh tht x 5 y 5 ) s 9 > fx eve f X s e ve 0 eve s V > x y f X b) s eve As ) 5 V ) s eessry V 5 Hee there exsts geertor x o f V suh tht x 5 eve ) V re both se 0 Se 0 the V for or wth 0 5 Hee there exsts geert or y of V suh tht y so 5 > x y f X o ) V x V x > f X b) s eve eve eessr eve ) V y xv > eve x f X ) ulty) 5 0 the for or ( 0) 5 0 (Por ulty) Let x be geertor of V suh tht x 5 0 (beuse of the Por Put 5 V 5 5 x 5 fx > Lemm If 5 Proof Let V suss my ses: eve ) V The there exst geertors of Copyrght 0 SRes

6 0 M R HILAL ET AL V suh tht ) s eve f X f X > f X b) s > eve f X ) V eve ) V b) V t > f 9 X eve he >f 9 X Lemm If 5 V Proof Put N V ) If N th e for ll ths mples the otrto 0 ) I f N We hve ) re bot h eve th e f X s eve 0 Let be some ge ertors of V wth therefore > f X b) re both f X s 0 so (for exmple) s >fx ) s eve s o (for exmple) > eve f X Lemm 9 If 5 f V hve fferet prtes V Proof Suppose tht hve the sme prty ) If re both eve ll re eve for 0 ) If re eessry (beuse tht ) but ths s mpos sble Proposto 0 If 5 f V m X Proof Put V Beus e the ulty of Por the ft tht hve fferet prtes the ft tht for ll Let suh tht s m t s evet tht s eve As H V m 0 Ths llows us to te p p wth p beuse f ot m H V Hee 0 0 Colue tht V 0 W tht m V eve tht m H W I [?] W s the l of m mml mo e () X () m Lemm If 5 f V oly oe m og or s V Proof Assume tht V re both eve beuse tht eessry There- fore f X 0 e f X s there exsts ge ertor of V suh t ht wth eve > f X Lemm If 5 f V V Proof Suppose 0 we ow from the ulty of Por tht 5 by the Lemms tht 5 We eue tht 5 tht w here ) If 5 As 5 5 so s eve for ll but ths mples tht 0 ) If eessry 0 Hee fx 5 Se re both eve the f X So s eve fo r ll but ths les to the otrto 0 ) If 0 Suppose tht 0 suss two se s ) re eve 0 b) 5 f X s eve but lso f X s eve Therefore s eve for ll 0 ) If 0 (P or ulty) Hee 5 f X So < Proposto If 5 f V X hve o e of the followg rht: p E: the totl spe of the fber bule wth Copyrght 0 SRes

7 M R HILAL ET AL e wher e re both eve Proof We ow by the preeet lemms by the Por ulty tht Put 5 0 W e stgush two ses: ) 0 re both beuse tht 0 Reple by put the mml moel of X s of the form x y y y wth y p < x < y q y x yy Hee X ~ E p q E s fbrto of the KS-omplex q s bse sp y y 0 y y x y ) 0 X hve the mml moel x y y y wth y y 0 y x e X wth re both eve Aowlegemets xy The uthors woul le to th the oymous revewers for ther ostrutve ommets sutble ves o erler rft of ths ppe r It s lso plesure to th Pul Goerss Kthry Hess for ther terest egrteful to Hroo Shg ourgemet The uthors re Toshhro Ymguh for the severl eml susso exhge before the submsso of ths pper REFERENCES [] G Bzzo V Muõz Rtol Homotopy Type of Nlmfols Up to Dmeso rxv: 000v 00 [] J B Freler S Hlper A Arthmet Chrterzto of the Rtol Homotopy Groups of Cert Spes Ivetoes Mthemte Vol 5 No 99 pp - o:000/bf09009 [] Y Felx S Hlper J-C Thoms Rtol Homotopy Theory Grute Texts Mthemts Vol 05 Sprger-Verlg New Yor 00 [] P Grffths J Morg Rtol Homotopy Theory Dfferetl Forms Progress Mthemts Brhäuser Bsel 9 [5] S Hlper Ftess the Mml Moels of Sullv Trstos of Amer Mthemtl Soety Vol 0 9 pp -99 [] I M Jmes Reue Prout Spes Als of Mthemts Vol No 955 pp 0-9 o:00/000 [] G M L Powell Ellpt Spes wth the Rtol Homotopy Type of Spheres Bullet of the Belg Mthemtl Soety Smo Stev Vol No 99 pp 5- [] H Shg T Ymguh The Set of Rtol Homotopy Types wth Gve Cohomology Algebr Homology Homotopy Appltos Vol 5 No 00 pp - Copyrght 0 SRes