Pre-Talbot ANSS. Michael Andrews Department of Mathematics MIT. April 2, 2013

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1 Pre-Talbot ANSS Mchael Andrews Deartment of Mathematcs MIT Arl 2, 203 The mage of J We have an unbased ma SO = colm n SO(n) colm n Ω n S n = QS 0 mang nto the -comonent of QS 0. The ma nduced by SO QS 0 on homotoy grous s called the J-homomorhsm. QS 0 From now on let s wor at an odd rme where the mage of J has a relatvely smle descrton: { 0 f n + 0 (mod 2( )) (ImJ) n = Z/ + f n + = 2( )s, s Let V be the Moore sace S 0 /. We have a self ma v : Σ 2( ) V V. The element α s / : S 2( )s Σ 2( )s V V S s of order and les n the mage of J. Trcery wth vanshng lnes shows that these elements le n hghest ossble Adams fltraton and there s a beautful story concernng how the mage of J shows u n the classcal Adams SS. We wll turn our attenton to how the mage of J s detected n the Adams-Novov SS. 2 A guess as to how α s / s detected Recall that we are worng at an odd rme and that we have constructed a sectrum BP whch s a retract of MU (). We wll see that = Z () [v, v 2, v 3,...] and so the cofber sequence S 0 S 0 V

2 gves rse the a SES of BP -comodules 0 / 0. Assumng that the namng conventon s sensble we mght guess that v / s a BP - comodule rmtve defnng an element v Cotor 0 BP (/, ) detectng an element whch we also call v : S 2( ) Σ 2( ) V v V. The ma V S s BP -null but t gves rse to a connectng homomorhsm; we should exect that the element α s / s detected by the mage of under δ : Cotor 0 BP ( /, ) Cotor BP (, ). Exlctly we are guessng that α s / s detected by ] [d Cotor BP BP (, ), where d denotes the coboundary ma n the cobar constructon Ω(, BP ). We now there exsts an element α s /+ such that α s /+ = α s /. We mght guess that ths s detected by an element [ ] d + Cotor BP (, ). So far t unclear that these elements are well-defned snce we don t now how to comute d. 3 Hazewnel s generators for (, BP ) Theorem (Hazewnel): The Hurewcz homomorhm π (BP ) H (BP ) s an njecton. There exst generators v, v 2, v 3,... for and m, m 2, m 3,... for π (BP ) = Z () [v, v 2, v 3,...] H (BP ) = Z () [m, m 2, m 3,...] such that v = m = 2( ) and under the ncluson we have where by conventon m 0 =. m + = +j= m v j+. Moreover: Usng the AHSS we see that the Hurewcz homomorhsm π (BP BP ) H (BP BP ) taes the form π (BP )[t, t 2, t 3,...] H (BP )[t, t 2, t 3,... ] 2

3 where t = 2( ). In artcular, π (BP BP ) and H (BP BP ) are flat (as left-modules) over π (BP ) and H (BP ), resectvely. By conventon, t 0 =. Imortant: The Hof algebrod (π (BP ), π (BP BP )) over Z () s determned by the Hof algebrod (H (BP ), H (BP BP )) over Z (). The H (BP )-bmodule structure of H (BP BP ) s gven by:. The left H (BP )-module structure on H (BP BP ) s the obvous one on H (BP )[t, t 2, t 3,...],.e. η L (m ) = m. 2. The rght H (BP )-module structure on H (BP BP ) s descrbed by η R (m ) = +j= m t j. Note: η R (m ) = m t 0 + m 0t = m + t so that η R (v ) = η R (m ) = m + t = v + t. Also, we have a commutng dagram H (BP ) H (BP ) (η L,η R ) H (BP BP ) η L η R H (BP BP ) H (BP BP ) H (BP BP ) H (BP ) H (BP BP ) and m m (m + t ) m = t (m + t ) m m + t m = t + t so that t (BP ) s a coalgebra rmtve. 4 d / + Recall that n the unreduced cobar constructon we have Ω 0 ( ; BP ) d Ω ( ; BP ) ψ R ( ) BP BP η R η L BP Recall that η L (v ) = v and η R (v ) = v + t. Thus n Ω( ; BP ) d( ) = η R ( ) η L ( ) = +j=s >0 ( s ) v j [t ]. Whenever > 0, + ( ) s dv and so s s a well-defned cocyle n Ω (BP +, BP ), determnng an element α s /+ Cotor BP (, ). 3

4 5 The chromatc sectral sequence We wsh to show the elements α s /+ generate the -lne of the ANSS. For ths we ll use the chromatc SS. We consder the followng n whch each down, u-rght s a SES. / v / /(, v ) v 2 /(, v ) [It s clear that we can construct the dagram above as -modules. It s less clear that all the objects have the structure of a rght BP -comodule. We ll assume ths for now.] We let M = v /(, v,..., v ), where we now have the conventon that v 0 =. Alyng Cotor BP BP (, ) we get an exact coule. Cotor t,u BP (/(,..., vs ), ) Cotor t,u BP (/(,..., vs ), ) Cotor t,u BP (M s, ) Cotor t,u BP (M s+, ) Here the dashed lne rases the degree of s relatve to what s ndcated. We get a SS wth E s,t,u = Cotor t,u BP (M s, ) s = Cotor s+t,u BP (, ), d r : E s,t,u r E s+r,t r+,u r. We wsh to use ths SS to comute the 0 and -lne of the E 2 -age of the ANSS. If M s a rght BP -comodule wrte H s (M) for Cotor s, BP (M, ). The above SS taes the form 6 H (M 0 ) H t (M s ) s = H s+t ( ). BP HQ s a wedge of HQ s. Thus HQ H Q BP slts and by defnton HQ s BP -njectve. We conclude that H (M 0 ) = Cotor BP BP (, ) = Cotor BP BP ( (SQ), ) = E 2 (HQ; BP ) s concentrated n degree 0 where t s equal to Q (and the u gradng s 0). 7 Some useful results 7. Prmtves We have already used that the followng dagrams commute x x BP BP η L BP = ψ R BP BP η R BP Ths tells us that BP -comodule rmtves are elements of er (η R η L ). 4 =

5 7.2 Landweber and Morava s a BP -comodule rmtve and so we can form the BP -comodule /. We have seen that (η R η L )(v ) = t BP and so v / s a BP -comodule rmtve. Part of a theorem due to Landweber and Morava says that H 0 ( ) = Z () and H 0 ( /) = F [v ]. Thus H 0 (v /) = F [v, v ]. 7.3 t BP Recall that t BP s a coalgebra rmtve. Thus, t defnes an element [t ] H (M) for any rght BP -comodule. Theorem: [t ] 0 n H (v /). 8 H 0 (M ) The SES of BP -comodules 0 v / a v / v / 0 a gves a LES 0 H 0 (v /) H 0 (M ) H 0 (M ) δ H (v /).... Gven a nonzero element x H 0 (M ) there exsts a 0 such that x 0 and + x = 0. Then x s n the mage of. Thus we can calculate H 0 (M ) by tang elements n the mage of and analysng how -dvsble they are; we now we cannot dvde an element by f the mage under δ s nonzero. We need a lemma. Lemma: x = vs + Ω(M ; BP ) s a cocycle. If s then δ[x] s nonzero. Proof: Note that n the statement of the lemma s s allowed to be negatve. Recall that d( ) = η R ( ) η L ( ) = +j=s >0 ( ) s v j [t ] n Ω( ; BP ). + ( ) s for > 0 and so n Ω( ; BP ), when we calculate ( ) d + 5

6 we actually obtan an element of Ω(, BP ). [Dvdng by commutes wth the dfferental snce s rmtve n.] Thus, n Ω( / ; BP ) and Ω(M ; BP ) ( ) d + = 0. Snce we are assumng that s odd, we have +2 ( ) s, whenever 2. Thus when we calculate ( ) d +2 n Ω( ; BP ) we obtan s [t ] lus an element of Ω( ; BP ). We conclude that δ[x] s s [t ]. v acts somorhcally on H (v /). Snce s and [t ] 0 we conclude δ[x] 0. We ve roved the lemma for s > 0. Settng s = n the frst comutaton and relacng by ( + ) we see that v + s rmtve mod +2. Thus v + : 2 / 2 / s a comodule ma. Ths means that multlcaton by v + = v commutes wth d on / + and / +2. Thus we have roved the lemma for all s 0. We re then done by the followng lemma. Lemma: Ω(M ; BP ) s a cocycle for all. Concluson: We now that the mage of s ] } {[ : s Z Z, N {0} } and we have analysed how -dvsble these elements are. U to a lttle thnng we have comuted H 0 (M ): t s generated as an abelan grou by } } + : s Z Z, N {0} : N. 6

7 9 Comutng the relevant art of the CSS We have a SES of abelan grous Localsng gves 0 Z Z Z/ 0. 0 Z () Q q Z/ 0. Snce we now that H 0 ( ) = Z () we have a commutng dagram H 0 (M 0 ) Q q d H 0 (M ) Z/ u-gradng-equal-to-0-art ncluded What about d : H 0 (M ) H 0 (M 2 ). Ths s just the comoste H 0 (v / ) H 0 ( /(, v )) H 0 (v 2 /(, v )). We see mmedately that } } + : s N N, N {0} : N s lled by d. On the other hand the mage of } + : s N N, N {0} under d s nonzero (snce H 0 (M) conssts of rmtves, n artcular, a submodule of M, there s no quotentng to worry about). Thus } E,0, = + : s N N, N {0}, whch loos suscously famlar (see sectons and 2). The element n H ( ) detected by [ ] + s reresented by d + = svs [t ] + +j=s > ( s ) v j [t ] Ω ( ; BP ) and so t s our frend α s /+. 7