The Main Theorem

Theorem 5.9. There is an isomorphism of spetral sequenes

E _∗ ( M )(LHP ^r _hS ¹ ) 1

u

∼ = E _∗ ( M )(LHP ^r ) ⊗ F p [u, u ^{− 1} ].

when re-indexing the olumns: ltration

pm

^{goes to ltration}

m

^for

m ∈ N

^.

Note: Thisimpliesthattheloalizedspetralsequene

E _∗ ( M )(LHP ^r _hS 1 ) ₁

u

ollapsesfromthe

E p

^{page, sine}

E _∗ ( M )(LHP ^r )

^{ollapsesfromthe}

E 1

^page.

To prove SF(6), we rst reall that by Theorem 5.6, the indued map

i ^∗ : E ₁

^odd

( M )(LHP ^r _hS 1 ) −→ E ₁

^odd

( M )(LHP ^r )

^{is surjetive. Sine every}

dif-ferentialin

E s ( M )(LHP ^r _hS 1 )

^{starting inodd total degree istrivial,the map}

i ^∗ : E _∞

^odd

( M )(LHP ^r _hS 1 ) −→ E _∞

^odd

( M )(LHP ^r )

^{is also surjetive. It is a}

gen-eral fat for spetral sequenes that the indued map ontheir limitsisthen

alsosurjetive, and this is easily seen by a ltrationargument. This means

that

i ^∗ : H

^odd

(LHP ^r _hS 1 ) −→ H

^odd

(LHP ^r )

^{is surjetive.}

We rst prove the Main Theorem for the odd part of the ohomology.

There are two kinds of

F p [u]

generators, torsion and free, and we need to use the

S ¹

^{transfer map}

τ

^{tond the rst kind. Let}

i : LHP ^r −→ ES ¹ × S ¹

LHP ^r = LHP ^r _hS 1

^{bethe inlusion. Thenit follows from} [Bökstedt-Ottosen ℄ Thm. 14.1 thatthe

S ¹

^{ationdierential}

d

^{is omposed as follows}

H ^{∗ +1} (LHP ^r )

τ

((

Q Q Q Q Q Q Q Q Q Q Q Q Q

d // H ^∗ (LHP ^r )

H ^∗ (LHP ^r _hS 1 )

i ^∗ n n n n n n n 66 n n

n n n

(56)

In general, for a spae

X

^{with an ation}

µ : S ¹ × X −→ X

^{, the map}

d

^is

given by

H ⁿ⁺¹ (X) −→ H ⁿ⁺¹ (S ¹ × X) −→ H ⁿ⁺¹ (X) ⊕ H ⁿ (X)

a 7→ µ ^∗ (a) 7→ (a, d(a))

where the last map is the Künneth formula. For ease of referene, in the

LemmabelowIhaveolletedallthefatsIneedabouttheationdierential.

Firstsome notation:

IF = IF (r, p) = { (4r + 2)i + 4j | δ ≤ j ≤ r, 0 ≤ i, p | (r + 1)i + j } \ { 0 } , IT = IT (r, p) = { (4r + 2)i + 4j | δ ≤ j ≤ r, 0 ≤ i, p ∤ (r + 1)i + j } ;

where

δ =

1, p ∤ r + 1

^;

0, p | r + 1

^.

Set

IA = IF ∪ IT

^{. Then denepowerseries by}

P _I (t) =

X ∞ n=0

a _n t ⁿ ,

^where

a _n =

1, n ∈ I (r, p)

^;

0, n / ∈ I (r, p)

^{. (57)}

for

I = IF , IT , IA

^{. By}[Bökstedt-Ottosen ℄ Lemma 11.4,

IF ∩ IT = ∅

^{, so}

weget

P _IA = P _IF + P _IT

^{. Also notethat by (54),}

P _H

^odd

_(LHP ^r ₎ (t) = 1

t P _IA (t).

⁽⁵⁸⁾

ThefollowingLemmaontheationdierentialisprovedin[Bökstedt-Ottosen℄

lemma11.6.

Lemma5.10(TheAtionDierential). Put

H ^∗ = H ^∗ (LHP ^r )

^andlet

k ∈ N

^.

(i)

^Ker

(d : H ^2k −→ H ^{2k − 1} )

^{is either a trivial or a} 1-dimensional vetor

spae. Itis non-trivial if and only if

2k ∈ IF (r, p)

^.

(ii)

^Im

(d : H ^2k −→ H ^{2k − 1} )

^{is either a trivial or a} 1-dimensional vetor spae. Itis non-trivial if and only if

2k ∈ IT (r, p)

^.

(iii)

^{The okernel of the map}

d : M

0 ≤ k ≤ (2r+1)mp − δ

H ^2k+2 −→ M

0 ≤ k ≤ (2r+1)mp − δ

H ^2k+1

has dimension

rm

^if

p ∤ r + 1

^{, and dimension}

(r + 1)m

^if

p | r + 1

^.

The next two Lemmasspeify the

F p [u]

^{generators for}

H ^∗ (LHP ^r _hS 1 ; F p )

^:

Lemma 5.11. There is a graded subgroup

T ^∗ ⊆ H

^odd

(LHP ^r _hS 1 )

^{suh that}

(i) u T ^∗ = 0

^.

(ii)

^{Therestritedinlusionmap}

i ^∗ | T ^∗ : H ^∗ (LHP ^r _hS 1 ) | T ^∗ −→ H ^∗ (LHP ^r )

^is

injetive.

(iii)

^{The image}

i ^∗ ( T ^∗ ) ⊆ H ^∗ (LHP ^r )

^{equals the image}

d(H ^{∗ +1} (LHP ^r )) ⊆ H ^∗ (LHP ^r )

^.

Proof. We use property

(iii)

^toonstrut

T ^∗

^{. Wehoose agradedsubgroup}

T ^∗ ⊆ H ^{∗ +1} (LHP ^r )

^{,suh that}

d

^maps

T ^∗

isomorphially ontoIm

d

^{. This we}

andosimplybyliftingeahgeneratorofIm

d ⊆ H ^∗ (LHP ^r )

^to

H ^{∗ +1} (LHP ^r )

^.

Now we put

T ^∗ = τ( T ^∗ )

^{. Then}

(iii)

^{follows by} onstrution, sine

i ^∗ ( T ^∗ ) = i ^∗ ◦ τ ^∗ ( T ^∗ ) = d( T ^∗ )

^{by the diagram (56). Also}

(ii)

^{holds, sine}

i ^∗

^restrited

to

T ^∗

^{orresponds to}

i ^∗ ◦ τ = d

^{restrited to}

T ^∗

^{, and we hose}

T ^∗

^{suh that}

d

^{was an} isomorphism of

T ^∗

^{onto its image. As for property}

(i)

^{, this holds}

beause

uτ = 0

^{aording to}[Bökstedt-Ottosen ℄ Thm. 14.1. This is beause thetransfer map

τ

^{appears rightafter}multipliationby

u

^{inthe Gysinexat}

sequene.

Remark 5.12. Bydenitionof

T ^∗

^{itfollows fromLemma5.10}

(ii)

^thatthe

non-trivialpart of

T ^∗

^sitsindegree

2k − 1

^{ifand onlyif}

2k ∈ IT (r, p)

^{. Using}

the notationin (57), we an write down the Poinaréseries of

T ^∗

^:

P _T ^∗ (t) = 1

t P _IT (t).

Lemma 5.13. There isa graded subgroup

U ^∗ ⊆ H

^odd

(LHP ^r _hS 1 )

^{suh that}

(i)

^{The omposition}

T ^∗ ⊕ U ^{∗  //} H

^odd

(LHP _hS ^r 1 ) ^{i ∗ //} H

^odd

(LHP ^r )

isan isomorphism.

(ii)

^{The restrition}

U ^{2i+1 //} H ²ⁱ⁺¹ (LHP ^r _hS 1 ) ^j

∗

// H ²ⁱ⁺¹ (( F pm ) hS ¹ )

is trivial if either

p | r + 1

^and

i > (2r + 1)pm

^{, or}

p ∤ r + 1

^and

i > (2r + 1)pm − 2

^.

Proof. Againwe rst speify a subgroup

U ^∗ ⊆ H

^odd

(LHP ^r )

^{, by demanding}

thatitmustbeaomplementarysubgroup of

i ^∗ ( T ^∗ )

^{, sothatwe have the}

F p

vetor spae isomorphism

H

^odd

(LHP ^r ) ∼ = i ^∗ ( T ^∗ ) ⊕ U ^∗

^{. The idea is to nd}

U ^∗ ⊆ H ^∗ (LHP ^r _hS 1 )

^{suh that}

i ^∗

^{maps it} isomorphially to

U ^∗

^{. This an be}

done sine

i ^∗

^{issurjetive by SF(6).}

We now use the Gysin sequene, see [Bökstedt-Ottosen℄ Thm. 14.1, to

makethe following diagramwith exat rows:

H ^{2i − 1} (LHP ^r _hS 1 ) ^{· u //}

j ^∗

H ²ⁱ⁺¹ (LHP _hS ^r 1 )

j ^∗

i ^∗

// // H ²ⁱ⁺¹ (LHP ^r )

H ^{2i − 1} (( F ^pm ) _hS ¹ ) ^{· u //} H ²ⁱ⁺¹ (( F ^pm ) _hS ¹ ) ^// H ²ⁱ⁺¹ ( F ^pm )

(59)

Thevertial maps

j ^∗

^{are surjetiveaording toSF(4). BySF(6),the upper}

horizontalmap

i ^∗

^{is surjetive.}

Under the assumptionin

(ii)

^{, weget fromSF(5)that}

H ²ⁱ⁺¹ ( F n , F n − 1 ) = E ₁ ^{n,2i+1 − n} = 0

^for

0 ≤ n ≤ pm

^{. Using the long exat sequene for the pair}

( F ⁿ , F ⁿ − 1 )

^for

n = 0, 1, . . . , pm

^{gives aseries of injetivemaps,}

H ²ⁱ⁺¹ ( F pm ) ֒ → H ²ⁱ⁺¹ ( F pm − 1 ) ֒ → · · · H ²ⁱ⁺¹ ( F 0 ) ֒ → H ²ⁱ⁺¹ ( F − 1 ) = 0.

This means

H ²ⁱ⁺¹ ( F ^pm ) = 0

^{. So}

U ²ⁱ⁺¹

^{is in the kernel of the right vertial}

map. To ensure that

U ²ⁱ⁺¹

^{is also in the kernel of the middle vertial map}

j ^∗

^{, we use diagram hase. The image}

j ^∗ ( U ²ⁱ⁺¹ )

^{maps to zero, so it omes}

from

H ^{2i − 1} ( F ^pm )

^{. The left}

j ^∗

^{map isonto this, so we an lift it,map it into}

H ²ⁱ⁺¹ (LHP ^r )

^{, and subtratitfromthe original}

U ²ⁱ⁺¹

^{. This givesa hoieof}

U ²ⁱ⁺¹

^{that satises both}

(i)

^and

(ii)

^.

Remark 5.14. By property

(i)

^of

U ^∗

^{, we an alulateits Poinaréseries}

P _U ^∗ (t) = P _H

^odd

_(LHP ^r ₎ (t) − P _T ^∗ (t) = 1

t (P _IA (t) − P _IT (t)) = 1

t P _IF (t),

where wehave used Remark 5.12 and (58).

Remark 5.15. We will need the dimension of parts of

U ^∗

^{. As}

T ^∗ ⊕ U ^{∗ i}

∗

∼ = H

^odd

(LHP ^r )

^{, and}

i ^∗ ( T ^∗ ) =

^Im

d ⊆ H

^odd

(LHP ^r )

^{, we an ompute the}

di-mension of

U ^∗

^{as the dimension of the okernel of the ation dierential}

d

^.

Forthis wean use Lemma 5.10

(iii)

^and

(iv)

^{, and get}

p ∤ r + 1 : dim M

k ≤ (2r+1)mp − 1

U ^{2k − 1}

= rm, p | r + 1 : dim M

k ≤ (2r+1)pm

U ^{2k − 1}

= (r + 1)m.

Nowwe an prove the MainTheorem for the odd degree ohomology:

Theorem 5.16. The map of

F p [u]

^-modules,

h 1 ⊕ h 2 : (F p [u] ⊗ U ^∗ ) ⊕ T ^∗ −→ H

^odd

(LHP ^r _hS 1 )

indued bythe inlusionsof

U ^∗

^and

T ^∗

^{, isan} isomorphism of

F p [u]

^-modules.

Expressed in terms of generators,

H

^odd

(LHP ^r _hS 1 )

^{is isomorphi as a}

graded

F p [u]

^{-module to}

M

2k ∈IF

F p [u]f 2k − 1 ⊕ M

2k ∈IT

(F p [u]/ h u i ) t 2k − 1 ,

where the lowerindex denotes the degree of the generators.

Proof. From Lemma 5.11

(i)

^{we see that}

T ^∗

^{is atually an}

F p [u]

^-submodule

of

H

^odd

(LHP _hS ^r 1 )

^{, and so the inlusion}

h 2 : T ^∗ −→ H

^odd

(LHP _hS ^r 1 )

^{is an}

F p [u]

^{-linear map. On the ontrary we just onsider}

U ^∗

^{as a subgroup, and}

makethe

F p [u]

^-module

F p [u] ⊗ U ^∗

^{. Thereis thenauniquewaytoextendthe}

inlusionof

U ^∗

^{to an}

F p [u]

^{-linear map}

h 1 : F p [u] ⊗ U ^∗ −→ H

^odd

(LHP _hS ^r 1 )

^.

First we remarkthat

h 1 ⊕ h 2

^{issurjetive. Tosee this weuse partof the}

Gysinexat sequene, see (59), wherethe rightmost zero is SF(6):

H ^{2i − 1} (LHP ^r _hS 1 ) −→ ^u H ²ⁱ⁺¹ (LHP _hS ^r 1 ) ⁱ

∗

−→ H ²ⁱ⁺¹ (LHP ^r ) −→ 0.

This is asequene of

F p

^{vetor spaes, so itsues to show that we an hit}

the image

u(H ^{2i − 1} (LHP ^r _hS 1 ))

^{and the okernel}

H ²ⁱ⁺¹ (LHP ^r _hS 1 )/ ker(i ^∗ ) ∼ =

H ²ⁱ⁺¹ (LHP ^r )

^{. The okernel an be hitaording to}

(i)

^{inLemma 5.13. We}

now use indution in the degree

2i + 1

^{. The indution start is trivial. We}

get indutively that the image

u(H ^{2i − 1} (LHP ^r _hS 1 ))

^{an be hit by}

u (F p [u] ⊗ U ^∗ ) ⊕ T ^∗

⊆ (F p [u] ⊗ U ^∗ ) ⊕ T ^∗

^{, wherethelastinlusionfollows fromLemma}

5.11. Soit remainstoshow that

h 1 ⊕ h 2

^{is injetive.}

The idea of the proof is now to show that map

h 1 ⊕ h 2

^{loalized away}

from

u

^{, whih wedenote}

(h ₁ ⊕ h ₂ )[ ¹ _u ]

^{,is injetive. Againby Lemma 5.11}

(i)

wesee thatwhenloalizingaway from

u

^,

T ^∗

^{vanishes. Sowelookat}

h 1

^,and

by Lemma 5.13 there isa ommutative diagram,

F p [u] ⊗ L

i U ^{2i+1 h 1 //}

id

⊗

^proj

H

^odd

(LHP _hS ^r 1 )

j ^∗

F p [u] ⊗ L

i ≤ (2r+1)pm − δ U ^{2i+1 h 1 //} H

^odd

(( F pm ) hS ¹ )

(60)

where

δ =

1, p ∤ r + 1

^;

0, p | r + 1

^.

The map

j ^∗

^{is surjetiveaording to SF(4).}

Loalizing away from

u

^{an be done by tensoring with}

F p [u, u ^{− 1} ]

^over

F p [u]

^{. Sine}

h 1 ⊕ h 2

^{is surjetive, and} loalization is exat,

(h 1 ⊕ h 2 )[ ¹ _u ]

^is

also surjetive. As noted,

h 2

^{vanishes when loalizing away from}

u

^{, so we}

onlude that

h 1 [ 1

u ] : F p [u, u ^{− 1} ] ⊗ U ^∗ −→ H

^odd

(LHP _hS ^r 1 )[ 1 u ]

issurjetive. When loalizing,weonlude from the diagram(60) that

h 1 [ 1

u ] : F p [u, u ^{− 1} ] ⊗ M

0 ≤ i ≤ (2r+1)pm − δ

U ²ⁱ⁺¹ −→ H

^odd

(( F ^pm ) _hS ¹ )[ 1 u ]

isalso surjetive.

To show

h ₁ [ _u ¹ ]

^{as injetive, we will prove that the domain and target}

spaes are isomorphi as abstrat modules. So we rst study the domain

of

h 1 [ ¹ _u ]

^{. The dimension of the}

U ^∗

^{part is alulated in Remark 5.15, and}

tensoring with

F p [u, u ^{− 1} ]

^{we obtain the rank:}

rank

F p [u, u ^{− 1} ] ⊗ M

0 ≤ i ≤ (2r+1)pm − δ

U ²ⁱ⁺¹

=

(r + 1)m, p | r + 1

^;

rm, p ∤ r + 1

^.

Turningtothetargetspaeof

h ₁ [ ¹ _u ]

^,

H

^odd

(( F pm ) _hS ¹ )[ _u ¹ ]

^{,weuseTheorem5.9:}

H

^odd

(( F pm ) hS ¹ )[ 1

u ] ∼ = H

^odd

( F m ) ⊗ F p [u, u ^{− 1} ]

⁽⁶¹⁾

Consequently,byCorollary5.8weanalulatetherankasan

F p [u]

^-module:

rank

H

^odd

(( F ^pm ) _hS ¹ )[ 1 u ] =

m(r + 1), p | r + 1

^;

mr, p ∤ r + 1

^.

So

h 1 [ ¹ _u ]

^{is a surjetive map between two free}

F p [u, u ^{− 1} ]

^{-modules of the}

same rank. Then

h 1 [ ¹ _u ]

^{must also beinjetive.}

All that remains is to show that

h ₁ ⊕ h ₂

^{is injetive. Atually it will be}

enoughtoshowthat

h 1 ⊕ h 2

^{isinjetiveforeah}

m

^{,sineagiven elementwill}

be in the domain of

h 1 ⊕ h 2

^{for a large enough}

m

^{. So onsider an element}

(a, t) ∈ F p [u] ⊗ L

i ≤ (2r+1)pm − δ U ²ⁱ⁺¹ ⊕ T ^∗

^{in the kernel of}

h ₁ ⊕ h ₂

^{. When}

loalizing,

t

^{vanishes, so}

c

^{loalized must be in the kernel of}

h 1

^loalized,

whih we have shown is injetive. This means

c

^{loalized is zero. But the}

loalizationmap on

F p [u] ⊗ U ^∗

^,

F p [u] ⊗ U ^∗

loalization

−→ F p [u, u ^{− 1} ] ⊗ ^F p [u] (F p [u] ⊗ U ^∗ ) ∼ = F p [u, u ^{− 1} ] ⊗ U ^∗

is injetive, so

c

^{is zero itself. This means}

t

^{is in the kernel of}

h 1

^{. And by}

Lemma 5.11,

h ₁

^{isinjetive, so}

t

^iszero.

The expression with generators follows diretly from the isomorphism

H

^odd

(LHP ^r _hS 1 ) ∼ = (F p [u] ⊗ U ^∗ ) ⊕ T ^∗

^{together with the omputation of the}

Poinaré series inRemarks 5.12 and 5.14.

Wean nowprove thegeneralMainTheorem, givinga omplete

desrip-tion of

H ^∗ (LHP ^r _hS 1 ; F p )

^:

Theorem 5.17. As a graded

F p [u]

^-module,

H ^∗ (LHP ^r _hS 1 ; F p )

^{is isomorphi}

to

F p [u] ⊕ M

2k ∈IF

F p [u]f 2k ⊕ M

2k ∈IF

F p [u]f 2k − 1 ⊕ M

2k ∈IT

(F p [u]/ h u i ) t 2k − 1 .

Here the lower index denotes the degree of the generator, and their names

are meant to suggest free and torsion generators.

Proof. First, note that when taking the odd part, we have already proved

this in Theorem 5.16. So it remains to show that

H

^even

(LHP ^r ; F p )

^{is a free}

F p [u]

^{-module with generators inthe stated degrees.}

FirstI arguewhy

H

^even

(LHP ^r _hS 1 ; F p )

^{is free,usingtheMorse spetral}

se-quene,

E _s ^{∗ , ∗} = E _s ^{∗ , ∗} ( M )(LHP ^r _hS 1 )

^{. BySF(1)andSF(2),}

E ₁

^evenisafree

F p [u]

-module,whihisonentratedin

E ₁ ^{pm, ∗}

^{. Sine bySF(3)all}non-trivial dier-entialsstartinevendegrees,

E _∞

^{evenisasubmoduleof}

E ₁

^{even. Notethat}

E ₁ ^{pm, ∗}

^is

anitelygenerated

F p [u]

^{-module. Sine}

F p [u]

^{isaprinipalidealdomain,the}

submodule

E _∞ ^{(pm, ∗ )}

^{even of the free}

F p [u]

^-module

E ₁ ^{(pm, ∗ )}

^{even is also free. Sine}

the spetral sequene

E s

^{onverges to}

H ^∗ (LHP ^r _hS 1 ; F p )

^,

H

^even

(LHP ^r _hS 1 ; F p )

is ltered by free

F p [u]

^{modules and is thus free itself. The generators are}

the generators of

E _∞

^even.

Now we must nd the degrees of the generators. We will ompute

E _∞

^even

intermsof Poinaréseries, anddedue the generatordegrees fromthis. The

Morse spetral sequene alone does not provide enough information, so we

ompare with Serre's spetralsequene for the bration

LHP ^r −→ LHP ^r _hS 1 −→ BS ¹ ,

that is,

H ^∗ (BS ¹ ; H ^∗ (LHP ^r , F p )) ⇒ H ^∗ (LHP ^r _hS 1 ; F p ).

Denote this spetral sequene by

E _s ^{∗ , ∗} ( S )

^{. Then}

E ₂ ^{∗ , ∗} ( S ) = H ^∗ (LHP ^r ; F p ) ⊗ F p [u]

^{. Aordingto(54)and(53),}

H ^∗ (LHP ^r ; F p )

^{has thefollowingform: the}

non-trivial part is one-dimensional in eah degree, and, apart from degree

zero,sitsindegrees that omeinpairsof odd-even, withatleast 2zero-rows

between the pairs. I have tried to diagramwhat this might look like below,

astar indiatinganon-trivialgroup.

E 2 ( S ) 8 ∗ ∗ ∗ · · ·

7 ∗ ∗ ∗ · · ·

4 ∗ ∗ ∗ · · ·

3 ∗ ∗ ∗ · · ·

0 ∗ ∗ ∗ · · ·

0 1 2 3 4 5 ...

E 3 ( S ) 8 ∗ ∗ ∗ · · ·

7 ∗ ∗ ∗ · · ·

4 · · ·

3 ∗ · · ·

0 ∗ ∗ ∗ · · ·

0 1 2 3 4 5 ...

We also see the only non-trivial

d 2

dierentials must be from the even to the odd row in the odd-even pairs. What happens when we pass to

E 3 ( S )

dependsonwhether

d 2

^iszerooranisomorphism(theonlypossibilities). If

d 2

iszero,theodd-even rowpairwillsurviveto

E 3

^,andif

d 2

^isanisomorphism, onlythe odd group inltration 0willsurvive to

E 3

^{, asindiated above.}

Here we an use a shortut: The dierential

d ₂

^{an be determined}

geo-metrially;itis atually given by the ationdierential. ByLemma 5.10

(i)

we then see that

d ^0,2k ₂ = 0

^{if and only if}

2k ∈ IF

^{. Then we an write down}

the Poinaré series of the

E ₃

^page:

P (E 3 ( S ))(t) = 1

1 − t ² + P (H

^odd

(LHP ^r ))(t) + P _IF (t)

1 − t ² + tP _IF (t)

1 − t ² .

⁽⁶²⁾

This mightnot lookvery helpful, but if we use (52) to alulate

P (E ₃

^even

( S ))(t) − 1

t P (E ₃

^odd

( S ))(t) = 1

1 − t ² + 1

t P (H

^odd

(LHP ^r ))(t) = 1

1 − t ² − t ² (1 − t ^4r )

(1 − t ⁴ )(1 − t ^4r+2 ) = 1 − t ^4r+4

(1 − t ⁴ )(1 − t ^4r+2 )

⁽⁶³⁾

we get a quantity that does not depend on

P _IF (t)

^.

LetusreturntotheMorsespetralsequene. UsingRemark2.10,wean

ompute the same quantity forthe

E 1 ( M )

^{page. For}

p ∤ r + 1

^{this yields}

P (E ₁

^even

( M ))(t) − 1

t P (E ₁

^odd

( M ))(t)

= 1 − t ^4r+4 + K(t)(1 − t ^4r )t ^4r+3 (1 − t ² )(1 − t ⁴ )

− 1 − t ^4r (1 − t ² )(1 − t ⁴ )

(1 − t ^4r+4 )t ²

1 − t ^4r+2 + K(t)t ^4r+3

= 1 − t ^4r+4 (1 − t ² )(1 − t ⁴ )

1 − (1 − t ^4r )t ² 1 − t ^4r+2

= 1 − t ^4r+4

(1 − t ⁴ )(1 − t ^4r+2 ) .

⁽⁶⁴⁾

Usingthe formulas for

p | r + 1

^{,thoughslightly dierent, alsogive the same}

quantity. As we wanted to ompute

E _∞ ( M )

^{, we really want to know this}

quantity for

E _∞ ( M )

^{. Sine by SF(3), all} non-trivialdierentialsin

E _∗ ( M )

goes fromeven toodd total degree, wehave

dim E _∞ ²ⁿ⁺¹ + dim M

k ≥ 1;i+j=2n+1

Im

(d k : E _k ^{i − k,j − k+1} −→ E _k ^i,j )

!

= dim E ₁ ²ⁿ⁺¹ .

From this we dedue

dim E _∞ ²ⁿ = dim E ₁ ²ⁿ − dim M

k ≥ 1;i+j=2n+1

Im

(d k : E _k ^{i − k,j − k+1} −→ E _k ^i,j )

!

= dim E ₁ ²ⁿ − dim E ₁ ²ⁿ⁺¹ + dim E _∞ ²ⁿ⁺¹ .

Expressing this by Poinaréseries yields

P (E _∞

^even

)( M ) − 1

t P (E _∞

^odd

)( M ) = P (E ₁

^even

)( M ) − 1

t P (E ₁

^odd

)( M )

Now by (63) and (64) we an onlude

P (E _∞

^even

)( M ) − 1

t P (E _∞

^odd

)( M ) = P (E ₃

^even

)( S ) − 1

t P (E ₃

^odd

)( S )

To onlude

P (E _∞

^even

)( M ) = P (E ₃

^even

)( S )

^{, we must show}

P (E _∞

^odd

)( M ) = P (E ₃

^odd

)( S )

^{. We an ompute}

P (E _∞

^odd

)( M )

^{by Theorem 5.16:}

P (E _∞

^odd

)( M ) = P (H

^odd

(LHP ^r _hS 1 )) = P ((F p [u] ⊗ U ^∗ ) ⊕ T ^∗ )

= 1

1 − t ² P _U ^∗ (t) + P _T ^∗ (t) = 1

t(1 − t ² ) P _IF (t) + 1

t P _IT (t),

whereI have used Remarks 5.12 and 5.14. Nowby Lemma5.10

(i)

^,

P (E ₃

^odd

( S ))(t) = P (H

^odd

(LHP ^r ))(t) + tP _IF (t)

1 − t ²

= 1

t P _IA (t) + t

1 − t ² P _IF (t) = 1

t(1 − t ² ) P _IF (t) + 1

t P _IT (t).

This allows us to onlude that

P (E _∞

^even

)( M ) = P (E ₃

^even

)( S )

^{, and we an}

ompute by (62),

P (E _∞

^even

)( M ) = P (E ₃

^even

)( S ) = 1

1 − t ² + P _IF (t) 1 − t ² ,

asstated inthe Theorem.

6

S ¹

^-equivariant

K

^{-theory of}

LCP ^r

ReallthattheMorsespetralsequeneomesfromthe

S ¹

-equivariantenergy ltration

CP ^r = F 0 ⊆ F 1 ⊆ · · · ⊆ F n ⊆ · · · ⊆ F ∞ = LCP ^r ,

⁽⁶⁵⁾

whih onsequently gives a ltration

{ ( F n ) hS ¹ } n

^of

LCP ^r _hS 1

^{. The Morse}

spetralsequene

E _∗ ( M )(LCP ^r _hS 1 )

ⁱⁿ

K

^{-theoryhas the following struture,}

Theorem 6.1. The Morse spetral sequene

E _r ^{∗ , ∗} ( M )(LCP ^r _hS 1 )

^onverging

to

K ^∗ (LCP ^r _hS 1 )

^{is a spetral sequene of}

K ^∗ (BS ¹ ) = Z[[t]]

^{-modules, and it}

has the following

E 1

^{page, using the}

Z/2Z

^{grading of}

K

^-theory:

E ₁ ^0,j =

Z[[t]] ⊗ Z Z[h]/ h h ^r i ,

^{j even;}

0,

^{j odd.}

E ₁ ^n,j =

0,

^{j even;}

Z[[t]] ⁽ⁿ⁾ ⊗ ^R Z[x, y ]/ h Q r , Q r+1 i ,

^{j odd. for}

n ≥ 1.

Here,

R = R(S ¹ ) = Z[U, U ^{− 1} ]

^{, and}

Z[[t]] ⁽ⁿ⁾

^{denotes the}

R

^{-module struture}

U 7→ (t + 1) ⁿ

^on

Z[[t]]

^{. The}

R

^{-module struture on}

Z[x, y]/ h Q r , Q r+1 i

^is

U 7→ (x − y)/(1 + y) + 1

^.

Proof. The method is exatly as in Theorem 5.1. The Morse spetral

se-quene is Theorem 4.4, and we use Theorem 3.7whih gives

K _hS ^∗ 1 (G(r) ⁽ⁿ⁾ )

^,

with the module strutures stated just below the Theorem. Finally, using

the

Z/2Z

^-gradingfromBott-periodiity,wesuppresstheThomisomorphism, and simplyget a shift fromeven to odd degree when

n ≥ 1

^.

Remark 6.2. Note that when

n = 1

^{, the}

S ¹

^{-ation is free on}

G(r)

^{, so}

G(r) hS ¹ ≃ ∆(r)

^{. So}

E ^1,

^odd

∼ = K ⁰ (∆(r)) ∼ = Z[x, y]/ h Q r , Q r+1 i

^{, with}

Z[[t]]

-module struture

t 7→ (x − y)/(1 + y)

^.

WeandepittheMorsespetralsequeneshematiallyasfollows,where

anempty spae denotes zero, and a

∗

^{denotes a} non-trivialmodule:

3 ∗ ∗

2 ∗ ∗ ∗ ∗

1 ∗ ∗

0 ∗ ∗ ∗ ∗

− 1 ∗ ∗

− 2 ∗ ∗ ∗ ∗

− 3 ∗ ∗

− 4 ∗ ∗ ∗ ∗

From the ongurationof this spetral sequene, we an immediately

estab-lishanumberof strutural fats. Reallthe notation

K _hS ^∗ 1 (X) = K ^∗ (X _hS ¹ )

^,

when

X

^{is an}

S ¹

^-spae.

Proposition 6.3. The Morse spetral sequene onverging to

K _hS ^∗ 1 (LCP ^r )

has the following properties:

(i)

^{The onlypossible}non-trivial dierentials start from olumn

0

^.

(ii) K _hS ⁰ 1 (LCP ^r )

^{isasubmoduleof}

K _hS ⁰ 1 ( F 0 ) = K ⁰ (BS ¹ ) ⊗ Z K ⁰ (CP ^r )

^{, and}

in partiular it is a free abeliangroup.

(iii)

^{The spetral sequene for the ltration}

{F ⁱ / F ⁰ } i

^has

K ^∗ (

^point

)

ⁱⁿ

ol-umn0, andthus itollapses. So

K ˜ _hS ⁰ 1 ( F ∞ / F 0 ) = 0

^{, and}

K _hS ¹ 1 ( F ∞ / F 0 )

isfree abelian.

Wewillalsoneedthetwistedase,i.etheMorse spetralsequeneforthe

(n)

^{-twisted ltration}

F 0 = F 0 ⁽ⁿ⁾ ⊆ F 1 ⁽ⁿ⁾ ⊆ · · · ⊆ (LCP ^r ) ⁽ⁿ⁾

^{,where we have}

Lemma 6.4. For the

(n)

^{-twisted ltration}

F 0 ⁽ⁿ⁾ ⊆ F 1 ⁽ⁿ⁾ ⊆ · · · ⊆ (LCP ^r ) ⁽ⁿ⁾

^,

the followingholds:

K ˜ _hS ⁰ 1 ( F 1 ⁽ⁿ⁾ / F 0 ) = 0

^{, and}

K ˜ _hS ¹ 1 ( F 1 ⁽ⁿ⁾ / F ⁰ ) ∼ = Z[[t]] ⁽ⁿ⁾ ⊗ ^R Z[x, y]/ h Q r , Q r+1 i .

Proof. Morse theory says that

F ¹ / F ⁰ ≃ T h(µ ⁻ ₁ )

^as

S ¹

^{-spaes, sine the}

l-trationis

S ¹

-equivariant. As aonsequene,

F 1 ⁽ⁿ⁾ / F 0 = ( F 1 / F 0 ) ⁽ⁿ⁾ ≃ (T h(µ ⁻ ₁ )) ⁽ⁿ⁾ = T h((µ ⁻ ₁ ) ⁽ⁿ⁾ ),

where the last equality is lear from the denition

T h(ξ) = D(ξ)/S(ξ)

^{. So}

by Thom isomorphism,

K ˜ _hS ¹ 1 ( F 1 ⁽ⁿ⁾ / F 0 ) ∼ = K _hS ⁰ 1 (G(r) ⁽ⁿ⁾ )

^{, whih by Theorem}

3.7is isomorphi to

Z[[t]] ⁽ⁿ⁾ ⊗ R Z[x, y]/ h Q _r , Q _r+1 i

^{. Likewise for}

K ˜ _hS ⁰ 1

^.

6.1 The rst dierential

We want to determine the rst dierential

d 1 : E ₁ ^{0, ∗} −→ E ₁ ^{1, ∗}

^{in the Morse}

spetral sequene onverging to

K _hS ^∗ 1 (LCP ^r )

^{. UsingRemark 6.2, we have a}

onretedesriptionofthe

E ₁

^{term,andwegetthefollowingexpliitformula}

for

d 1

^:

Theorem 6.5. The rst dierential

d 1

ⁱⁿ

E _∗ ( M )(LCP ^r _hS 1 )

^{is the}

Z[[t]]

-module homomorphism

d 1 : Z[[t]] ⊗ Z[h]/h ^r+1 −→ Z[x, y]/ h Q r , Q r+1 i

givenby

d 1 (h ^j ) = x ^j − y ^j

^for

j = 0, 1, . . . , r

^.

Proof. The rst dierentialis indued by the boundary map

δ

^below:

( F ⁰ ) _hS ^{1 //} ( F ¹ ) _hS ^{1 //} ( F ¹ ) _hS ¹ /( F ⁰ ) _hS ^{1 δ //} Σ(( F ⁰ ) _hS ¹ )

where

Σ

^{denotes the (redued)} suspension. FromMorse theory (40) wehave

( F 1 ) hS ¹ /( F 0 ) hS ¹ ≃ T h((µ ⁻ ₁ ) hS ¹ )

^{, where}

µ ⁻ ₁

^{is the negative bundle over}

X = G(r)

^{, and wehave the diagram}

S((µ ⁻ ₁ ) hS ¹ ) ^//

D((µ ⁻ ₁ ) hS ¹ ) ^//

T h((µ ⁻ ₁ ) hS ¹ ) ^//

∼ =

ΣS((µ ⁻ ₁ ) hS ¹ )

( F 0 ) hS ¹ // ( F 1 ) hS ¹ // ( F 1 ) hS ¹ /( F 0 ) hS ¹ δ

// Σ(( F 0 ) hS ¹ )

Thevertialmaps fromthesphere-and disbundlesare given by the owof

theenergyfuntional;wereturntothemlater. First,sine

µ ⁻ ₁

^isan

S ¹

^-vetor

bundle,we an assumethat the Riemmanian metrionit is

S ¹

-invariant,so that

S((µ ⁻ ₁ ) _hS ¹ ) = ES ¹ × S ¹ S(µ ⁻ ₁ )

^,and

D((µ ⁻ ₁ ) _hS ¹ ) = ES ¹ × S ¹ D(µ ⁻ ₁ )

^{. Then}

T h((µ ⁻ ₁ ) _hS ¹ ) ∼ = ES ₊ ¹ ∧ S ¹ T h(µ ⁻ ₁ )

^,see [Bökstedt-Ottosen ℄ Lemma5.2, and we get the diagram

ES ¹ × S ¹ S(µ ⁻ ₁ ) ^//

id

× (f + ⊔ f − )

ES ¹ × S ¹ D(µ ⁻ ₁ ) ^//

ES ₊ ¹ ∧ S ¹ T h(µ ⁻ ₁ )

∼ =

ES ¹ × ^{S 1} F ^{0 //} ES ¹ × ^{S 1} F ^{1 //} ES ₊ ¹ ∧ S ¹ F 1 / F 0

This means we an simplyignore the

ES ¹

^{-fator, and onsider the diagram}

S(µ ⁻ ₁ ) ^//

f + ⊔ f ⁻

D(µ ⁻ ₁ ) ^//

T h(µ ⁻ ₁ ) ^//

∼ =

ΣS(µ ⁻ ₁ )

Σf + ∨ Σf ⁻

F ^{0 //} F ^{1 //} F 1 / F 0 δ

// Σ F ⁰

BytheproofofProp. 4.2,

µ ⁻ ₁

^{isatrivialreallinebundle,and overageodesi}

γ ∈ X

^,weanparametrize

µ ⁻ ₁

^as

Riγ ^′

^{. Therefore thespherebundle}

S(µ ⁻ ₁ ) = X + ⊔ X ₋

^{isadisjointunionoftwoopiesofthe basespae}

X

^,wheretheber

is

(X + ) γ = +iγ ^′

^and

(X ₋ ) γ = − iγ ^′

^{. The map}

f _± : X _± −→ F 0

^{isgiven bythe}

owoftheenergyfuntional: Forageodesi

γ ∈ X

^,

f _± (γ)

^{givestheendpoint}

in

F ⁰ = CP ^r

^{for the owlines in diretion}

± iγ ^′

^{. Sine}

µ ⁻ ₁

^is 1-dimensional, the Thomspae

T h(µ ⁻ ₁ )

^{isjustthe suspension}

ΣX

^{of the base spae}

X

^{, and}

ΣS(µ ⁻ ₁ ) = ΣX + ∨ ΣX ₋

^{. Themap}

δ : F 1 / F 0 −→ Σ F 0

^{isnowtheomposition}

δ : F ¹ / F ^{0 ∼ = //} Σ(X) ^// ΣX + ∨ ΣX ₋ ^{Σf + ∨ Σf − //} Σ F 0 .

⁽⁶⁶⁾

Here, the last mapfolds the twosummands in the wedge.

We now investigate the maps

f _± : G(r) −→ CP ^r

^{. Reall from (4) that}

the simple losed geodesi

γ

ⁱⁿ

CP ^r

^{determined by}

[v, w] ∈ P V 2

^{is given by}

the map

P V 2 −→ G(r), [v, w] 7→ q ◦ c(x, v),

where

c(x, v)(t) = cos(πt)x + sin(πt)v

^for

t ∈ [0, 1]

^{, and}

q : S ^2r+1 −→ CP ^r

is the projetion. Suh a

γ

^{is a geodesi on a}

CP ¹ = P { v, w } ⊆ CP ^r

^{, and}

wean give

P { v, w }

homogeneousoordinates,

[a v , a w ] = q(a v v + a w w)

^,and

map

P { v, w } −→ C ∪ {∞} , [a v , a w ] 7→ a v

a w

.

We see that

γ

^{under this map is the urve}

t 7→ ^cos(πt) _sin(πt) = _tan(πt) ¹ ∈ C ∪ {∞}

for

t ∈ [0, 1]

^{, i.e. the real line traversed in the negative diretion, from}

+ ∞

^to

−∞

^{. It is now lear that the ow in diretion}

+iγ ^′

^{will end in}

− i ∈ C ∪ {∞}

^{, or} homogeneous oordinates

√ 1

2 [1, i] ∈ P { v, w }

^{, so}

f + (γ) =

√ 1

2 [1, i] ∈ P { v, w }

^{. The ow in diretion}

− iγ ^′

^{ends in}

i ∈ C ∪ {∞}

^{, so}

f ₋ (γ) = ^{√ 1} ₂ [1, − i] ∈ P { v, w }

^.

Having determined

f _±

^{, we an now alulate the indued map}

f _± ^∗

^on

K ⁰ (CP ^r ) ∼ = Z[h]/ h h ^r i

^{,so we need onlydetermine}

f _± ^∗ (h)

^{, where}

h = [H] − 1

and

H ց CP ^r

^{is the standard line bundle. We do this by} determining the pullbak

f _± ^∗ (H)

^{. From the preeding paragraph we see that the berof}

f ₊ ^∗ (H)

^{over asimple losedgeodesi}

γ

^{determined by}

[v, w] ∈ P V 2

^{is exatly}

all the points on the line given by

√ 1

2 (v + iw)

^{. Reall that the line bundle}

X

^{was dened as the pullbak of the standard bundle}

γ ₁ ց P(γ 2 )

^{under the}

omposite

G(r) −→ P V 2 −→ P V g 2 −→ P(γ 2 ),

γ 7→ [v, w] 7→ ^{√ 1} ₂ [v + iw, v − iw] 7→ C(v + iv) ⊆ Cv ⊕ Cw

It follows that

f ₊ ^∗ (H) = X

^{, so}

f ₊ ^∗ (h) = x

^{. Likewise we get}

f ₋ ^∗ (h) = y

^,

beause

Y

^{is the pullbak of the omplement of}

γ 1

ⁱⁿ

γ 2

^{. Sine}

f _± ^∗

^{is a ring}

homomorphism,weget

f ₊ ^∗ (h ^j ) = x ^j

^,and

f ₋ ^∗ (h ^j ) = y ^j

^{. From(66),weannow}

ompute

d ₁ (h ^j )

^{. Whenfoldingthe maps,theseondsuspensioninthewedge}

ΣX + ∨ ΣX ₋

^{has the} orientationreversed,so weobtain

d 1 (h ^j ) = x ^j − y ^j

^.

In the Morse spetral sequene

E _∗ ( M )((LCP ^r ) ⁽ⁿ⁾ _hS 1 )

^{for the}

(n)

^-twisted

ltration,the rst dierentialisamap

d ⁽ⁿ⁾ ₁ : K _hS ^∗ 1 (Σ F 0 ) −→ K ˜ _hS ^∗ 1 ( F 1 ⁽ⁿ⁾ / F 0 )

^,

f. Lemma6.4.

Lemma 6.6. The rst dierential in

E _∗ ( M )((LCP ^r ) ⁽ⁿ⁾ _hS 1 )

^{is the map of}

Z[[t]]

^{-modules given by}

d ⁽ⁿ⁾ ₁ : Z[[t]] ⊗ Z[h]/

h ^r+1

−→ Z[[t]] ⁽ⁿ⁾ ⊗ ^R Z[x, y]/ h Q r , Q r+1 i , d ⁽ⁿ⁾ ₁ (h ^j ) = x ^j − y ^j ,

^for

j = 0, 1, . . . , r.

Proof. Usingthesame diagramasinthe proofof Theorem6.5above,wesee

that the geometry of this situation is exatly the same, so the ow map is

idential tothe one omputed before.

From(44),thepowermap

P ^j

^{givesamapofthefollowingexatsequenes,}

givinga ommutative diagram:

F 0 //

id

F 1 ^(j) //

P j

F 1 ^(j) / F 0 δ ^(j) ₁

//

P j

Σ F 0

id

F 0 // F ∞ // F ∞ / F 0 δ

// Σ F 0

where

δ ₁ ^(j)

^{denotes the boundary map whih induesthe rst dierential}

d ^(j) ₁

inthe Morse spetralsequene

E _∗ ( M )((LCP ^r ) ^(j) _hS 1 )

^{. So the dierential}

d ^(j) ₁

determined inLemma 6.6an alsobewritten asthe omposite map

d ^(j) ₁ : K _hS ¹ 1 (Σ F ⁰ ) −→ ^δ K ˜ _hS ¹ 1 ( F ∞ / F ⁰ ) ^P

∗

−→ j K ˜ _hS ¹ 1 ( F 1 ^(j) / F ⁰ ).

⁽⁶⁷⁾

In document Afhandling (Sider 65-79)