Afhandling

Cohomology of the Free Loop Spae

of a Projetive Spae

Ph.d. Thesis

Mia Hauge Dollerup

Thesis advisor: Marel Bökstedt

July 1, 2009

Department of Mathematis

Aarhus University

1 Projetive spae and geodesis 8

1.1 The quaternions . . . 8

1.2 Spaesof geodesis . . . 8

1.3 Fibrationsinvolvingspaes of geodesis . . . 12

1.4 Homotopy orbits of spaes of geodesis . . . 14

2 Cohomology of spaes of geodesis in

HP ^r

¹⁷ 2.1 The parametrizedgeodesis . . . 17

2.2 The unparametrized geodesis . . . 21

2.3 Equivariantohomologyof spaes of geodesis . . . 30

3

K

^{-theory of spaes of geodesis in}

CP ^r

³⁵ 3.1 The unparametrized geodesis . . . 35

3.2 Equivariant

K

^{-theory of spaes of geodesis . . . . . . . . . . 38}

4 The free loop spae and Morse theory 47 4.1 The negative bundle . . . 49

4.2 The power map . . . 51

4.3 The Morse theory spetral sequene . . . 52

5

S ¹

-equivariant ohomology of

LHP ^r

⁵⁸ 5.1 The Morse spetral sequenes . . . 58

5.2 The MainTheorem . . . 65

6

S ¹

-equivariant

K

^{-theory of}

LCP ^r

⁷⁵ 6.1 The rst dierential. . . 76

6.2 The MainTheorem for

r > 1

^{. . . . . . . . . . . . . . . . . . . 79}

6.3 The MainTheorem for

r = 1

^{. . . . . . . . . . . . . . . . . . . 84}

Notation 91

Referenes 92

Introdution

The free loopspae

LX

^{of a spae}

X

^{is the spae of ontinuous maps from}

S ¹

^to

X

^{. The irle group}

S ¹

^{ats on}

LX

^{by rotation, and we study the}

spaeofhomotopy orbits,

LX _hS ¹ = ES ¹ × S ¹ LX

^{,sometimesalled theBorel}

onstrution. The main method for understanding this spae will be Morse

theory on the energy funtional, whih to a losed urve assoiates its en-

ergy. This version of Morse theory has been studied by W. Klingenberg in

[Klilngenberg1℄. Asonewouldexpet,theritialpointsofthisfuntionalare

the losed geodesis of

X

^{, soknowing those willbe animportantingredient}

inunderstanding

LX _hS ¹

^{viaMorse theory.}

Inthispaperwestudy

LX hS ¹

^{forapartiularspae,namelytheprojetive}

spae

X = FP ^r

^{, where}

F = C

^or

F = H

^{. The goal is to determine the}

ohomology of

LHP ^r _hS 1

^{and the omplex}

K

^{-theory of}

LCP ^r _hS 1

^{. This is}

alled

S ¹

-equivariantohomology(or

K

^-theory)of

LFP ^r

^{. In general,weget}

a map

ES ¹ × S ¹ LX −→ BS ¹

byprojetionontherstfator. Foraohomologytheory

h ^∗

^{,wethereforeget}

a map

h ^∗ (BS ¹ ) −→ h ^∗ (LX _hS ¹ )

^{, so}

h ^∗ (LX _hS ¹ )

^{beomes a}

h ^∗ (BS ¹ )

^-module.

The methods of Morse theory require the use of Thom isomorphism, whih

destroystheprodutstruture,soweannothopetoalulate

h ^∗ (LFP ^r _hS 1 )

^as

a ring. But the

h ^∗ (BS ¹ )

^{-module struture is preserved by the Morse theory}

mahinery, so the aim is to alulate

h ^∗ (LFP ^r _hS 1 )

^{as an}

h ^∗ (BS ¹ )

^-module,

where

h ^∗

^{iseither singularohomology}

H ^∗

^{or omplex}

K

^-theory

K ^∗

^.

We will now outline our main results. For

X = HP ^r

^{, we study the}

ohomology with

F p

^{-oeients of}

LX hS ¹

^{, where}

F p = Z/pZ

^{, and obtain a}

omplete desription as an

H ^∗ (BS ¹ ; F p ) = F p [u]

^-module:

Theorem 1. As a graded

H ^∗ (BS ¹ ; F p ) = 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 the index sets

IF

^and

IT

^{are knowndisjointsubsets of}

{ (4r + 2)i + 4j | 0 ≤ j ≤ r, i ≥ 0 }

^.

In partiular, there is at most one generator in eah degree.

For

X = CP ^r

^{, we study the omplex}

K

^{-theory of}

LCP ^r _hS 1

^{, and obtain}

Theorem 2. As a

K ^∗ (BS ¹ ) = Z[[t]]

^-module,

K ⁰ (LCP ^r _hS ¹ ) = K ⁰ (BS ¹ ) = Z[[t]] .

As an abelian group,

K ¹ (LCP ^r _hS 1 )

^is torsion-free.

This isone of the rst alulationsof

K (LM _hS ¹ )

^{for a}non-trivial mani- fold

M

^{. Theresultisquitesurprisingwhenomparedto}

H ^∗ (LCP ^r _hS 1 )

^,whih

has a lotof torsion aording to[Bökstedt-Ottosen ℄.

Unfortunately, we have not been able to determine

K ¹ (LCP ^r _hS 1 )

^{as a}

K ^∗ (BS ¹ )

^{-module. As a partial resultin this diretion, we have}

Theorem 3. There is a spetralsequene of

K ^∗ (BS ¹ ) = Z[[t]]

^{-modules on-}

verging strongly to

K ^∗ (LCP ^r _hS 1 )

^{, whih has}

E ₁

^page,

E ₁ ^0,j =

Z[[t]] ⊗ ^Z Z[h]/ h h ^r i ,

^{j even;}

0,

^{j odd.}

E ₁ ^n,j =

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

^{j odd;}

0,

^{j even.}

The rst dierential

d 1

^{is given by}

d 1 (p(t) ⊗ h ^j ) = p(t) ⊗ (x ^j − y ^j )

^{, where}

p(t) ∈ Z[[t]]

^.

Theorem2statesthat

K ⁰ (LCP ^r _hS 1 )

^{is(almost)trivial,while}

K ¹ (LCP ^r _hS 1 )

is free abelian. This is rather similar to the well-known ase of

K ⁰ (BG)

^as

the ompletion of the representation ring

R(G)

^{for a ompat Lie group}

G

^,

while

K ¹ (BG) = 0

^{. This is a lassial result of M. Atiyah. One an also}

ompare to e.g. [Freed-Hopkins-Teleman℄, who nd

K _τ ^∗ (LBG)

^{as the om-}

pletion of ertain representations of the loopgroup

LG

^{, although it should}

be remarked that they onsider

K

^{-theory twisted by a ohomologylass}

τ

^,

and not

S ¹

-equivariant

K

^{-theory as we do. Still,this promptsthe following}

Conjeture. The exists a representation theory type group, suh that

K ¹ (LCP ^r _hS 1 )

^{isa ompletion of this group.}

The outline of this paper is as follows: The paper onsists of two main

parts,eahdividedinthreesetions. Therst setionofeahparttreatsthe

general theory needed and investigates the relevant spaes and strutures,

while the next two setions are more omputational and deal, respetively,

with the ohomologyfor

F = H

^{, and the}

K

^{-theory for}

F = C

^.

Setion 1 investigates

FP ^r

^{and its geodesis, obtaining some useful -}

brations. We onsider both the spae of parametrized and unparametrized

geodesis;the latter beingthe quotient of the former under the ation of

S ¹

by rotation.

Setion 2 alulates the ohomology of the above spaes using Serre's

spetral sequene for the brations found in setion 1. We then turn to

S ¹

-equivariant ohomology of the spae of parametrized geodesis, via two brationsandthe non-equivariantohomologyresultsfromthepreviousse-

Setion 3 obtains similar results for

K

^{-theory. We use the Atiyah-}

Hirzebruh spetral sequene along with the known ohomology results for

CP ^r

^{to determine the}

K

^{-theory of the spae of} unparametrized geodesis.

The

S ¹

-equivariant

K

^{-theory is determined using the same brations as for}

ohomology,butthemethodisdierent,employingtheresultofAtiyahabout

K

^{-theory of lassifying spaes.}

Setion 4 studies of the free loop spae,

LFP ^r _hS 1

^{. First we explain the}

workings of Morse theory in this setting, then we apply this to

LFP ^r

^and

LFP ^r _hS 1

^{toget the so-alledMorse spetral sequene.}

Setion 5 is dediated to proving Theorem 1. The method is losely

baseduponasimilaralulationbyM.BökstedtandI.Ottosenintheirpaper

StringCohomologyGroupsofComplexProjetiveSpaes,[Bökstedt-Ottosen℄.

We extrat a lotof informationabout the Morse spetral sequene, its size,

itsdierentials,andtherelationbetweentheequivariantandnon-equivariant

ase. All this information is brought together to prove the Main Theorem

for ohomology, Theorem 1 above. But even then, it is neessary to turn

to other soures of information to omplete the proof. One is loalization,

the other is omparison with the Serre spetral sequene alsoonverging to

H ^∗ (LHP ^r _hS 1 )

^.

Setion6isdediatedtoprovingTheorem2. Themethodsherearequite

dierent, relying on the fat that the Morse spetral sequene in Theorem

3 has a rather speial onguration, whih implies that all its non-trivial

dierentials start from the zeroth olumn. A very important point is the

alulation of the rst dierential

d 1

^{. The entral idea is then to twist the}

rotationationof

S ¹

^{withapositiveinteger,whihgivesnew Morse spetral}

sequenes related to the standard one. This gives enough information to

prove Theorem

2

^.

For the reader's onveniene, we have assembled a table of notation at

the end of this doument.

Aknowledgements. Finally,it is apleasure to thank my advisor, Marel

Bökstedt, forhishelp throughinnumerable fruitfuldisussions,whihadded

many new insights and ideas to this projet. Also, I would like to thank

Jørgen Tornehave for his time and valuable input when standing in as my

advisor for one year.

1 Projetive spae and geodesis

1.1 The quaternions

I start by introduing the quaternions,

H

^{, as an assoiative algebra of real}

dimension

4

^{,generated by}

1, i, j, k

^{with the following}multipliation rules:

i ² = j ² = k ² = − 1, ij = − ji = k, jk = − kj = i, ki = − ik = j.

Itshouldbestressed,eventhoughitisobviousfromthe aboverelations,that

H

^{is not} ommutative. If one wants to be onrete, one an realize

H

^{as a}

subalgebra of

M 2 (C)

^{generated over}

R

^{by (inthe matrix entries,}

i = √

− 1

^):

i =

i 0 0 − i

, j =

0 1

− 1 0

, k =

0 i i 0

.

Itisstraightforward tohekthe abovemultipliationrules. Similartoom-

plexonjugation, there is an

R

^{-linear map, alsoalled} onjugation,

H −→ ^∗ H

z = x 0 + x 1 i + x 2 j + x 3 k 7→ z ^∗ = x 0 − x 1 i − x 2 j − x 3 k,

satisfying the usual rule

(zw) ^∗ = w ^∗ z ^∗

^{. In the matrix} desription, this is preisely the usual

∗

^{-operation of taking the onjugate transpose. This an}

beusedtodeneaninnerprodut

h z, w i H = w ^∗ z

^{,whoserealpartistheusual}

inner produt on

R ⁴

^{. Noting that}

h z, z i ^H ∈ R

^{we an then dene a norm}

| z | = p

h z, z i H

^{. This satises}

| zw | = | z | | w |

^and

| z ^∗ | = | z |

^{. The unit sphere}

in

H

^{is usually denoted}

Sp(1) = { z ∈ H | | z | = 1 }

^{, and this is anonially}

identied with

S ³

^{. Finally we note that if}

z 6 = 0

^then

z

^{is invertible this}

is most easily seen by using the matrix desription, whih gives an expliit

inverse, and heking thatthis belongs to

H

^.

Weantakethediretprodutof

H

^{withitselftoform}

H ^r

^{. Theoperations}

h· , ·i ^H

^and

|·|

^from

H

^{areextendedto}

H ^r

^{intheusualway: For}

z = (z 1 , . . . , z r )

and

w = (w 1 , . . . , w r )

^{, we set}

h z, w i H =

X r j=1

h z j , w j i H , | z | = q

| z 1 | ² + . . . + | z r | ² .

1.2 Spaes of geodesis

Let

F

^denoteeither

C

^or

H

^{. Toeasethenotationwedenotetheunitspherein}

F

^by

S(F)

^{. Wedenetheprojetivespae}

FP ^r

^{astheset ofall}

1

-dimensional

F

^-subspaes

zF

^of

F ^r+1

^{, for}

z ∈ F ^r+1

^{. We dene the projetion map}

π : F ^r+1 \ { 0 } −→ FP ^r

⁽¹⁾

z = (z ₀ , . . . , z _r ) 7→ [z ₀ , . . . , z _r ] = zF,

so

π(z) = zF

^{is the subspae spanned by}

z

^{. Note that for}

F = H

^{it is im-}

portantthatwespeify whihside wemultiplyon; Ihavehosen tomultiply

from the right. We give

FP ^r

^{the quotient topology from}

π

^{. To show that}

FP ^r

^{is a smooth manifold of real dimension}

2r

^(resp.

4r

^{) for}

F = C

^(resp.

F = H

^{), we display the expliit harts}

h j : U j = { [z 0 , . . . , z r ] ∈ FP ^r | z j 6 = 0 } −→ F ^r , h _j ([z ₀ , . . . , z _r ]) = (z ₀ z _j ^{− 1} , . . . , z \ _j z _j ^{− 1} , . . . , z _r z _j ^{− 1} ),

where the hat denotes omission;the harts have inverses

h ⁻ _j ¹ (w ₁ , . . . , w _r ) = [w ₁ , . . . , 1, . . . , w _r ],

with the

1

^atthe

j

^thplae.

Example 1.1. We willshow

HP ¹

^{is dieomorphi to}

S ⁴

^{. This an be seen}

by stereographi projetion. Think of

S ⁴ ⊆ R ⁵ = R × H

^{with north pole}

p + = (1, 0)

^{and south pole}

p ₋ = ( − 1, 0)

^. Stereographi projetion are the maps

ψ _± : S ⁴ \ { p _± } −→ H,

whih takes a point

(t, z )

ⁱⁿ

S ⁴

^{to the} intersetion of the line through

(t, z)

and

p _±

^with

0 × H

^{. This iseasily omputed:}

ψ + (t, z) = z

1 − t , ψ ₋ (t, z) = z 1 + t ,

and arelearly smoothmaps. Now wewanttoompose

ψ +

^and

ψ ₋

^withthe

h ⁻ _j ¹

^{toget two maps to}

HP ¹

^{. When we dothis, wewould like the two maps}

toagree when

t ∈ ] − 1, 1[

^{. To ahieve this, wereplae}

ψ ₊

^{with itsonjugate}

ψ ₊ ^∗ (t, z) = ₁ ^z ₋ ^∗ _t

^{. Doing this, we get maps,}

S ⁴ \ { p + } ^ψ

∗

−→ + H ^h

− 1

−→ 0 HP ¹ , S ⁴ \ { p ₋ } −→ ^{ψ −} H ^h

− 1

−→ 1 HP ¹ ,

given by

(t, z) 7→

1, z ^∗

1 + t

, (t, z) 7→

z 1 − t , 1

.

By multiplying the rst expression from the right by

z

1 − t

^{and using that}

1 = | (t, z) | = t ² + | z | ² = t ² + z ^∗ z

^{, we see that these two maps agree when}

t ∈ ] − 1, 1[

^{, so they ombine toa} dieomorphism

S ⁴ −→ HP ¹

^.

We an modify the projetion map

π

^{in (1)toa map}

π : S(F ^r+1 ) −→ FP ^r

where

S(F ^r+1 ) ⊆ F ^r+1

^{is the unit sphere. This an be used to desribe the}

tangent bundle of

FP ^r

^{. Speially for}

z ∈ S(F ^r+1 )

^{there is an}

F

^-linear

isometry,

t z : (zF) ^⊥ ⊆ T z S(F ^r+1 ) −→ ^{π ∗} T π(z) FP ^r ,

where

(zF) ^⊥ = { w ∈ F ^r+1 | h w, z i ^F = 0 }

^{. This map satises}

t _zλ (wλ) = t _z (w)

^for

λ ∈ S(F).

⁽²⁾

The above properties of

FP ^r

^{are rather elementary, and the reader an}

see e.g. [Madsen-Tornehave℄ Chapter 14for proofs of the results in the ase

of

CP ^r

^.

Consider the Riemannian metri on

FP ^r

^{given by the real part of the}

inner produt on

F ^r+1

^{. This is the standard metri on}

FP ^r

^{, and we will}

use a metri

g

^{whih is a salar multiple of this metri. Take the unique}

onnetion on

T (FP ^r )

^{ompatible with this metri, alled the} Levi-Civita onnetion. We now dene

G(r) = G(FP ^r )

^{as the spae of} parametrized, simple, losed geodesis

f : [0, 1] −→ FP ^r

^{with respet to this onnetion.}

The salar determining

g

^{is speied by requiring that suh a geodesi has}

length 1 with respet to

g

^{. Note that every geodesi in}

FP ^r

^{is losed: The}

group of

F

-orthogonal matries (

U (r + 1)

^or

Sp(r + 1)

^, respetively) ats transitively on

HP ^r

^{, so it is onlyneessary to hek it for one geodesi, e.g.}

on

FP ¹ ⊆ FP ^r

^,andsine

CP ¹ ∼ = S ²

^and

HP ¹ ∼ = S ⁴

^{, allgeodesis on}

FP ¹

^are

known to be losed.

We also onsider the set of

n

^{times iterated geodesis}

G _n (r)

^{for every}

integer

n ≥ 1

^{, whose elements}

γ : [0, 1] −→ FP ^r

^{are given by}

γ(t) = f(nt)

forsome

f ∈ G(r)

^{, wherewe makethe obvious}identiationof the intervals

[j − 1, j]

^with

[0, 1]

^for

j = 2, . . . , n

^{. There isanationon}

G _n (r)

^by

S ¹

^given

by rotation;expliitly,

S ¹ × G n (r) −→ G n (r) (e ^2πiθ , f (t)) 7→ f (t − θ).

Wean twistthe rotationationon

G(r)

^byaninteger

n ≥ 1

^{,and wedenote}

the resulting

S ¹

^-spae

G(r) ⁽ⁿ⁾

^:

S ¹ × G(r) ⁽ⁿ⁾ −→ G(r) ⁽ⁿ⁾

⁽³⁾

(e ^2πiθ , f(t)) 7→ f (t − nθ).

This ation is the rotation ation preomposed with the

n

^{th power map}

P ⁿ : S ¹ −→ S ¹

^,

P ⁿ (z) = z ⁿ

^{in omplex notation. Then}

G n (r)

^and

G(r) ⁽ⁿ⁾

are isomorphias

S ¹

^{-spaesvia theobviousmap}

G(r) ⁽ⁿ⁾ −→ G n (r)

^{given by}

f (t) 7→ f(nt)

^{, so from now on, we will hiey use}

G(r) ⁽ⁿ⁾

^{instead of}

G _n (r)

^.

We also onsider the quotient

∆(r) = S ¹ \ G(r)

^{under the rotation ation,}

whih is the spae of oriented, unparametrized, simple, losed geodesis on

FP ^r

^.

We now want to get a more onrete desription of

G(r)

^and

∆(r)

^{, fol-}

lowing [Bökstedt-Ottosen ℄, Ÿ2. Let

V 2 = V 2 (F ^r+1 )

^{be the Stiefel manifold of}

F

-orthonormal2-frames in

F ^r+1

^,so

V 2 =

(v, w) ∈ F ^r+1 × F ^r+1 | k v k = k w k = 1, h v, w i F = 0 ,

and let

P V 2

^{be the quotient manifold by the right diagonal}

S(F)

^ation,

(v, w) ∗ z = (vz, wz)

^{. On}

V 2

^{we have a left ation of}

S ¹

^{by rotation by an}

angle

θ

^{: For}

θ ∈ R

^{, the ation is}

_w ^v

7→ R(θ) _w ^v

, where

R(θ) =

cos(θ) − sin(θ) sin(θ) cos(θ)

For eah

n ∈ N

^{, we an dene an ation of}

S ¹

^on

P V ₂

^{, and we denote the}

resulting

S ¹

^{-spae by}

P V ₂ ⁽ⁿ⁾

^:

S ¹ × P V ₂ ⁽ⁿ⁾ −→ P V ₂ ⁽ⁿ⁾ ; e ^2πiθ ∗ [x, y] = [R(nπθ)(x, y)].

This gives awell-dened

S ¹

^-ationon

P V 2

^{, beause we multiplythe matrix}

R

^{onthe left,while}

P V ₂ = V ₂ /

^diag

S(F)

^{, wherewemultiplyontheright. We}

an now make an

S ¹

-equivariant dieomorphism

ϕ ₁ : P V ₂ ⁽ⁿ⁾ −→ G(r) ⁽ⁿ⁾

⁽⁴⁾

[x, y] 7→ π ◦ c(x, y)

where

π : S(F ^r+1 ) −→ FP ^r

^{is the projetion, and}

c(x, y )

^{isthe simple losed}

geodesi startingat

x

^{in diretion}

y

^{; expliitly,}

c(x, y)(t) = cos(πt)x + sin(πt)y,

^for

t ∈ [0, 1].

This iswell-dened,and abijetionbeauseeverygeodesion

FP ^r

^islosed.

Clearly,

ϕ 1

^isa dieomorphism,and it isstraightforward to hek that it is

S ¹

-equivariant, using the trigonometriformulas.

Another very useful model for

G(r)

^is

S(τ ) = S(T (HP ^r ))

^{, the sphere}

bundle of the tangent bundle

τ

^of

FP ^r

^{. There isa} dieomorphism

ψ : P V 2 −→ S(τ )

[x, y] 7→ t x (y) ∈ T π(x) FP ^r

This is well-dened beause of (2), and we an give an expliit inverse:

Given

y ∈ T π(x) FP ^r

^,

ψ ^{− 1} (y) = [x, t ⁻ _x ¹ (y)]

^{. Thus we an give}

S(T (FP ^r ))

a rotation ation of

S ¹

^{, namely the ation that makes this} dieomorphism

S ¹

-equivariant. Combining this with (4), we have an

S ¹

-equivariant dieo- morphism

ψ ^{− 1} ◦ ϕ ₁ : S(τ ) −→ G(r).

⁽⁵⁾

The last desription only works for

CP ^r

^{. Going bak to}

P V 2 (C ^r+1 )

^{, we}

rst hange oordinates as follows

ϕ ₂ : P V ₂ (C ^r+1 ) −→ g P V ₂ (C ^r+1 ), [x, v] 7→

x + iv

√ 2 , x − iv

√ 2

.

Here

P V g 2

^is

P V 2

^{equipped the}

S ¹

^{-ationindued fromthis hangeofoordi-}

nates. Itis easilyomputed that the ation of

θ ∈ [0, 1]

^is

θ ∗ [a, b] = [za, zb]

where

z = e ^πiθ ∈ S ¹

^.

We are interested in

∆(CP ^r )

^{, i.e. we divide out the rotation ation.}

Therefore we now onsider the following spae: Let

γ 2

^{be the standard 2-}

dimensional bundle over the Grassmannian Gr

2 (C ^r+1 )

^of

2

^{-planes in}

C ^r+1

^,

and let

p : P(γ 2 ) −→

^Gr

² (C ^r+1 )

^{be the assoiated projetive bundle. Then}

P(γ 2 ) = { V 1 ⊆ V 2 ⊆ C ^r+1 | dim C (V j ) = j }

^{. Wean make a}dieomorphism,

ϕ 3 : S ¹ \ P V g 2 (C ^r+1 ) −→ P(γ 2 ), [a, b] 7→

^span

C { a } ⊆

^span

C { a, b } .

This is well-dened, but only for

F = C

^{. In onlusion we get a omposite}

S ¹

-equivariant dieomorphism

ϕ : ∆(CP ^r ) ^ϕ

− 1

1 // S ¹ \ P V 2 (C ^r+1 ) ^{ϕ 2 //} S ¹ \ P V g 2 (C ^r+1 ) ^{ϕ 3 //} P(γ 2 ).

⁽⁶⁾

1.3 Fibrations involving spaes of geodesis

We are going to ompute the ohomology and

K

^{-theory of the spaes}

G(r)

and

∆(r)

^{. In ohomology, our most important tool will be Serre's spe-}

tralsequene. I will write down the most important part; for the omplete

formulationand proof, see e.g [Hather2 ℄ Thm 1.14 pp.

Theorem 1.2 (Serre's Spetral Sequene). Let

F −→ X −→ B

^{be a -}

bration, with

B

^a path-onneted CW omplex, and

π 1 (B )

^{ating trivially}

on

H ^∗ (F ; G)

^{. Then there is a spetral sequene}

{ E _r ^p,q , d r }

^{onverging to}

H ^∗ (X; G)

^with

E ₂ ^p,q ∼ = H ^p (B; H ^q (F ; G)).

If

G = R

^{is a ring, then there is a produt}

E _r ^p,q × E _r ^s,t −→ E _r ^p+s,r+t

^{, and the}

dierentials are derivations, i.e.

d(xy) = (dx)y + ( − 1) ^p+q x(dy)

^{. For}

r = 2

the produt is

( − 1) ^qs

^{times the standard up produt. The produt struture}

on

E _∞

^{oinide with that indued by the up produt on}

H ^∗ (X; R)

^.

Forthe denitionofabration,and theuseful fatthatberbundlesare

brations, see [Hather1 ℄, p. 375 and Prop. 4.48.

Thereisasimilarresultforabrationin

K

^{-theory,butIamhiey going}

tousetheimportantspeialasewherethebrationis

∗ −→ X −→ X

^,alled

the Atiyah-Hirzebruhspetralsequene, see [Atiyah-Hirzebruh ℄:

Theorem1.3(Atiyah-HirzebruhSpetralSequene). Let

X

^{be aniteCW}

omplex. Then there is a spetral sequene

{ E _r ^p,q , d _r }

^{onverging to}

K ^∗ (X)

with

E ₂ ^p,q ∼ = H ^p (X; K ^q ( ∗ )).

We will need a way tobuild brations fromother brations, and this is

provided by the following theorem.

Theorem1.4. Let

F −→ X −→ B

^{bea bration,andassumethatthegroup}

G

^{ats freely on}

X

^{. Then,}

(i)

^Ifthe

G

^{-ationpreservesthebres,}

F/G −→ X/G −→ B

^isabration.

(ii)

^If

G

^{ats freely on}

B

^{, then}

F −→ X/G −→ B/G

^{is a bration.}

Proof. This follows from the fat that

G −→ X −→ X/G

^{is a bration,}

whihis aonsequene of the slie theorem, [Bredon ℄ Thm. 5.4.

To applythe spetral sequenes, wemust knowsome brationsinvolving

the spaes of geodesis. Firstby denition we have the bration

S ¹ −→ G(r) −→ ∆(r).

⁽⁷⁾

For the appliation of Serre's spetral sequene, note that the base is 1-

onneted. This an be seen from the long exat sequene of homotopy

groups, usingthat

G(r) ∼ = S(τ )

^{is 1-onneted.}

Then there is the map

P V 2 (F ^r+1 ) −→

^Gr

² (F ^r+1 )

indued by the map

V ₂ (F ^r+1 ) −→

^Gr

2 (F ^r+1 )

^,

(x, y) 7→ { xλ + yµ | λ, µ ∈ F }

^,

whih is well-dened on

P V 2

^{. The bre is}

P V 2 (F ² )

^{. Bythe} dieomorphism (4), this means we havethe bration

G(1) −→ G(r) −→

^Gr

² (F ^r+1 ).

Sinethe left

S

^{ationonthe totalspae isfreeand preservesthe bres,we}

an divide by it in the total spae and bre, by Theorem 1.4

(i)

^obtaining

the bration

∆(1) −→ ∆(r) −→

^Gr

² (F ^r+1 ).

⁽⁸⁾

Againwe note that the base is 1-onneted.

1.4 Homotopy orbits of spaes of geodesis

In this setion we are going to study the so-alled homotopy orbits of the

spaes of geodesis we have studied so far. For this denition we need the

following onepts: Let

G

^{be a group, and suppose we have a ontratible}

spae with a free

G

^{ation. It turns out that all suh spaes are homotopy}

equivalent, so we an dene

EG

^{to be any suh spae. We an then dene}

BG = EG/G

^{to bethe lassifying spae of}

G

^{. Notethat this isa working}

denition; atually

BG

^{is dened for a ategory, but this is all I willneed.}

For

G = S ¹

^{we nd}

ES ¹ ≃ S ^∞

^{, sine this is} ontratible. Thus we get

BS ¹ ≃ S ^∞ /S ¹ = CP ^∞

^.

Denition1.5. Let

X

^{beatopologialspaewith a(left)ationof}

S ¹

^{. We}

denethe spae of homotopy orbits of

X

^by

X hS ¹ = ES ¹ × S ¹ X = ES ¹ × X/

(e, tx) ∼ (et, x), t ∈ S ¹ .

Projetion onthe rst fator givesamap

X hS ¹ −→ BS ¹

^{, and fora oho-}

mologytheory

h ^∗

^{(weonsiderohomologyand}

K

^{-theory),wegetanindued}

map

h ^∗ (BS ¹ ) −→ h ^∗ (X _hS ¹ ).

As explained in the introdution, this gives

h ^∗ (X hS ¹ )

^{the struture of an}

h ^∗ (BS ¹ )

^-module.

Reall that

G(r)

^{is the spae of simple} parametrized geodesis with the free left ation of

S ¹

^{given by rotation. The spae of}

n

^{-times iterated}

geodesis,

G n (r)

^{, we have identied as an}

S ¹

^{-spae with}

G(r) ⁽ⁿ⁾

^{, whih is}

G(r)

^{with therotationationtwistedby the}

n

^thpowermap

P n : S ¹ −→ S ¹

^,

see (3).

horizontal maps are brations with 1-onneted base spaes:

G(r)

BC n // ES ¹ × S ¹ G(r) ^{(n) //}

∆(r)

BS ^{1 B P n //} BS ¹

Here

C _n ⊆ S ¹

^{denotes the group of}

n

^{th roots of unity.}

Proof. To see that the vertial map is a bration, use the produt bundle

G(r) ⁽ⁿ⁾ −→ ES ¹ × G(r) ⁽ⁿ⁾ −→

^pr

¹ ES ¹

^{, and divide out by the free ation of}

S ¹

onboth total spae and base,aording toTheorem 1.4

(ii)

^{. Usingthe long}

exat homotopy sequene for the bration

S ¹ −→ ES ¹ −→ BS ¹

^{shows that}

the base

BS ¹

^{is 1-onneted.}

The horizontal bration is built up in steps: We start with the produt

bre bundle,

ES ¹ −→ ES ¹ × G(r) ⁽ⁿ⁾ −→

^pr

² G(r) ⁽ⁿ⁾ .

Clearly,

C n ⊆ S ¹

^{ats freely on}

ES ¹ × G(r) ⁽ⁿ⁾

^{, preserving the bres. So by}

Theorem 1.4

(i)

^{, dividing out by}

C n

^{in the total spae and bre yields the}

bration:

BC n −→ ES ¹ × ^{C n} G(r) ⁽ⁿ⁾ −→ G(r) ⁽ⁿ⁾ .

Weget

ES ¹ /C n = BC n

^beause

ES ¹

^{is a ontratiblespae uponwhih}

C n

ats freely, and so

ES ¹ ≃ EC n

^{. Now onsider the quotient group}

S ¹ /C n

^,

whih isisomorphi to

S ¹

^{by the}

n

^{'th powermap. Sine}

C n

^{ats triviallyon}

G(r) ⁽ⁿ⁾

^{, we have an ation of}

S ¹ /C _n

^on

G(r) ⁽ⁿ⁾

^{. By denition, this ats on}

G(r) ⁽ⁿ⁾

^{exatly as}

S ¹

^{ats on}

G(r)

^{, so}

(S ¹ /C n ) \ G(r) ⁽ⁿ⁾ ∼ = S ¹ \ G(r)

^{. By}

Theorem1.4

(ii)

^{,dividingoutbythis freeationinthe totaland basespaes}

gives usthe bration

BC n −→ ES ¹ × ^{C n} G(r) ⁽ⁿ⁾

/(S ¹ /C n ) −→ S ¹ \ G(r).

Now

ES ¹ × ^C n G(r) ⁽ⁿ⁾

/(S ¹ /C n ) ∼ = ES ¹ × S ¹ G(r) ⁽ⁿ⁾

^{,bythe denitionofthe}

ations, sowe get the desired bration. As noted inSetion 1.3, the base is

1

^-onneted.

To get the ommutative square, note that we have the homotopy equiv-

alene pr

2 : ES ¹ × G(r) −→ G(r)

^{, sine}

ES ¹

^is ontratible. Sine this is an

S ¹

^{map and}

S ¹

^{ats freely on both spaes, we an use [tomDiek℄}

Prop. 2.7 to onlude that

ES ¹ × S ¹ G(r) −→ S ¹ \ G(r) = ∆(r)

^{is also a}

homotopy equivalene. The upper vertial map in the square is dened as

pr

2 : ES ¹ × S ¹ G(r) −→ ∆(r)

^{usingthis homotopy}equivalene. Fortheiden- tiation

S ¹ /C n

^with

S ¹

^{above,we usedthe}

n

^thpowermap

P n : S ¹ −→ S ¹

^,

so for the diagram to ommutate, the lower horizontal map

BS ¹ −→ BS ¹

must also be the one indued by

P ⁿ

^{. Note: This is well-dened on}

BS ¹

beause

S ¹

^is ommutative.

Remark 1.7. If we let

n = 1

^{, the vertial bration beomes}

G(r) −→

ES ¹ × S ¹ G(r) −→ BS ¹

^{. Asnotedintheproof,}

ES ¹ × S ¹ G(r) −→ S ¹ \ G(r)

^is

ahomotopyequivalene. So, uptohomotopy,wehaveinpratieabration

G(r) −→ ∆(r) −→ BS ¹ .

⁽⁹⁾

2 Cohomology of spaes of geodesis in

HP ^r

2.1 The parametrized geodesis

Inthissetionwendthe ohomologyofthespaeofparametrizedgeodesis

on

HP ^r

^,

G(r) = G(HP ^r )

^{, followed by some Lemmasneessary to determine}

thespaeoforiented,unparametrizedgeodesis,

∆(r) = ∆(HP ^r ) = S ¹ \ G(r)

^.

Theorem 2.1. As a graded ring,

H ^∗ (G(HP ^r ); Z) ∼ = Z[y, τ ]/

(r + 1)y ^r , y ^r+1 , τ ² ,

where

y ∈ H ⁴ (G(HP ^r ); Z)

^and

τ ∈ H ^4r+3 (G(HP ^r ); Z)

^.

Let

p

^{be a prime number. Then}

H ^∗ (G(HP ^r ); F p ) ∼ =

F p [y, σ]/ h y ^r+1 = 0, σ ² = 0 i , p | r + 1;

F p [y, τ ]/ h y ^r = 0, τ ² = 0 i , p ∤ r + 1.

where

y ∈ H ⁴ (G(HP ^r ); F p )

^,

σ ∈ H ^{4r − 1} (G(HP ^r ); F p )

^,

τ ∈ H ^4r+3 (G(HP ^r ); F p )

^.

Proof. We use the dieomorphismfrom(5),

G(r) ∼ = S(τ )

^{, where}

S(τ)

^isthe

sphere bundleof the tangentbundle,

S ^{4r − 1} −→ S(τ ) −→ HP ^r .

Sine

HP ^r

^{is 1-onneted, we an use Serre'sspetral sequene,}

H ^p (HP ^r ; H ^q (S ^{4r − 1} )) ⇒ H ^p+q (S(τ ))

⁽¹⁰⁾

(here the oeients will be

Z

^{atrst, and}

F p

^{toprove the lastpart) whih}

has the following

E 2

^page:

4r − 1 σ yσ y ² σ y ^r σ

0 1 y y ² y ^r

0 4 8 . . . 4r

We an see for dimensional reasons that there an only be one non-trivial

dierential,namely

d 4r (σ)

^{. Forthespherebundle,itisageneraltheoremthat}

this dierentialismultipliation by the Euler harateristi of the manifold,

here

HP ^r

^{, so}

d 4r (σ) = (r + 1)y ^r

^{. This is proved in} [Milnor-Stashe℄, Cor.

11.12and Thm. 12.2. This is an injetivemap

Z −→ Z

^{, sowhen passing to}

the

E 4r+1

^{page, the result is}

4r − 1 0 yσ y · yσ y ^{r − 2} · yσ y ^{r − 1} · yσ

0 1 y y ² y ^{r − 1} y ^r

0 4 8 . . . ^4r − 4 4r

Asmentioned,therearenoothernon-trivialdierentials,sothisis

E _∞

^{. Also,}

therearenoextensionproblemssinethereisatmostonenon-trivialgroupon

eahdiagonal

p + q = n

^,so

yσ

^denesalassin

H ^4r+3 (S(τ); Z)

^whihweall

τ

^{. We an then read o the lasses}

y ∈ H ⁴ (S(τ ); Z)

^and

τ ∈ H ^4r+3 (S(τ ); Z)

with the relations

y ^r+1 = 0

^,

(r + 1)y ^r = 0

^,and

τ ² = 0

^.

Toprovetheresultwith

F p

^{oeients,weusethesamespetralsequene}

(10), now with

F p

^{-oeients. In ase}

p | r + 1

^,

d 2 (σ) = 0

^{, so there are no}

non-trivial dierentials, and

E _∞ = E 2

^{. As above, there are no extension}

problems, and

σ

^{denes an element in}

H ^{4r − 1} (S(τ); F p )

^{. So we an read o}

thedesiredresult. Inase

p ∤ r + 1

^,

r + 1

^{isaunit in}

F p

^,so

d 2 : F p σ −→ F p y ^r

is an isomorphism. So when passing to the

E 4r+1

^{page, these two groups}

disappear. The result follows.

Now we an deal with the smallest ase,

HP ¹

^{, whih we have shown in}

Example1.1isdieomorphito

S ⁴

^{. This isgoingtobeuseful, sine wehave}

the bration

∆(HP ¹ ) −→ ∆(HP ^r ) −→

^Gr

² (H ^r+1 )

^from(8).

Lemma 2.2.

H ^∗ (∆(HP ¹ ); Z) ∼ = Z[x, t]/

2t − x ² , t ² ,

where

x ∈ H ² (∆(HP ¹ ); Z)

^and

t ∈ H ⁴ (∆(HP ¹ ); Z)

^.

Proof. Weusethebration

S ¹ −→ G(HP ¹ ) −→ ∆(HP ¹ )

^fromthe

S ¹

^ation.

Hereweknowtheohomologyofthebreandthetotalspae,thelatterfrom

Theorem 2.1,

H ⁿ (G(HP ¹ )) =

 



Z, n = 0, 7

^;

Z/2Z, n = 4

^;

0,

^else.

We an use the Serre'sspetralsequene,

H ^p (∆(HP ¹ ); H ^q (S ¹ ; Z)) ⇒ H ^p+q (G(HP ¹ ); Z),

to nd the ohomology of the base. Let

σ ∈ H ¹ (S ¹ )

^{denote a generator.}

The

E 2

^{page has only two non-zero rows. We see that the only possible}

non-trivial dierentials are

d ₂

^{, so}

E ₃ = E _∞

^{. We know the total spae has}

nothing in degree 1, so there must be zero at

(1, 0)

^{sine this annot be}

killedby anything. So

H ¹ (∆(HP ¹ )) = 0

^{, whih meansthereiszero at}

(1, 1)

^,

too. Also,

σ

^{must be killed by an outgoing} dierential, so

d ^0,1 ₂

^{is injetive.}

Atually it must be an isomorphism, otherwise something would survive in

degree2,and thereisnothing. Sowehave a

H ² (∆(HP ¹ ); Z) ∼ = Z

^generated,

say, by

x = d ₂ (σ)

^{. Letus take a look atthe}

E ₂

^{page as we knowit now:}

1 σ 0 σx ? ? ? ? ? ? · · · 0 1 0 x ? ? ? ? ? ? · · · 0 1 2 3 4 5 6 7 8 · · ·

Continuinginthisfashionweseethereiszeroat

(3, 0)

^sine

H ³ (G(HP ¹ ); Z) = 0

^{, and so also at}

(3, 1)

^{. Likewise, there are zeroes at}

(5, 0)

^and

(5, 1)

^{. Now}

onsider

d ^2,1 ₂

^{. This must be injetive, sine it starts in degree 3, where the}

total spae has nothing. Also,

d ^2,1 ₂

^{ends at}

(4, 0)

^{, and must besuh that we}

get

H ⁴ (G(HP ¹ ); Z) = Z/2Z

^{when taking the okernel of it. This means it}

must be multipliation by

± 2

^{; we might as well say 2 for} onreteness. So

H ⁴ (∆(HP ¹ ); Z) ∼ = Z

^{generated by some}

t

^{, whih we an hoose suh that}

d ₂ (σx) = 2t

^{. A quik summary:}

1 σ 0 σx 0 σt 0 ? ? ? · · · 0 1 0 x 0 t 0 ? ? ? · · · 0 1 2 3 4 5 6 7 8 · · ·

Nowwehavegottensomethingat

(4, 1)

^{,butthetotalspaehaszeroindegree}

5,so

σt

^{must bekilledby the outgoingdierential}

d ^4,1 ₂

^{. Againitmust bean}

isomorphism. Note that by the derivation property of

d 2

^,

d(σt) = d(σ)t − σd(t) = d(σ)t = xt

so

xt

^{is agenerator of}

H ⁶ (∆(HP ¹ ); Z)

^{. This givesusa}

Z

^at

(6, 1)

^generated

by

σxt

^{. Now to see what further happens, we note that}

∆(HP ¹ )

^{is at most}

7

-dimensional, sine

G(HP ¹ ) = S(T (HP ¹ ))

^{is a}

7

^{-manifold. So we know}

that

H ^∗ (∆(HP ¹ ); Z)

^{is zero above degree}

7

^{. This means that}

σxt

^annot

be killed, soit survives to

E _∞

^{, meaningthere an be nothing else in degree}

7

^{. So from olumn 7 and onwards there are zeroes in the}

E 2

^{page. Now we}

know the full story:

1 σ 0 σx 0 σt 0 σxt 0 0 · · ·

0 1 0 x 0 t 0 xt 0 0 · · ·

0 1 2 3 4 5 6 7 8 · · ·

Toget tothe bottomof the multipliativestruture wealulate:

2t = d(σx) = d(σ)x − σd(x) = d(σ)x = x ² .

Fordimensional reasons

t ² = 0

^{, and all other relationsome from these two}

(e.g.

x ³ = x ² · x = 2xt

^{). This proves the result.}

Wenowturntothegeneralaseof

∆(r)

^{. Wehavethe brationfrom(8),}

∆(HP ¹ ) −→ ∆(HP ^r ) −→

^Gr

2 (H ^r+1 ).

So in order to apply Serre's spetral sequene, we need to know the oho-

mology of Gr

2 (H ^r+1 )

^{. This is taken are of by the following Lemma, whih}

isthe quaternion version of [Bökstedt-Ottosen ℄ Thm. 3.1:

Lemma 2.3. For

r ≥ 1

^,

H ^∗ (

^Gr

2 (H ^r+1 ); Z) ∼ = Z[p 1 , p 2 ]/ h ϕ r , ϕ r+1 i ,

where

p 1 , p 2

^{arethePontryaginlassesofthestandardbundle}

γ 2 ց

^Gr

2 (H ^r+1 )

^,

and

ϕ i = ϕ i (p 1 , p 2 )

^{is the polynomial given indutively by}

ϕ 0 = 1, ϕ 1 = p 1 , ϕ i = − p 1 ϕ i − 1 − p 2 ϕ i − 2 ,

^for

i ≥ 2.

Proof. We use a result of Borel, [Borel℄ Prop. 31.1. Let

γ 2 ց

^Gr

² (H ^r+1 )

denote the standard 2-dimensional bundle, i.e. the bre over

V ⊆ H ^r+1

is

V

^{. Let}

p _i

^,

i ≥ 0

^{be the Pontryagin lasses,}

p _i ∈ H ⁴ⁱ (

^Gr

₂ (H ^r+1 )

^{, whih}

satisfy

p i = 0

^for

i > 2

^{, sine}

γ 2

^is 2-dimensional. Let

γ ¯ r − 1

^{denote its}

(r − 1)

-dimensional orthogonal omplement, i.e. the bre over

V ⊆ H ^r+1

^is

V ^⊥ ⊆ H ^r+1

^{. Denote the Pontryagin lasses of this bundleby}

p ¯ _j

^,

j ≥ 0

^,

p ¯ _j ∈ H ^4j (

^Gr

2 (H ^r+1 ))

^{,and note that}

p ¯ j = 0

^for

j > r − 1

^{. Then}

γ 2 ⊕ γ ¯ r − 1 ∼ = ε ^r+1

^,

thetrivialbundleofdimension

r + 1

^{. ThesumformulaforPontryagin lasses}

gives the relations

X

i+j=k

p i p ¯ j = ¯ p k + ¯ p k − 1 p 1 + ¯ p k − 2 p 2 = 0,

^for

k > 0

⁽¹¹⁾

Borel'stheoremstatesthat

H ^∗ (

^Gr

2 (H ^r+1 ); Z)

^{isgeneratedbythePontryagin}

lasses of

γ 2

^and

¯ γ r − 1

^{, subjet tothe relations mentioned above:}

H ^∗

^Gr

2 (H ^r+1 ); Z ∼ = Z[p i , p ¯ j | i, j > 0]/ h{ p i } i>2 , { p ¯ j } j>r − 1 , X

i+j=k

p i p ¯ j

k>0 i .

By(11) we see thatwean indutivelyexpress

p ¯ _k

^{asa polynomialin}

p ₁

^and

p 2

^{. Callthat polynomial}

ϕ k

^{, so}

p ¯ k = ϕ k (p 1 , p 2 )

^{, and we get from (11)}

ϕ 0 = 1, ϕ 1 = p 1 , ϕ i = − p 1 ϕ i − 1 − p 2 ϕ i − 1 , i ≥ 2.

Then we get

H ^∗ (

^Gr

2 (H ^r+1 ); Z) ∼ = Z[p 1 , p 2 , p ¯ j | j > 0]/

*

{ p ¯ j } _{j>r − 1} , X

i+j=k

p i p ¯ j

k>0

+

∼ = Z[p 1 , p ₂ , p ¯ ₁ , p ¯ ₂ , . . .]/ D

{ p ¯ _j } _{j>r − 1} , { p ¯ _k − ϕ _k (p ₁ , p ₂ ) } k>0

E

∼ = Z[p 1 , p 2 ]/ h ϕ k | k ≥ r i .

From the indutive formula for

ϕ k

^{it is seen that}

h ϕ k | k ≥ r i = h ϕ r , ϕ r+1 i

^,

and this proves the lemma.

2.2 The unparametrized geodesis

Reall

H ^∗ (BS ¹ ) ∼ = H ^∗ (CP ^∞ ) ∼ = Z[u]

^where

u

^{has degree 2; a fat that an}

be dedued from

H ^∗ (CP ⁿ ) ∼ = Z[u]/ h u ⁿ⁺¹ i

^.

Theorem 2.4. The spae of unparametrized oriented geodesis,

∆(HP ^r )

^,

has the following ohomology:

H ^∗ (∆(HP ^r ); Z) ∼ = Z[x, t]/ h Q r , Q r+1 i ,

where

x ∈ H ² (∆(HP ^r ); Z)

^{istheimageofthegenerator}

u ∈ H ^∗ (BS ¹ ) ∼ = Z[u]

and

t ∈ H ⁴ (∆(HP ^r ); Z)

^.

Q k

^for

k ∈ N

^{isapolynomialin}

x

^and

t

^indutively

given by

Q 0 = 1, Q 1 = 2t − x ² , Q s = (2t − x ² )Q s − 1 − t ² Q s − 2 ,

^for

s ≥ 2.

Note that Lemma 2.2 is a speial ase of this with

r = 1

^:

Q 1 = 2t − x ²

^,

and

Q 2 = (2t − x ² )Q 1 − t ² ≡ t ² (mod Q 1 )

^{. TheproofofTheorem2.4for}

HP ^r

isnot atalllikethe

CP ^r

^{ase, sine}

∆(HP ^r )

^{isnot isomorphito}

P(γ 2 )

^{, and}

the proof willtakequite sometime. First we show that the ohomologyisa

polynomial algebragenerated by lasses

x

^and

t

^{asinthe Theorem, modulo}

ertain relations. It will follow from Lemma 2.3 that the polynomials

Q r

^,

Q r+1

^{are among these relations. Then we use a purely algebrai ounting}

argumentto showthat there an beno further relations.

Proposition 2.5 (Theorem 2.4, Part1). There is a surjetive map

Z[x, t]/ h Q r , Q r+1 i ։ H ^∗ (∆(HP ^r ); Z).

Proof of Theorem 2.4,Part 1. We write down the

E 2

^{page of the Serre's}

spetral sequene for the bration

∆(HP ¹ ) −→ ∆(HP ^r ) −→

^Gr

² (H ^r+1 )

^,