The Main Theorem for r = 1

and assume

n | d ⁽ⁱ⁾ ₁ (a j )

^{for all}

i

^{. This holdsfor}

j = 1

^{. Then}

a j ∈ Z[[t]] ⊗ L j

^,

and we onsider the image of under

d ⁽ⁱ⁾ ₁

^:

Z[[t]] ⊗ ^R L j d ⁽ⁱ⁾ ₁

−→ Z[[t]] ⁽ⁱ⁾ ⊗ ^R M j ,

f _j x ^j + . . . + f _r x ^r 7→ f _j (x ^j − y ^j ) + . . . + f _r (x ^r − y ^r ).

Byassumption,

f j (x ^j − y ^j ) + . . . + f r (x ^r − y ^r ) = nb

^forsome

b

^{. Nowwe use}

the projetion

π j : M j −→ M j /M j+1

^{, whihindues amap}

Z[[t]] ⁽ⁱ⁾ ⊗ R M j π j

−→ Z[[t]] ⁽ⁱ⁾ ⊗ R M j /M j+1 , f j (x ^j − y ^j ) + . . . + f r (x ^r − y ^r ) = nb 7→ f j (x ^j − y ^j ) = n · π j (b).

We wish to map

M _j /M _j+1 −→ Z

^{. For now, assume}

r > 1

^{. If}

j > 1

^{, we use}

the map

q

^{from (69), (70). Sine}

q(x ^j − y ^j ) = 1

^for

j > 1

^{, we get}

Z[[t]] ⁽ⁱ⁾ ⊗ R M j /M j+1

−→ q Z[[t]] ⁽ⁱ⁾ ⊗ R Z,

f j (x ^j − y ^j ) = n · π j (b) 7→ f j = n · qπ j (b).

⁽⁷²⁾

If

j = 1

^{,we use the well-dened map}

q ₁ (x) = 1

^,

q ₁ (y) = 0

^{,and get the same}

result. The onlusion is that

f j (t) ∈ n · Z[[t]] ⁽ⁱ⁾ ⊗ ^R Z

^{for all}

i

^{. By Lemma}

6.7this implies

f j (t) ∈ nZ[[t]]

^{. Sine}

a j = f j (t)x ^j + a j+1

^{, indutively we get}

n | d ⁽ⁱ⁾ ₁ (a j+1 )

^{for all}

i

^{. This nishestheindution step. Thisindution shows}

that

n | f j (t)

^{for all}

j = 1, . . . , r

^{, so}

a ∈ nK _hS ¹ 1 (Σ F ⁰ , ∗ )

^.

Now take

r = 1

^{. Then}

j = 1

^{. We use the map}

q : M 1 /M 2 −→ Z

^from

(69). Then in (72), we get instead

2f 1 (t) ∈ n · Z[[t]] ⊗ ^R Z

^{. By Lemma 6.7,}

2f 1 (t) ∈ nZ[[t]]

^{, and}

2a ∈ nK _hS ¹ 1 (Σ F 0 , ∗ )

^.

Lemma 6.14. There is an

S ¹

^{transfer map}

τ

^on

K

^{-theory, whih ts into}

the followingexat sequene,

−→ K ⁰ (X) −→ ^τ K _hS ¹ 1 (X) −→ ^ϕ K _hS ¹ 1 (X) −→ ^q K ¹ (X) −→ ^τ K _hS ⁰ 1 (X) −→

where

K ^∗ (BS ¹ ) = Z[[t]]

^{, and the map}

ϕ

^is multipliation by

− t

^.

Proof. Let

T −→ BS ¹

^{denote the standard omplex line bundle, as usual.}

Let

p : ES ¹ × S ¹ X −→ BS ¹

^{,beprojetionontherstfator, andlet}

ξ = p ^∗ T

denotethe pullbak. As in(48), we use the ober sequene,

S(ξ) −→ D(ξ) −→ T h(ξ).

As shown in (49),

S(ξ) ∼ = ES ¹ × X ≃ X

^{. The long exat sequene on}

K

^{-theory beomes, using the Thom} isomorphism,f. [Atiyah℄ Cor. 2.7.3,

K ^{∗− 1} (X) −→ ^δ K ^∗ (ES ¹ × S ¹ X) −→ ^ϕ K ^∗ (ES ¹ × S ¹ X) −→ K ^∗ (X) −→ ^δ

The map

ϕ

^{is given by} multipliation with

Λ ₋ 1 (T ) = 1 − T = − t

^{, sine}

T

^is

alinebundle. Wedenethe

S ¹

^{transfer map}

τ

^{tobethe boundary map}

δ

ⁱⁿ

the long exat sequene.

By exatness, Im

(τ ) =

^Ker

(ϕ)

^{, and sowe willneed the kernel of}

t

^:

Lemma 6.15. The kernel of the map givenby multipliation by

t

^,

t : Z[[t]] ^(k) ⊗ R Z[x]/x ² −→ Z[[t]] ^(k) ⊗ R Z[x]/x ²

is

Zp k − 1 (t)x

^{, where}

(t + 1) ^k − 1 = tp k − 1 (t)

^.

Proof. Firstwerelate thekernelof

t

^{tothe kernelof}

u : M −→ M

^{(this part}

holds for all

r

^{). Reall}

R = R(S ¹ ) = Z[U, U ^{− 1} ]

^{, and let}

u = U − 1

^{. Then}

M

^{is an}

R

^{-module by}

u 7→ (x − y)/(1 + y)

^{, and}

Z[[t]] ^(k)

^{is an}

R

^{-module by}

u 7→ (t + 1) ^k − 1

^{. Consider the exat sequene}

0 ^// Z[[t]] ^{t //} Z[[t]] ^// Z ^// 0.

Tensoring with

M

^over

R

^{yieldsthe exat sequene}

0 ^//

^Tor

^R ₁ (Z, M ) ^// Z[[t]] ^(k) ⊗ R M ^{t //} Z[[t]] ^(k) ⊗ R M

To ompute Ker

(t) ∼ =

^Tor

^R ₁ (Z, M )

^{, we use the following free resolution of}

Z

over

R

^:

0 ^// R ^{u //} R ^// Z ^// 0.

Again, we tensorover

R

^with

M

^{and nd}

0 ^//

^Tor

^R ₁ (Z, M) ^// R ⊗ ^R M ^{u //} R ⊗ ^R M ^// Z[[t]] ^(k) ⊗ R Z ^// 0.

so Tor

R

1 (Z, M ) ∼ =

^Ker

(u)

^{. All we need to know is how to translate from}

Ker

(u)

^toKer

(t)

^{. The followingdiagram,}

0 ^//

^Ker

(u) ^//

R ⊗ R M ^{u //}

p k− 1 (t) ⊗

^id

R ⊗ R M ^//

Z[[t]] ^(k) ⊗ R Z

id

0 ^//

^Ker

(t) ^// Z[[t]] ^(k) ⊗ ^R M ^{t //} Z[[t]] ^(k) ⊗ ^R M ^// Z[[t]] ^(k) ⊗ ^R Z

isommutative,sine

tp k − 1 (t) = (t + 1) ^k − 1 = u

^{. Fromthis diagram,wesee}

that Ker

(t) = p k − 1 (t)

^Ker

(u)

^.

So all that remains is to determine Ker

(u)

^{. This an be done for any}

r

^,

but it is espeially easy when

r = 1

^{, and}

M = Z[x]/x ²

^{, where}

u1 = 2x

^and

ux = 0

^{. Clearly Ker}

(u) = Zx

^{, and so Ker}

(t) = Zp k − 1 (t)x

^.

Wean now provethe MainTheorem inase

r = 1

^:

Proof of Theorem6.13. Bythe exat sequene

K _hS ¹ 1 (Σ F ⁰ , ∗ ) −→ ^δ K ˜ _hS ¹ 1 ( F ∞ / F ⁰ ) −→ K _hS ¹ 1 ( F ∞ ) −→ 0,

we see that

K _hS ¹ 1 (LCP ¹ ) = K _hS ¹ 1 ( F ∞ )

^{is isomorphi to the okernel Cok}

(δ)

of

δ

^{. Sine}

r = 1

^,

K _hS ¹ 1 (Σ F 0 , ∗ ) = Z[[t]] · h

^{, solet}

f(t) ∈ Z[[t]]

^{begiven, and}

assumethat

δ(f (t)h)

^{is divisibleby}

2

^{. We willshow thatthis implies}

f(t)

^is

divisible by

2

^{,meaning that there isno}

2

^{-torsion inCok}

(δ)

^.

For ontradition, assume that

f (t)

^{is not divisible by}

2

^{. Then, without}

loss of generality,

f(t)

^{has the form}

t ^l g(t)

^{, where}

g(t) = 1 + tp(t)

^{for some}

p(t) ∈ Z[[t]]

^{. Here}

l

^{is the rst exponentin}

f(t)

^{with anodd oeient,and}

so

2 | δ(f(t)h)

^ifandonlyif

2 | δ(t ^l g(t)h)

^{. Then}

g (t)

^isaunitin

Z[[t]]

^,sosine

δ

^{is a}

Z[[t]]

^-module homomorphism,

2 | δ(t ^l g (t)h)

^{if and only if}

2 | δ(t ^l h)

^.

We haveshown that if

δ(f (t)h)

^{is divisibleby}

2

^,but

f(t)

^{is not divisible by}

2

^,then

δ(t ^{N − 1} h)

^{is alsodivisible by}

2

^forall

N > l

^.

We will now show that this leads to a ontradition if

N = 2 ⁿ > l

^.

Consider the omposite map, whih we all

d ^(N) ₂

^,

K _hS ¹ 1 (Σ F ⁰ , ∗ ) ^{δ //} K ˜ _hS ¹ 1 ( F ∞ / F ⁰ ) ^// K ˜ _hS ¹ 1 ( F ^2N / F ⁰ ) ^P

∗

N // K ˜ _hS ¹ 1 ( F 2 ^{(N )} / F ⁰ )

Then

d ^(N) ₂ (t ^{N − 1} h)

^{is divisible by}

2

^{, sine}

δ(t ^{N − 1} h)

^{is. We will} investigate

d ^(N) ₂ (t ^{N − 1} h)

^{via the following diagram:}

Σ F 0 ^oo ( F 1 / F 0 ) ^(2N)

'' O O O O O O O O O O O

( F ¹ / F ⁰ ) ^{(N) //}

OO

( F ² / F ⁰ ) ^{(N ) //}

gg OOO

OOO OOO OOO O

( F ² / F ¹ ) ^(N)

The maps into

Σ F ⁰

^{are the ones induing the various} dierentials in the Morse spetral sequenes. The map

( F 1 / F 0 ) ^(2N) −→ ( F 2 / F 1 ) ^{(N )}

^{is simply}

the omposite of the two other maps inthe triangle

( F 1 / F 0 ) ^{(2N) P}

(N)

2 // ( F 2 / F 0 ) ^{(N) //} ( F 2 / F 1 ) ^{(N )} .

On

S ¹

-equivariant

K

^{-theory this beomes}

K _hS ¹ 1 (Σ F ⁰ ) ^d

(2N) 1 //

d ^(N) ₁

d ^(N) ₂

))

S S S S S S S S S S S S S S S

K ˜ _hS ¹ 1 (( F ¹ / F ⁰ ) ^(2N) )

K ˜ _hS ¹ 1 (( F 1 / F 0 ) ^(N) ) ^{oo i} K ˜ _hS ¹ 1 (( F 2 / F 0 ) ^(N) )

k

OO

K ˜ _hS ¹ 1 (( F 2 / F 1 ) ^{(N )} )

oo j E N

ii SSS

SSSS SSSS SSSS

(73)

with the lowerrow short exat (

i

^{surjetive and}

j

^{injetive). When}

N = 2 ⁿ

^,

we have

p N − 1 (t) = t ^{− 1} ((t + 1) ^{2 n} − 1) = t ^{N − 1} + 2q(t),

for some polynomial

q(t)

^{, sine allbinomial oeients}

² _j ⁿ

are divisible by

2

^for

j 6 = 0, 2 ⁿ

^{. Sinewe havededued that}

d ^(N ₂ ⁾ (t ^{N − 1} h)

^{isdivisible by}

2

^{, we}

therefore get

d ^(N) ₂ (p _{N − 1} (t)h)

^{is also divisible by}

2

^{, say}

d ^(N) ₂ (p _{N − 1} (t)h) = 2a

forsome

a

ⁱⁿ

K ˜ _hS ¹ 1 (( F 2 / F 0 ) ^(N) )

^{. ByLemma5.3weseethat}

d ^(N) ₁ (p _{N − 1} (t)h) = 2p N − 1 (t)x

^{. Sine the diagram(73) is} ommutative, we get

i(a) = p N − 1 (t)x

^,

sine the group

K ˜ _hS ¹ 1 (( F 1 / F 0 ) ^(N) )

^istorsion-free, see Lemma6.4.

We now use the

S ¹

^{transfer, see Lemma 6.14. We an hoose a transfer}

lass

e ∈ K ¹ ( F ¹ / F ⁰ )

^{, suh that}

τ(e) = p N − 1 (t)x

^{by Lemma 6.15. We an}

liftthis transfer lassto

e ¯ ∈ K ¹ ( F 2 / F 0 )

^{, so}

i(τ(¯ e)) = τ (e) = p N − 1 (t)x

^{. Thus}

we have an element

w = a − τ(¯ e) ∈ K ˜ _hS ¹ 1 (( F 2 / F 0 ) ^{(N )} )

^with

i(w) = 0

^{. By}

exatness of thelowerrowin(73), thereisanelement

z ∈ K ˜ _hS ¹ 1 (( F ² / F ¹ ) ^(N) )

with

j(z) = w

^{. By} ommutativity of (73), we get

E N (z) = k(w) = k(a − τ(¯ e)) = k(a) − k(τ(¯ e)),

so let us ompute this. Sine

2a = d ^(N) ₂ (p N − 1 h)

^{, we see that}

k(2a) =

d ^(2N ₁ ⁾ (p N − 1 h) = 2p N − 1 x

^{, and sine}

K ˜ _hS ¹ 1 (( F ¹ / F ⁰ ) ^(2N) )

^is torsion-free,

k(a) =

p _{N − 1} x

^{. But}

k(τ (¯ e))

^{isin the image of the transfer map, so by Lemma 6.15,}

k(τ (¯ e)) = mp 2N − 1 (t)x

^{for some}

m ∈ Z

^{. In onlusion,}

E N (z) = k(w) = (p N − 1 (t) − mp 2N − 1 (t))x.

⁽⁷⁴⁾

Toinvestigatethis equality,wewillneed touse

F 2

^{-oeients,and to}

deter-mine the map

E N

^{. This isdone in the followinglemmas:}

Lemma 6.16. As

K ^∗ (BS ¹ ) = Z[[t]]

^-modules,

K ˜ _hS ¹ 1 (( F ¹ / F ⁰ ) ^{(2 k )} ; F 2 ) ∼ = (F 2 [t]/t ^{2 k} )1 ⊕ (F 2 [t]/t ^{2 k} )x.

Proof. As explainedin the beginning,

K ˜ _hS ¹ 1 (( F ¹ / F ⁰ ) ^{(2 k )} ; F 2 ) ∼ = Z[[t]] ^{(2 k )} ⊗ ^R M ⊗ ^Z F 2 ,

where

M = Z[x]/x ²

^,and

u1 = 2x

^,

ux = 0

^{. Soweseethat}

M ⊗ ^Z F 2 = F 2 ⊕ F 2

istrivial asan

R = Z[U, U ^{− 1} ]

^{-module. So}

Z[[t]] ^{(2 k )} ⊗ R M ⊗ Z F 2 = (Z[[t]] ^{(2 k )} ⊗ R F 2 )1 ⊕ (Z[[t]] ^{(2 k )} ⊗ R F 2 )x.

On

Z[[t]] ^{(2 k )}

^,

u

^atsas

(t + 1) ^{2 k} − 1 ≡ t ^{2 k} (

^mod

2)

^{. Therefore,}

Z[[t]] ^{(2 k )} ⊗ R F 2 = F 2 [t]/t ^{2 k}

^{. This shows the Lemma.}

Lemma 6.17. The map

E _N

^is multipliation by

1 − (t + 1) ^N

^.

Proof. Wemust determinethemap induedby

( F 1 / F 0 ) ^{(2N )} −→ ( F 2 / F 1 ) ^{(N )}

^,

whih is the

(N )

^{-twistingof the omposite map}

( F 1 / F 0 ) ⁽²⁾ −→ F ^{P 2} 2 / F 0 −→ F 2 / F 1 .

We willrst study this untwistedase. The indued map, allit

E

^{, isgiven}

asfollows:

K ˜ _hS ¹ 1 ( F ² / F ¹ ) ^{∼ = //}

K ˜ _hS ¹ 1 (T h(µ ⁻ ₂ )) ^{Φ 2 //}

K _hS ⁰ 1 (G 2 (r))

E

˜

K _hS ¹ 1 (( F ² / F ¹ ) ⁽²⁾ ) ^{∼ = //} K ˜ _hS ¹ 1 (T h((µ ⁻ ₁ ) ⁽²⁾ )) ^{Φ 1 //} K _hS ⁰ 1 (G(r) ⁽²⁾ )

wherethe rstisomorphismsare Morse theory,and the

Φ j

^{denotethe Thom}

isomorphisms (the index indiates whih negative bundle). Also,

G 2 (r)

^is

the geodesis of length2, whih asan

S ¹

^{-spae isisomorphi to}

G(r) ⁽²⁾

^,the

(2)

^{-twistedspae of simple losedgeodesis of length 1.}

This isaspeialase of thefollowinggeneralsituation: Forabundle and

a subbundle,

ξ ⊆ η

^{, overa spae}

X

^{, the following diagramommutes}

K ˜ ^∗ (T h(η)) ^// K ˜ ^∗ (T h(ξ))

K ^∗ (X)

Φ η ∼ =

OO

Λ // K ^∗ (X)

Φ ξ ∼ =

OO

The vertial maps are the Thom isomorphisms. Then the indued map on

K

^{-theory of the base spae is given by} multipliation by the Euler lass

Λ = Λ ₋ 1 (η − ξ)

^{of the bundle}

η − ξ

^{, i.e. the} (orthogonal) omplement of

ξ

inside

η

^.

Sowe need the negative bundle

µ ⁻ ₂ = ε 2 ⊕ ν 2

^over

G 2 (r)

^,see Proposition 4.2. I have given

ε

^and

ν

^{anindex, so one an} distinguish between them for

µ ⁻ ₂

^and

µ ⁻ ₁

^{. Now}

(µ ⁻ ₁ ) ⁽²⁾

^{isnotapriori asubbundleof}

µ ⁻ ₂

^{, butsine}

µ ⁻ ₁ = ε 1

wherethe

S ¹

^{ationistrivialonthebers,weseethat}

(µ ⁻ ₁ ) ⁽²⁾ = ε 2

^asbundles

over

G(r) ⁽²⁾ ∼ = G 2 (r)

^{, sothat}

µ ⁻ ₂ − (µ ⁻ ₁ ) ⁽²⁾ = ν 2 = ν

^{, where}

ν

^{is the omplex}

bundlefound inthe proof ofProposition4.2. From here, weknowthatfor a

geodesi

f

^{of length 2,} parametrizedas

f(t)

^for

t ∈ [0, 1]

^{, the berof}

ν

^over

f

^{is given by}

g(t)if ^′ (t)

^for

t ∈ [0, 1]

^{, where}

g ∈

^span

R { cos(2πt), sin(2πt) }

^.

The rotationation of

S ¹

^{isgiven by, for}

θ ∈ [0, 1]

^:

θ ∗ (f (t), cos(2πt)if ^′ (t)) = (f (t − θ), cos(2πt − 2πθ)if ^′ (t − θ))

and similarly for

sin(2πt)

^{. The omplex struture}

J

^{found in the proof of}

Proposition4.2 is

J (cos(2πt)) = sin(2πt)

^.

Nowletusomparethis tothebundle

T

^{,i.e. thebundleomingfromthe}

standard representation of

S ¹

^{. Ignoring the}

S ¹

^ation,

T

^{is just a produt}

bundle

G 2 (r) × C

^{. The}

S ¹

^{ation of}

θ ∈ [0, 1]

^{is given by}

θ ∗ (f (t), c) = (f(t − θ), e ^2πiθ c),

^for

t ∈ [0, 1].

Wewillnow onstrut a map

ϕ : T −→ ν

^{, given by}

ϕ(f, c)(t) = (f(t), c cos(2πt)if ^′ (t)).

We hek that this is

S ¹

^{-equivariant, i.e. that the following diagram}

om-mutes (it sues to hek

c = 1

^):

T ^{ϕ //}

θ ∗

ν

θ ∗

(f (t), 1) ^{ϕ //}

_

θ ∗

(f (t), cos(2πt)if ^′ (t))

_

θ ∗ ?

T ^{ϕ //} ν (f (t − θ), e ^2πiθ ) ^{ϕ //} (f(t − θ), e ^2πiθ cos(2πt)if ^′ (t − θ))

This ommutes, sine

e ^2πiθ = cos(2πθ) + i sin(2πθ)

^{is multipliedon}

cos(2πt)

as

e ^2πiθ cos(2πt) = cos(2πθ) cos(2πt) + sin(2πθ)J(cos(2πt))

= cos(2πθ) cos(2πt) + sin(2πθ) sin(2πt)

= cos(2π(t − θ))

by the trigonometri formula. So

ϕ

^is

S ¹

-equivariant. Then

ϕ

^{denes an}

isomorphism of

S ¹

^{bundles, sine it is learly an} isomorphism on the bers.

We haveshown

µ ⁻ ₂ − (µ ⁻ ₁ ) ⁽²⁾ = ν ∼ = T

^.

Now let us look at the

(N)

^{-twisted ase. We get again}

(µ ⁻ ₁ ) ^{(2N )} = ε 2N

^,

and so

(µ ⁻ ₂ ) ^(N) − (µ ⁻ ₁ ) ^(2N) = ν ^{(N )} ∼ = T ^(N)

^{, by the above} isomorphism. Now,

T ^{(N )}

^{is the bundle with}

S ¹

^ationof

θ ∈ [0, 1]

^{given by}

θ ∗ (f (t), c) = (f (t − θ), (e ^2πit ) ^N c),

^for

t ∈ [0, 1].

This shows that this is the same bundle as

T ^N

^{, so the map}

E N

^is

multi-pliation by the Euler lass of

T ^N

^{, and sine this is a line bundle, we get}

Λ _{− 1} (T ^N ) = 1 − T ^N = 1 − (t + 1) ^N

^.

Using the previous two lemmas, we an now investigate equation (74)

in

K ˜ _hS ¹ 1 (( F ¹ / F ⁰ ) ^{(2N )} ; F 2 )

^{, where}

N = 2 ⁿ

^{. As already noted,}

p N − 1 (t) ≡ t ^{N − 1} (

^mod

2)

^{, and so the left-hand side of (74) is}

(t ^{N − 1} − mt ^{2N − 1} )x

mod-ulo 2. The right-hand side is

E N (z) = (1 − (t + 1) ^{2 n} )z ≡ − t ^{2 n} z(

^mod

2)

^.

So

(t ^{N − 1} − mt ^{2N − 1} )x = − t ^N z ∈ (Z[t]/t ^2N )1 ⊕ (Z[t]/t ^2N )x

Clearly,thisisimpossible,sinetheterm

t ^{N − 1} x

^{annotbeanelledby}

− t ^N z

in

(Z[t]/t ^2N )1 ⊕ (Z[t]/t ^2N )x

^{. This gives a} ontradition, so the given

f

^we

startedwith must be divisible by

2

^{. This proves the Theorem.}

Notation

Inthistableanbefoundsomeofthefrequentlyusednotationinthis paper:

≃

^{(between topologialspaes): homotopy} equivalent.

F C

^or

H

^.

G(r)

^{The spae of simple} parametrizedlosed geodesis on

FP ^r

^.

Sometimes written

G(HP ^r )

^or

G(CP ^r )

^{tobe spei.}

G _n (r)

^{The spae of} parametrizedlosed geodesis of length

n

^,

an be obtained by iterating

n

^{times the elements of}

G(r)

^.

∆(r)

^{The quotient}

S ¹ \ G(r)

^{under the rotationation of}

S ¹

^.

EG

^{A ontratiblespae with a freeation of the group}

G

^;

unique up to homotopy.

BG EG/G

^{,the lassifying spaeof}

G

^.

X hS ¹ ES ¹ × ^{S 1} X

^{, where}

X

^{is an}

S ¹

^-spae.

K _hS ^∗ 1 (X) K ^∗ (X _hS ¹ )

^.

K _hS ^∗ 1 (X, ∗ )

^{The relativegroup}

K ^∗ (X hS ¹ , BS ¹ )

^.

T

^{The standard omplex linebundle over}

BS ¹ = CP ^∞

^{, or its}

pullbak to

X hS ¹

^{underthe map pr}

1 : ES ¹ × S ¹ X −→ BS ¹

^.

Also used forthe lass of this bundle inK-theory.

t

^{the lass}

T − 1

^{, see}

T

^.

F ⁿ E ^{− 1} (] − ∞ , n ² ])

^,the

n

^{th term inthe Morse ltration.}

µ ⁻ _n

^{the negativebundle for the ritial manifold}

G _n (r)

^.

Referenes

[Atiyah℄ M. Atiyah, K-theory,Benjamin, New York,1967.

[Atiyah2℄ M. Atiyah, Charaters and ohomology of nite groups, Publ.

Math. InstitutdesHautes ÉtudesSientiques,Paris(1961),247-289.

[Atiyah-Hirzebruh℄ M. Atiyah, F. Hirzebruh,Vetor bundles and

homoge-neousspaes,Proeedings ofSymposiuminPureMathematis,Vol3,

Amer. Math. So., 1961.

[Atiyah-MaDonald℄ M.Atiyah, I.MaDonald,Introdution toommutative

Algebra, Addison-Wesley, 1969.

[Bökstedt-Ottosen℄ M. Bökstedt, I. Ottosen, String Cohomology Groups of

Complex Projetive Spaes, University of Aarhus, Preprint Series no

8, 2006.

[Bökstedt-Ottosen2℄ M. Bökstedt, I. Ottosen, The Suspended Free Loop

Spae of a Symmetri Spae, University of Aarhus, Preprint Series

no 18,2004.

[Borel℄ A. Borel, Sur la ohomologie des espaes brés prinipaux et des

es-paes homogènes de groupes de Lie ompats. Ann. Math. 57, 1953,

p. 115-207.

[Bredon℄ G.Bredon, Introdutionto Compat Transformation Groups,

Aa-demi Press, 1972.

[Freed-Hopkins-Teleman℄ D.Freed, M.Hopkins, C. Teleman, Twisted

equiv-ariant

K

^{-theory with omplex oeients, Journal of Topology 2008}

1(1), 16-44.

[Hather1℄ A. Hather, Algebrai Topology, Cambridge University Press,

2002.

[Hather2℄ A. Hather, Spetral Sequenes in Algebrai Topology, A.

Hather's homepage (www.math.ornell.edu/

∼

^hather/).

[Klilngenberg1℄ W. Klingenberg, Letures on losed geodesis, Grundlehren

der Math. Wiss.230, Springer Verlag, 1978.

[Klingenberg2℄ W. Klingenberg, The spae of losed urves on a projetive

In document Afhandling (Sider 84-93)