写像空間と評価写像の代数的模型について

写像空間と評価写像の代数的模型について

栗林 勝彦 信州大学理学部

1.

CW

,

S _∗ ( − )

.

S _∗ ( − )

,

.

(

)

.

,

.

,

,

(

)

.

.

,

,

,

.

この稿を通じて「空間」は連結な

CW

.

2.

空間の模型を定義するために

,

¹ .

定義

2.1.

φ

C

. C

W e

Cof

,

(C1),(C2),(C3)

C

[24]

. W e, Cof

' // , ^{// //}

,

,

.

(C1) C

f : A → B

g : B → C

f , g, gf

W e

W e

.

f, g ∈ Cof

f g ∈ Cof

.

(C2)

i : A → B, j : A → C

Cof

push-out A

²² j

²² // ⁱ // B

j

²²

C i

// B `

A C

が存在し

, i ⁰ , j ⁰

Cof

.

j ∈ W e

j ⁰ ∈ W e

.

(C3)

f : A → B

i ∈ Cof , q ∈ W e

f = qi

² .

対象

R

i : R ^{// ' //} B

r : B ^// R

ri = 1 _R

, R

.

φ → A

A

.

Quillen

[43]

Baues

[3]

する

.

1 § 2

[24]

Hess

.

,

”

”

.

Hess[24]

.

2

.

1

さてシリンダー圏

C

, (C2)

(id+id) : A `

A → A

.

(C3)

,

A `

A ^{// i //} IA _' ^{p //} A

を得る

.

IA

³ .

定義

2.2.

C

f, g : A → B

,

IA

H : IA → B

f + g = H ◦ i

, f

g

(f ' g)

.

ファイブラントかつコファイブラント対象のなす充満部分圏

C cf

'

[3, II, § 2].

,

.

2.3. T

, C

.

M : T → C

M : h( T ) → h( C )

⁴ . T

X,

f

M (X)(

), M (f)(

)

X

,

f

.

3.

この章ではいくつかのシリンダー圏またはモデル圏に付随して現れる空間の代数的模型 について紹介する

⁵ .

3.1.

.

Q

(Sullivan[45]

).

, Bousfield-Gugenheim[5]

,

A P L

,

D

∆ S

∆ : D → ∆ S , Ω : ∆ S → D

を

∆A := Hom _D (A, A P L ), ΩK := Hom ∆ S (K, A P L )

.

X

, A _{P L} (X) := ΩS _∗ (X)

.

A

| A |

∆A

: | A | = | ∆A | .

[5].

h( Q

,

,

)

∼ = A

( · )

²²

h( Q

).

|·| OO

Q

⁶

W e,

F ib, W e ∩ F ib

LLP(left lifting property)

Cof

[5].

また知られているように

, Q

Quillen

Q

(

)Lie

h(DGL 0 )

る

.

,

h(DGL ₀ )

[44].

上述の圏の同値性を用いて

,

.

[21], pull-back

[46], [47], L-S

[16]

.

.

3

.

4

h( C )

C

W e

C

(

)

[3, (3.6)Proposition], [15, 6.2. Theorem].

5

. Goerss[18]

.

.

6

.

2

3.2. Lie

, Adams-Hilton

.

Dwyer[14], Hess[24]

tame

Z

Lie

.

Lie

.

R _∗ = { R _k } k ≥ 0

R _k := Z [ _n ¹ ; 2n ≤ k + 2]

.

CW

, W e

π _k+1 (f ) ⊗ R _k

f

, Cof

CW 1 (R _∗ )

.

Lie

, W e

H _k (ϕ) ⊗ R _k

Lie

ϕ

, Cof

Lie

L → L `

L (V )

,

DGL 0 (R _∗ )

.

L (V )

V

Lie

. tame

M : h(CW ₁ (R _∗ )) −−−→ ^{∼ =} h(DGL ₀ (R _∗ ))

.

X

X = (pt.) ∪ ( S

j ∈ J e _j ), deg e _j > 1

M (X)

M (X) = ( L (x j ; j ∈ J ), d), deg x j = deg e j − 1

Lie

,

U

U M

Adams-Hilton

[24].

p

, S = Z [(p − 1)! ^{− 1} ]

.

Anick

([2, Theorem 8.5])

M ⊗ S

p

CW

S

Lie

DGL 0 (S)

.

Anick[1], Dupont, Hess[12][13]

pull-back

,

Adams-Hilton

.

3.3. TV-

. 1.1

Sullivan

TV-

型

[23]

.

(

)

W e

, Cof

A → A `

T (V )

,

DA

[3,

(7.10) Theorem].

T (V )

V

る

.

DA

(

)

,

⁷ .

M : h( ? ) −−−→ ^{∼ =} h(DA).

Munkholm[40]

Eilenberg-Moore

shc-

(strongly homotopy commutative algebra)

. shc-

A

,

bar

B, cobar

Ω

µ _A : ΩB(A ⊗ A) → ΩB(A)

が存在するものとして定義される

.

shc-

”

”

⁸ .

shc-

.

Munkholm

shc-

.

Ndombol, Thomas[41]

shc-

TV-

, shc-

型や

shc-formal

.

shc-

ても展開出来るようになった

⁹ .

shc-

pull-back

[29], TV-

[32]

Steenrod

shc-

, π-shc-

(Ndombol-Thomas[42])

.

7

8

,

. 9

.

3

3.4.

( E -

).

E

E - A lg

W e

, F ib

, Cof

W e ∩ F ib

LLP

E - A lg

(C1), (C3)

¹⁰ , (C2)

, A

¹¹ .

, ( E - A lg) c = ( E - A lg) cf

,

¹² .

E

F ^p

E _∞

(

) E _∞ F ^p

, Mandell

p-adic

[38]

は

,

E - A lg

¹³ .

.

h(

p-

,

, p-

)

∼ = C

( · ; F

)

²²

h(E _∞ F ^p - A lg

)

|·|

OO

C ^∗ ( − ; F ^p )

F ^p

.

| · |

.

,

A _{P L}

E _∞ F p -

C ^∗ (∆[ • ]; F p )

| A | := Hom _{E−A lg} (A, C ^∗ (∆[ • ]; F p ))

¹⁴ .

Mandell

E _∞

E _∞ F p

.

たい空間の性質に従いオペラドを使い分けることも可能であるが

,

.

, n

Berger, Fresse[4]

により

,

Chataur[10]

,

.

4.

BPS

この章ではオペラド上の代数を思い出し

,

.

K

.

.

定義

4.1. E = {E (r) } r ≥ 0

K -

. (1)

E (r)

K

Σ r

.

(2)

γ : E (k) ⊗ E (r ₁ ) ⊗ · · · ⊗ E (r _k ) → E (r ₁ + · · · +r _k )

(unital),

,

¹⁵ .

例

4.2. (The endomorphism operad) (V, d)

K

.

End _V (n)

End _V (n) := Hom _K (V ^{⊗ n} , V ), δf = df − ( − 1) ^{deg f} f d

.

γ : End _V (r) ⊗ End _V (i 1 ) ⊗ · · · ⊗ End _V (i r ) → End _V (i 1 + · · · + i r )

を

γ(α, β 1 , ..., β r ) = α ◦ (β 1 ⊗ · · · ⊗ β r )

End V := { End V (n) }

.

4.3.

E = {E (r) } r ≥ 0

E (r)

K [Σ _r ]-

acyclic

E

E _∞ -

.

定義

4.4. E

, A

(

)

.

E → End _A

, A

E -

.

10 [38, Proposition 2.6]

11 [38, Corollary 2.9]

[15, 3.14 Proposition (ii)]

. 12

§ 4

.

13

E - A lg

[38, Characterization Theorem].

14

∆[ • ]

∆[n] := Hom

( , [n])

. 15

[28]

4

FACT ( McClure-Smith[39], Berger-Fresse[4] ¹⁶ ) C ^∗ ( − ; K )

K

.

E _∞ K -

E

,

X

C ^∗ (X; K )

E

.

C ^∗ ( − ; K )

E

:

C ^∗ ( ; K ) : ∆ S −−−→ E - A lg.

以下では

F (X, Y )

X

Y

空間または

,

X

Y

¹⁷

.

写像空間のオペラド模型を述べる前に

,

Lannes

.

Hom _{E - A lg} (( : B) _{E - A lg} , D) ∼ = Hom _E - A lg ( , B ⊗ D)

定理

4.5. (Fresse [17] ¹⁸ , Chataur-K[11, Theorem 1.3]) Y

n-

,

p-

¹⁹ ,

, X

n ≥ dim X

.

E - A lg

:

C ^∗ ( F (X, Y ); F p ) ^quasi-iso ' ( E (V ) :C ^∗ (X; F p )) _{E -Alg}

ただし

, E (V ) → ^' C ^∗ (Y ; F ^p )

Y

(cofibrant replacement )

. E (V )

,

Lannes

.

”

”

.

4.6

, Bousfield, Peterson, Smith[6]

BPS

.

mod p BPS

,

Steenrod

.

(

)P ^s

p > 2

βP ^s , s ∈ Z

Adem relations

B

,

Steenrod

. Mandell

[38, Theorem 1.4]

B /(Id − P ⁰ )

Steenrod

A

.

B

F p -

K - F p

.

F p -

H ^∗ ( )

.

定理

4.6. [11, Theorem 1.4]

4.5

, H ^∗ ( F (X, Y ))

2,3

{ E r , d r }

:

E ₂ ^{s, ∗} ∼ = L _s (H ^∗ (Y ) : H ^∗ (X)) _{K - F}

p

.

さらに

{ E r , d r }

X

Y

.

, L s ( − : H ^∗ (X)) _K - _F

K - F ^p

Lannes

( − : H ^∗ (X)) _K - F

s

.

定理

4.7. [11, Theorem 1.5] A

B

A -

.

s

, L _s (B : A) _K - F

L _s (B : A) _S - F

B -

.

この定理の系として

,

mod p BPS

2

.

.

命題

4.8. [11, Theorem A.7] mod p BPS

{ E r , d r }

,

B

(

A

)

.

(1) β ^ε P ⁱ d r = ( − 1) ^ε d r β ^ε P ⁱ ,

16 Eilenberg-Zilber

Z (the “endomorphism operad” [25])

C

(X )

.

E

-

Barratt-Eccles

E

Z

[4]: E (r) :=

C

(the bar construction of Σ

) → Z (r) := Nat(C

( )

, C

( )) 17 F (X, Y )

= Hom

(X × ∆[ • ], Y )

.

18 Fress

F (X, Y )

F

π

(X)

. 19 mod p

5

(2)

β ^ε P ⁱ ∈ B

β ^ε P ⁱ : E r ^s,t → E r ^{s,t+2i(p − 1)+ε} , (3) Cartan formula

が成立する

.

{ E r , d r }

, B -

H ^∗ ( F (X, Y ))

.

(edge homomorphism)

.

4.9. [11, Proposition 4.6]

4.5

,

ev ₀ ^∗ : H ^∗ (Y ) → H ^∗ ( F (X, Y )))

mod p BPS

{ E _r , d _r }

(edge homomorphism)

,

:

H ^∗ (Y ) ^ev

∗0

//

θ ²²

H ^∗ ( F (X, Y )) E ₂ ^{0, ∗} = L ₀ (H ^∗ (Y ) : H ^∗ (X)) _K - F

// // E ₃ ^{0, ∗ // //} · · · ^{// //} E _∞ ^{0, ? ∗} .

OO

E ₂ ^{0, ∗}

H ^∗ ( F (X, Y ))

B -

[11, Remark 4.7].

H ^∗ ( F (X, Y ))

B -

Lannes

[11, Theorem 6.4, Example 6.7].

5.

私達は写像空間の

E -

.

,

²⁰ .

Lannes

. Brown, Szczarba[7]

で

Lannes

(

[6], [20]

).

したがって先の問いに対する解答を得るために

,

.

.

∧ V

V

.

V

V ^even

V ^odd

∧ V = Q [V ^even ] ⊗ E(V ^odd )

連結可換微分代数

A

( ∧ V, d) → ^' A

v ∈ V

dv ∈ ∧ V

(

)

.

A

.

( ∧ V, d)

A

,

A = A _{P L} (X)

( ∧ V, d)

X

.

F (U, X; f )

f : U → X

F (U, X )

. Brown, Szczarba[7]

による写像空間

F (U, X; f)

.

D

Lannes

( − : B) _D

:

Hom _D ((A : B) _D , C ) ∼ = Hom _D (A, B ⊗ C)

Brown, Szczarba[7]

A = ( ∧ V, d)

(A : B) _D ∼ =

∧ (V ⊗ B _∗ )

,

B _∗

B

²¹ .

今

α : A = ( ∧ V, d) → ^' A P L (X)

X

, β : B → ^' A P L (U )

U

.

.

dim ⊕ ^q ≥ 0 H ^q (U ; Q ) < ∞

dim ⊕ ⁱ ≥ 2 π i (X) ⊗ Q < ∞ .

(A : B) _D

E

.

f : U → X

E

0-

∆(E) 0

u

[26, Section 3],

u

[7, Section 6]

E

20 mod p BPS

E

”

”

.

. 21

B

= B

6

のイデアルを

M _f

.

[7, Theorem 1.5]

(

ξ

).

E _{a minimal} ^'

model

//

π ²² ²²

(A : B) _D

adjoint of id

D

// Ω∆((A : B) _D ) ^{' //} A P L ( F (X, Y ))

²²

E/M f '

ξ // A _{P L} ( F (X, Y ; f))

F (X, Y ; f )

E/M f

.

F (X, Y ; f )

B-S

.

5.1. [30][8][26]

m(ev) : A = ( ∧ V, d) → (E/M f , δ) ⊗ B

x ∈ A

,

m(ev)(x) = X

j

( − 1) ^{τ( | b}

^{| )} π ◦ r(x ⊗ b _{j ∗} ) ⊗ b _j ,

と定義する

,

τ (n) = [(n + 1)/2], r : (A : B ) _D → E

, { b j } ^j

B _∗

.

m(ev)

,

:

A _{P L} (X) ^A

^(ev) // A _{P L} ( F (U, X; f ) × U ) A P L ( F (U, X; f ) ⊗ A P L (U )

OO '

( ∧ V, d)

' α

OO

m(ev) // (E/M _f , δ) ⊗ B

' OO ξ ⊗ β

特に

, ( ∧ V, d)

(X, ∗ )

, ι(v) = v ⊗ 1

ι : ( ∧ V, d) → (E/M _u , δ)

は評価写像

ev 0 : F (U, X; f ) → X; ev 0 (f ) = f ( ∗ )

.

ここで構成した

B-S

”

”

,

.

Gottlieb

,

.

6.

(Gottlieb

)

ev 0 : F (U, X ; f ) → X

. n

Gottlieb

G n (U, X ; f )

n

π _n (X)

.

G _n (U, X; f) := Im { ev _{0 ∗} : π _n ( F (U, X ; f )) → π _n (X) } .

G n (X, X; id X )

G n (X)

.

この

Gottlieb

G _n (X)

,

n

G _n (X) = 0

, X

.

,

X

²²

dim G 1 (X) ⊗ Q

[22].

Gottlieb

.

G n ( − )

.

Gottlieb

.

f _∗ : G _n (U ) → G _n (U, X ; f )

が成り立つので

, Gottlieb

G _n (U, X; f )

[35], [48], [49].

有理

Gottlieb

G n (X) ⊗ Q

, Halperin-F´ elix [16]

.

L-S

Gottlieb

22

T

X

(k ≥ 0).

.

7

非常に興味深い

. Halperin-F´ elix

Gottlieb

derivations

.

Lupton-Smith

[36], [37],

Gottlieb

G _n (U, X; f ) ⊗ Q

.

X

,

Gottlieb

.

私たちの研究手法は写像空間

F (U, X; f )

B-S

, X

,

G n (U, X; f)

.

X

,

Lupton-Smith

.

1

Gottlieb

.

6.1.

. G

. Γ ₁ G = G

, { xyx ^{− 1} y ^{− 1} | x ∈ G, y ∈ Γ _{j − 1} G }

G

Γ j G = [G, Γ j − 1 G]

.

G

G = Γ 1 G ⊃ Γ 2 G ⊃ · · · ⊃ Γ j G ⊃ · · ·

Γ _q G/Γ _q+1 G

(Γ _q /Γ _q+1 )G

.

定理

6.1. [26, Theorem 1.7] f : U → X

,

(i)(ii)

. (i) f

ψ-

π _ψ ¹ (f ) : π _ψ ¹ (X) → π ¹ _ψ (U )

. (ii) U

CW

X

dim ⊕ i ≥ 2 π _i (X) < ∞

.

,

(1) (Γ _k /Γ _k+1 )π ₁ (X) ^] 6 = 0

i (i < k)

, dim ³

Γ _i G ₁ (U, X; f) ±

Γ _i+1 π ₁ (X) ∩ Γ _i G ₁ (U, X; f ) ´

⊗ Q ≤ dim(Γ _i /Γ _i+1 )π ₁ (X) ⊗ Q − 1.

(2) ([π 1 (X), π 1 (X)]/Γ 3 π 1 (X)) ^] 6 = 0

dim

³

G 1 (U, X ; f ) ±

[π 1 (X), π 1 (X)] ∩ G 1 (U, X ; f )

´ ⊗ Q ≤ dim H 1 (X; Q ) − 2.

上の定理は

Gottlieb

”

”

.

次に具体的計算への応用を述べる

. n

T ⁿ

S ¹

S ¹ → X _f → T ⁿ

を考える

.

f : T ⁿ → K( Z , 2)

H ² (T ⁿ ; Z ) ∼ = [T ⁿ , K( Z , 2)]

H ¹ (T ⁿ ; Z )

{ t i } 1 ≤ i ≤ n

,

ρ _f = X

i<j

c ij t i t j

と表す

.

(n × n)-

A _f

A _f = ¡ c ⁰ _ij ¢

と定義する

.

c ⁰ _ij = c _ij (i < j), c ⁰ _ij = − c _ji (i > j), c ii = 0

.

A f

Q

,

rankA f

.

.

定理

6.2. [26, Theorem 1.9] G ₁ (X _f ) ∼ = Z ^{⊕ (1+n − rankA}

⁾ .

6.2.

.

F (U, X ; f )

E/M _f

§ 5

,

(

)

.

さらに

Gottlieb

.

(Γ i /Γ i+1 )π 1 ( F (U, X; f ))

ev

²² ²²

ev

,, Y

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

M := Γ i G 1 (U, X; f )/(Γ i+1 π 1 (X) ∩ Γ i G 1 (U, X; f)) ^{// //} (Γ i /Γ i+1 )π 1 (X).

この図式の双対に

[5]

6.1

,

6.2

.

8

7.

写像空間や評価写像の模型に関連して

, § 6

. (i) Q

A

HC _∗ ^(k) (A)

[34]

.

k

X

,

HC _∗ ^(k) (A P L (X)) ∼ = H _S ^∗

( F ( T ^k , X); Q )

²³ .

,

5.1

[9]

.

(ii)[33]

,

F ∗ (X, Y )

与えられている

. Y

X

Lie

B-S

( § 5

)

.

(iii)[26]

Gottlieb

G _∗ (U, X ; f )

π _∗ (X)

π _∗ (X)

”

”

.

5.1

.

6.1,6.2

.

(iv)

M

aut 1 (M )

. aut ₁ (M ) = F (M, M ; id _M )

§ 5

B-S

. [31]

(M, a) ²⁴

, H ^∗ (B aut 1 (M); Q )

Kedra-McDuff µ-

[27] ²⁵

B-S

. H ^∗ (Baut ₁ (M ); Q )

Eilenberg-Moore

5.1

.

さらに私たちが写像空間や付随する空間の持つ正標数的

(

)

, § 4

(1)

, (2)

,

.

.

References

[1] D. Anick, A model of Adams-Hilton type for fiber squares, Illinois J. Math. 29(1985) 463-502.

[2] D. Anick, Hopf algebra up to homotopy, J. Ameri. Math. Soc. 2(1989), 417-453.

[3] H. Baues, Algebraic Homotopy, Cambridge University Press, Cambridge, 1988.

[4] C. Berger and B. Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos.

Soc. 137(2004), 135-174.

[5] A. K. Bousfield and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Memoirs of AMS 179(1976).

[6] A. K. Bousfield, C. Peterson and L. Smith, The rational homology of function spaces, Arch. Math.

52(1989), 275-283

[7] E. H. Brown Jr and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer.

Math. Soc. 349(1997), 4931-4951.

[8] U. Buijs and A. Murillo, Basic constructions in rational homotopy theory of function spaces, Ann.

Inst. Fourier (Grenoble) 56(2006), 815-838.

[9] D. Burghelea and M. Vigu´ e-Poirrier, Cyclic homology of commutative algebras. I, Lecture Notes in Math., 1318, Springer, Berlin, 1988, 51-72.

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[11] D. Chataur and K. Kuribayashi, An operadic model for a mapping space and its associated spectral sequence, J. Pure and Appl. Algebra, 210(2007), 321-342.

[12] N. Dupont and K. Hess, Noncommutative algebraic models for fibre squares, Math. Ann. 314(1999) 449-467.

[13] N. Dupont and K. Hess, How to model the free loop space algebraically, Math. Ann. 314(1999) 469-490.

[14] W. G. Dwyer, Tame homotopy theory, Topology 18(1979), no. 4, 321-338.

23 S

F ( T

, X)

S

. 24 M

2n-

, a ∈ H

(M ; Q )

a

6 = 0

.

25

M-

a

e a

,

R

M

e a

k

Kedra-McDuff µ-

.

9