講演集 pdf file

第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
Chain level operations in string topology via de Rham
chains
)∗
(
1.
1
Chas-Sullivan [2]
Batalin-Vilkovisky
1.1. Chas-Sullivan
(Chas-Sullivan)
LM := C (S 1 , M )
R/Z
LM
L
H∗ ( · ) := H∗+d ( · )
e : L → M; γ →
" γ(0)
M d
S 1 :=
∞
L e ×e L := {(γ, γ # ) ∈ L × L | γ(0) = γ # (0)}
j : L e×e L → L×L
◦ : H∗ (L)⊗2
×
◦
c : L e×e L → L
! H∗+2d (L × L)
p ∈ M
p
H∗ (M ) → H∗ (LM )
j
(concatenation)
Gysin
j!
! H∗+d (L e ×e L)
H∗ (c)
! H∗ (L).
i : M → LM
∩
H∗ (M )
H∗ (i)(x) ◦ H∗ (i)(y) = H∗ (i)(x ∩ y)
H∗ (i) :
(x, y ∈ H∗ (M ))
H∗ (i)
M
1.2. Batalin-Vilkovisky
L
S1
r : S 1 × L → L r(t, γ)(θ) := γ(θ − t)
∆x := H∗ (r)([S 1 ] × x)
{ , } : H∗ (L)⊗2 → H∗+1 (L)
∆ : H∗ (L) → H∗+1 (L)
∆ =0
2
{a, b} := (−1)|a| ∆(a ◦ b) − (−1)|a| ∆a ◦ b − a ◦ ∆b
{,}
◦
Lie
{a, b ◦ c} = {a, b} ◦ c + (−1)|b|(|a|+1) b ◦ {a, c}
∗
25800041
e-mail: [email protected]
1
[12]
C∞
51
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
2
V
Batalin-Vilkovisky(BV)
Gerstenhaber
BV
f D(0)
(V, ◦, ∆)
BV
Gerstenhaber
n≥1
D(n)
H∗ (f D)
H∗ (D)
◦, ∆, { , }
(V, ◦, { , }) Gerstenhaber
{z ∈ C | |z| ≤ 1}
disjoint n
1 ×n
f D(n) := D(n) × (S )
D(0)
D = (D(n))n≥0
f D = (f D(n))n≥0
D
fD
Gerstenhaber
H∗ (D)
V
(Hom(V ⊗n , V ))n
Cohen [3]
V
BV
H∗ (f D) V
Getzler [6]) H0 (D(2)), H1 (D(2)), H1 (f D(1))
1
◦, { , }, ∆
V
V
V
H∗ (D)
1.3.
3
“Chain level string topology”
[14]
Massey
X
A∞
C ∗ (X)
dga
algebra)
Leibniz
V
A∞
!
k+l=m+1
1≤i≤k
m≥1
k≥1
cup
dga(differential graded
2−k
µk : V ⊗k → V
±µk (v1 ⊗ · · · ⊗ µl (vi ⊗ · · · ⊗ vi+l−1 ) ⊗ · · · ⊗ vm ) = 0
µk = 0 (∀k ≥ 3)
A∞
dga
A∞
∗
C (X)
A∞
H (X)
A∞
(µk )k≥1
µ1 = 0
µ2 cup
µ3
Massey
µ3
[11] Section 9
Poincaré
H ∗ (M )
cup
H∗ (M )
H∗ (LM )
H∗ (M )
H∗ (LM )
A∞
(µk )k≥1
∗
M
Z
2
3
1
[12]
52
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
• µ1 = 0
•
µ2
k
∗
C (M )
A∞
µk (H∗ (M )⊗k ) ⊂ H∗ (M )
(µk )k≥1
H∗ (M )
4
Floer
Hochschild
2. Floer
"
T ∗ M := q∈M Tq∗ M
HF∗ (T ∗ M )
C∞
M
Floer
πM : T ∗ M → M
T ∗M
1
λM
(q ∈ M, p ∈ Tq∗ M, X ∈ T(q,p) T ∗ M )
λM (X) := p((πM )∗ (X))
dλM
C
H : T ∗M → R
T ∗M
Hamilton
AH : C ∞ (S 1 , T ∗ M ) → R
#
AH (γ) :=
γ ∗ λM − H(γ(t)) dt
(γ ∈ C ∞ (S 1 , T ∗ M ))
∞
4
S1
AH
AH
H
H
Hamilton
AH
AH
C× → T ∗ M
5
Cauchy-Riemann
Morse
Floer
T ∗M
Floer
CP 1 \ {0, 1, ∞} → T ∗ M
◦ : HF∗ (T ∗ M )⊗2 → HF∗ (T ∗ M )
C× → T ∗ M
Floer
∗
∆ : HF∗ (T M ) → HF∗+1 (T ∗ M )
BV
Floer
H
∗
C
×
HF∗ (T M )
CP \ {0, ∞}
1
pair-of-pants
S1
(HF∗ (T ∗ M ), ◦, ∆)
2.1
Liouville
6
4
5
6
M
∗
Riemann
T M
Liouville
symplectic (co)homology
[13]
H(q, p) := |p|2
dλM
Floer
53
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
Floer
BV
BV
7
2.2 M
HF∗ (T ∗ M ) ∼
= H∗ (LM )
BV
HF∗ (T ∗ M ) ∼
Viterbo
= H∗ (LM )
Salamon-Weber, Abbondandolo-Schwarz
pairof-pants
Abbondandolo-Schwarz
Kragh
Abouzaid [1]
[1]
Liouville
Floer
BV
Lie
Jacobi
1
CP
4
Riemann
Floer
1
CP
k
Riemann
Floer
k≥2
Floer
A∞
L∞
4
H∗ (LM )
A∞
L∞
HF∗ (T ∗ M )
A∞
L∞
3. Hochschild
Gerstenhaber [5]
3.1
Gerstenhaber
A
A
∞
C
AjM
dga
HH −∗ (A, A)
Hochschild
dga
A∞
j
0≤j≤d
8
dga
M
AjM
j
=0
M
R
d
A∗M
I : H∗ (LM ) → HH −∗ (AM , AM )
M
Gerstenhaber
Gerstenhaber
I
3.1
4.2
3.1
3.2
C
C
P
dg
P → End(C)
7
8
M
H∗ (P)
LM
P
C dgP
H∗ (C)
θM
54
dg
End(C) := (Hom(C ⊗n , C))n≥0
C
dg
P C
HF∗ (T ∗ M ) ∼
= H∗ (LM : θM )
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
O
3.2 dg
H∗ (O) ∼
= H∗ (D)
(i):
(ii):
HH −∗ (A, A)
A
−∗
CH (A, A)
3.3
Hochschild
O
O
(i)
Gerstenhaber
D
C∗ (D) D
dg
Deligne
Deligne
(McClure-Smith, Kontsevich-Soibelman,
Voronov, Tamarkin, Berger-Fresse, Kaufmann)
Kontsevich-Soibelman
A∞
[11] Section 13.3.15
3.2
4.
D
1.2
fD
fP
4.1 ([7], [8]) dg
P
H∗ (P) ∼
= H∗ (D)
(i):
H∗ (f P) ∼
= H∗ (f D)
∼
=
H∗ (P)
"
H∗ (f P)
(ii):
dga A
CH −∗ (A, A)
(iii):
C∞
HH −∗ (A, A)
dg P
∼
=
! H∗ (D)
"
! H∗ (f D).
Gerstenhaber
dg f P
M
Hochschild
C∗LM
Φ : H∗ (LM ) ∼
= H∗ (C LM )
• BV
J : C∗LM → CH −∗ (AM , AM )
• dg P
H∗ (J) ◦ Φ : H∗ (LM ) → HH −∗ (AM , AM )
I 3
(ii)
A∞
3.2
dga
4.1
(iii)
55
3.2
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
4.2
I : H∗ (LM ) → HH −∗ (AM , AM )
4.1
M
Gerstenhaber
Merkulov, Tradler
H∗ (LM )
−∗
H∗ (M )
H∗ (i)
1.3
l1 = 0
dga
! H −∗ (M )
dR
"
H∗ (ι)
! H (C LM ).
∗
∼
=
H∗ (LM )
H∗ (M )
µ2
(lk )k≥1
lk |H∗ (M )⊗k = 0
∼
=
"
H∗ (LM )
µ1 = 0
C∗LM
ι : A (M ) →
dga
C∗LM
A∞
(µk )k≥1
A (M )
−∗
H∗ (LM )
L∞
k ≥1
l2
5.
C∗LM
4.1
5.1
C∗LM
M
d
5.2
C∞
5.3
L = LM = C ∞ (S 1 , M )
5.1.
H∗ (L)⊗2 → H∗+d (L e ×e L)
M
x ∩ y ∈ Ck+l−d (M )
y
C d−∗ (M )
C∗ (M )
5.2.
x
x ∈ Ck (M ), y ∈ Cl (M )
C∗LM
[7]
Fukaya [4]
[7]
de Rham
“approximate de Rham chain”
C∗LM
1:
Moore
Lk
56
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
C∞
M
L0 M
Moore
L0 M := {(γ, T ) | T ∈ R≥0 , γ ∈ C ∞ ([0, T ], M ), γ(0) = γ(T ),
γ (m) (0) = γ (m) (T ) = 0 (∀m ≥ 1)}
C∞
γ (m) γ m
concatenation
Moore
k
k≥1
Lk M := {(γ, t1 , . . . , tk , T ) | (γ, T ) ∈ L0 M,
γ (m) (tj ) = 0 (∀m ≥ 1,
j ∈ {0, . . . , k}
ej (γ, t1 , . . . , tk , T ) :=
Lk
ik : M → L k M
2: Lk
n
U ∈U
(a): ϕ
(b):
C∞
p∈M
$
1 ≤ ∀j ≤ k)}
e j : Lk M → M
γ(0)
(j = 0)
γ(tj ) (1 ≤ j ≤ k)
p
Lk M
9
0 ≤ t1 ≤ · · · ≤ tk ≤ T,
0
Lk
Rn
ϕ : U → Lk
(U, ϕ)
P(Lk )
Un
(a), (b)
U :=
%
Un
ϕ := (γ ϕ , tϕ1 , . . . , tϕk , T ϕ )
tϕ1 , . . . , tϕk , T ϕ ∈ C ∞ (U )
{(u, t) | u ∈ U, 0 ≤ t ≤ T ϕ (u)} → M ; (u, t) "→ γ ϕ (u)(t) C ∞
ϕj := ej ◦ ϕ : U → M
j = 0, . . . , k
(U, ϕ) ∈ P(Lk ), (V, ψ) ∈ P(Ll )
i ∈ {1, . . . , k}
U ϕi ×ψ0 V := {(u, v) ∈ U × V | ϕi (u) = ψ0 (v)}
ϕ ∗i ψ : U ϕi ×ψ0 V → Lk+l−1
(ϕ ∗i ψ)(u, v) := 0
(b)
U
i−1 i
1
ϕ(u)
ψ(v)
i+l
k+l−1
i+l−1
ϕi (u) = ψ0 (v)
◦i : P(Lk ) × P(Ll ) → P(Lk+l−1 )
(U, ϕ) ◦i (V, ψ) := (U ϕi ×ψ0 V, ϕ ∗i ψ)
9
n≥1
(plot)
Diffeological space
K.T. Chen
Differentiable space
57
Souriau
P(Lk )
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
3: Lk
de Rham
de Rham
CNdR
:=
&
'
(U,ϕ)∈P(Lk )
A∗c (U )
R
N
(
dim U −N
Ac
(U ) /ZN
R
U
ZN
(U, ϕ, π! ω) − (V, ϕ ◦ π, ω)
∞
C
C∗dR (Lk )
(U, ϕ) ∈ P(Lk ), V ∈ U , π : V → U
π!
∂
∂[(U, ϕ, ω)] := [(U, ϕ, dω)]
2
∂ =0
de Rham
5.1
well-defined
Lk de Rham
C∗dR (Lk )
de Rham
k≥0
(i):
Lk → L0 ;
(γ, t1 , . . . , tk , T ) "→ (γ, T )
C∗dR (Lk ) → C∗dR (L0 )
(ii):
H∗ (C∗dR (L0 )) ∼
= H∗ (LM : R)
1≤i≤k
LM
C∞
dR
dR
dR
◦i : C∗+d
(Lk ) ⊗ C∗+d
(Ll ) → C∗+d
(Lk+l−1 )
l≥0
[(U, ϕ, ω)] ◦i [(V, ψ, η)] := ±[(U ϕi ×ψ0 V, ϕ ∗i ψ, ω × η)]
dR
CL := (C∗+d
(Lk ))k≥0
dg
dR
10
[(M, i1 , 1)] ∈ Cd (L1 )
Z/(k + 1)Z Lk
dR
C∗ (Lk )
CL
(cyclic) dg
LM
4:
C∗
11
5.2 O = (O(k))k≥0
µ ◦1 µ = µ ◦2 µ
µ ◦1 ε = µ ◦ 2 ε = 1 O
10
11
M
Euclid
µ ∈ O(2)0
ε ∈ O(0)0
1O O
dg
O
µ
M ∈U
Gerstenhaber-Voronov
58
∂µ = 0,
∂ε = 0,
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
O
dg
∂µ : Õ∗ → Õ∗−1
Õ∗ :=
µ
∂µ (xk )k≥0 := (∂xk )k≥0 +
&!
k−1
j=1
(Õ∗ , ∂µ )
End(A)
!
End(A)
±xk−1 ◦j µ +
j=1
O(k)∗+k
±µ ◦j xk−1
(
k≥1
A dga
End(A)
A
A
k=0
5.2
Hochschild
CL
3
µ := [(M, i2 , 1)],
µ
2
!
)∞
ε := [(M, i0 , 1)]
Z/3Z
CL(2)
* ∗ , ∂µ )
C∗LM := (CL
dR
C∗LM → C∗+d
(L0 ); (xk )k "→ x0
5.1 (ii)
k=0
5.1 (i)
H∗ (C LM ) ∼
= H∗+d (C∗dR (L0 )) ∼
= H∗ (LM : R)
k≥0
4.1(iii)
Φ
dR
Jk : C∗+d (Lk ) → Hom−∗ (A⊗k
M , AM )
Jk ([(U, ϕ, ω)])(η1 ⊗ · · · ⊗ ηk ) := ±(ϕ0 )! (ω ∧ ϕ∗1 η1 ∧ · · · ∧ ϕ∗k ηk )
(Jk )k≥0 : CL → End(AM ) dg
4.1(iii)
J : C∗LM → CH −∗ (AM , AM )
I
5.3. C∗LM
O
dg
(x ◦ y)k :=
l+m=k
{, }
∼
H∗ (LM ) = H∗ (C LM )
5.3 dg
(a): (O, µ)
!
◦
Õ∗
µ
±(µ ◦1 xl ) ◦1 ym ,
◦ (Õ∗ , ∂µ )
dg Lie
dg
(x ∗ y)k :=
dga
H∗ (J) ◦ Φ
!
∗
l+m=k+1
1≤i≤l
±xl ◦i ym
{x, y} := x ∗ y ± y ∗ x
O = CL
◦
{, }
P
fP
P
4.1 (i)
P
dg
59
(Õ∗ , ∂µ )
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
(b): (O, µ, ε)
O(2)0 O(2)
fP
5.4
5.3
dg
Z/3Z
P
(Õ∗ , ∂µ )
µ∈
[8]
[15]
P
fP
Kaufmann [9], [10]
cacti
5.3(a) End(A)
CH (A, A) dg P
C∗LM
dg f P
−∗
HH (A, A)
Gerstenhaber
A
−∗
H∗ (LM )
4.1
dga
Hochschild
(b) CL
BV
[1] M. Abouzaid, Symplectic cohomology and Viterbo’s theorem, arXiv:1312.3354.
[2] M. Chas, D. Sullivan, String Topology, arXiv:math/9911159.
[3] F. R. Cohen, The homology of Cn+1 -spaces, n ≥ 0, The homology of iterated loop spaces,
207–351, Lecture Notes in Math. vol. 533, Springer-Verlag, Berlin-New York, 1976.
[4] K. Fukaya, Application of Floer homology of Lagrangian submanifolds to symplectic topology, Morse theoretic methods in nonlinear analysis and in symplectic topology, 231–276,
NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006.
[5] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78
(1963), 267–288.
[6] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories,
Comm. Math. Phys. 159 (1994), no. 2, 265–285.
[7] K. Irie, Transversality problems in string topology and de Rham chains, arXiv:1404.0153.
[8] K. Irie, A chain level Batalin-Vilkovisky structure in string topology and decorated cacti,
arXiv:1503.00403.
[9] R. M. Kaufmann, On spineless cacti, Deligne’s conjecture and Connes-Kreimer’s Hopf
algebra, Topology 46 (2007), no. 1, 39–88.
[10] R. M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti, Math.
Res. Lett. 15 (2008), no. 5, 901–921.
[11] J.-L. Loday, B. Vallette, Algebraic Operads, Grundlehren der Mathematischen Wissenschaften, 346. Springer, Heidelberg, 2012.
Gorenstein
,
61
[12]
, 2014.
[13] P. Seidel, A biased view of symplectic cohomology, Current developments in mathematics,
2006, 211–253, Int. Press, Somerville, MA, 2008.
[14] D. Sullivan, String topology background and present state, Current developments in mathematics 2005, 41–88, Int. Press, Somerville, MA, 2007.
[15] B. Ward, Maurer-Cartan Elements and Cyclic Operads, arXiv:1409.5709.
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