ON CONFIGURATION SPACE INTEGRAL OF SMOOTH SPHERE BUNDLES Tadayuki WATANABE RIMS, Kyoto University Apr. 02, 2008 Aarhus, CTQM Workshop “Finite Type Invariants, Fat graphs and Torelli-Johnson-Morita Theory” 1
1. INTRODUCTION FUNDAMENTAL PROBLEM: - Classification of smooth M -bundles, or - Determine the homotopy type of B Diff( M ) (Diff( M ) = { ϕ : M → M, C ∞ -diffeom } ; C ∞ -topology). REMARK bijec { smooth M -bundles over B } / isom ← → [ B, B Diff( M )] 2
1. INTRODUCTION HISTORY: ( M : (homology) sphere) (S. Smale) B Diff( S 2 ) ≃ BO 3 , ? B Diff( S 3 ) ≃ BO 4 ? ( ← Affirmative, A. Hatcher ) 3
1. INTRODUCTION HISTORY: ( M : (homology) sphere) (S. Smale) B Diff( S 2 ) ≃ BO 3 , ? B Diff( S 3 ) ≃ BO 4 ? ( ← Affirmative, A. Hatcher ) (J. Milnor) B Diff( S 6 ) ≃ / BO 7 (existence of exotic S 7 ). 4
1. INTRODUCTION HISTORY: ( M : (homology) sphere) (S. Smale) B Diff( S 2 ) ≃ BO 3 , ? B Diff( S 3 ) ≃ BO 4 ? ( ← Affirmative, A. Hatcher ) (J. Milnor) B Diff( S 6 ) ≃ / BO 7 (existence of exotic S 7 ). (S. Novikov) B Diff 0 ( S 7 ) ≃ / BSO 8 . 5
1. INTRODUCTION HISTORY: ( M : (homology) sphere) (S. Smale) B Diff( S 2 ) ≃ BO 3 , ? B Diff( S 3 ) ≃ BO 4 ? ( ← Affirmative, A. Hatcher ) (J. Milnor) B Diff( S 6 ) ≃ / BO 7 (existence of exotic S 7 ). (S. Novikov) B Diff 0 ( S 7 ) ≃ / BSO 8 . (F. Farrell, W. Hsiang) i << d (stable range) π i ( B Diff( S d )) ⊗ Q ∼ = π i ( BO d +1 ) ⊗ Q ⊕ ( Q or 0) . 6
1. INTRODUCTION HISTORY: ( M : (homology) sphere) (S. Smale) B Diff( S 2 ) ≃ BO 3 , ? B Diff( S 3 ) ≃ BO 4 ? ( ← Affirmative, A. Hatcher ) (J. Milnor) B Diff( S 6 ) ≃ / BO 7 (existence of exotic S 7 ). (S. Novikov) B Diff 0 ( S 7 ) ≃ / BSO 8 . (F. Farrell, W. Hsiang) i << d (stable range) π i ( B Diff( S d )) ⊗ Q ∼ = π i ( BO d +1 ) ⊗ Q ⊕ ( Q or 0) . (K. Igusa) The extra Q can be detected by “higher FR torsion”. 7
1. INTRODUCTION PROBLEM (D. Burghelea): Is π i ( B (Diff( S d ) /O d +1 )) ∼ = π i ( B Diff( D d , ∂ )) finite? for each fixed ( i, d ). (M. Kontsevich) (non-stable) M : ’singularly framed’ odd-dim HS → H ∗ ( � CSI (graph homology) ∗ − B Diff( M ); R ) . (G. Kuperberg, D. Thurston) dim M = 3, 3-valent CSI ∈ H 0 ( ⊔ M � B Diff( M ); (certain space of graphs)) is a universal FTI of Ohtsuki, 31 stable d Goussarov-Habiro. 23 non-stable� We give a higher-dim. generalization 15 ?? of this to understand non-stable. 7 3 i 0 4 8 12 16 20 24 28 32 36 40 • : πi ( B Diff( Dd, ∂ )) infinite 8
1. INTRODUCTION PROBLEM (D. Burghelea): Is π i ( B (Diff( S d ) /O d +1 )) ∼ = π i ( B Diff( D d , ∂ )) finite? for each fixed ( i, d ). (M. Kontsevich) (non-stable) M : ’singularly framed’ odd-dim HS → H ∗ ( � CSI (graph homology) ∗ − B Diff( M ); R ) . (G. Kuperberg, D. Thurston) dim M = 3, 3-valent CSI ∈ H 0 ( ⊔ M � B Diff( M ); (certain space of graphs)) is a universal FTI of Ohtsuki, 31 stable d Goussarov-Habiro. 23 non-stable� We give a higher-dim. generalization 15 ?? of this to understand non-stable. 7 3 i 0 4 8 12 16 20 24 28 32 36 40 • : πi ( B Diff( Dd, ∂ )) infinite 9
1. INTRODUCTION PROBLEM (D. Burghelea): Is π i ( B (Diff( S d ) /O d +1 )) ∼ = π i ( B Diff( D d , ∂ )) finite? for each fixed ( i, d ). (M. Kontsevich) (non-stable) M : ’singularly framed’ odd-dim HS → H ∗ ( � CSI (graph homology) ∗ − B Diff( M ); R ) . (G. Kuperberg, D. Thurston) dim M = 3, 3-valent CSI ∈ H 0 ( ⊔ M � B Diff( M ); (certain space of graphs)) is a universal FTI of Ohtsuki, 31 stable d Goussarov-Habiro. 23 non-stable� We give a higher-dim. generalization 15 ?? of this to understand non-stable. 7 3 i 0 4 8 12 16 20 24 28 32 36 40 • : πi ( B Diff( Dd, ∂ )) infinite 10
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.1. SPACE OF GRAPHS G 2 n := span Q { conn. v-ori. 3-valent graphs, 2 n -vertices } . A 2 n := G 2 n / IHX , AS . IHX = - AS = - 11
� 2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.2. COMPACTIFICATION OF CONFIGURATION SPACE M : (homology) (2 k + 1)-sphere with a fixed pt ∞ ∈ M . C n ( M \ ∞ ) := { ( x 1 , · · · , x n ) ∈ ( M \ ∞ ) × n | x i � = x j ( i � = j ) } , C n ( M \ ∞ ) := Fulton-MacPherson-Kontsevich compactification “= Bℓ Σ ( M × n ) real blow-up” of C n ( M \ ∞ ) . blow incl . down � C 2 ( M ) M × M M × M \ Σ 12
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.3. FUNDAMENTAL FORM ω ON C 2 ( M \ ∞ ) -BUNDLE Given a ( D 2 k +1 , ∂ )-bundle π : E → B (with P → B assoc principal), C n ( π ) : EC n ( π ) → B EC n ( π ) := P × Diff( D 2 k +1 ,∂ ) C n ( S 2 k +1 \ ∞ ) ∼ → R 2 k +1 × E given, then If a trivialization (framing) τ E : T fib E − ∃ ω ∈ Ω 2 k dR ( EC 2 ( π )) closed form s.t. ω | ∂ fib EC 2 ( π ) = Sτ ∗ E Vol S 2 k ∈ Ω 2 k dR ( ∂ fib EC 2 ( π )) . � ∼ → S 2 k × E Sτ E : ∂ fib EC 2 ( π )”=” S ( T fib E ) − � ∂ � � 2 k +1 Γ( k + 3 2 ) Vol S 2 k = j =1 x j i dx 1 ∧ · · · ∧ dx 2 k +1 (2 k +1) · π (2 k +1) / 2 ∂x j 13
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.3. FUNDAMENTAL FORM ω ON C 2 ( M \ ∞ ) -BUNDLE Given a ( D 2 k +1 , ∂ )-bundle π : E → B (with P → B assoc principal), C n ( π ) : EC n ( π ) → B EC n ( π ) := P × Diff( D 2 k +1 ,∂ ) C n ( S 2 k +1 \ ∞ ) ∼ → R 2 k +1 × E given, then If a trivialization (framing) τ E : T fib E − ∃ ω ∈ Ω 2 k dR ( EC 2 ( π )) closed form s.t. ω | ∂ fib EC 2 ( π ) = Sτ ∗ E Vol S 2 k ∈ Ω 2 k dR ( ∂ fib EC 2 ( π )) . � ∼ → S 2 k × E Sτ E : ∂ fib EC 2 ( π )”=” S ( T fib E ) − � Vol S 2 k : S 2 k Vol S 2 k = 1 , SO 2 k +1 -invariant 14
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.4. FROM GRAPHS TO DIFFERENTIAL FORMS We define a linear map Φ : G 2 n → Ω 6 nk dR ( EC 2 n ( π )) by � pr → EC 2 ( π )) ∗ ω. Φ(Γ) := ω e , ω e := ( EC 2 n ( π ) e dR ( EC 2 n ( π )) → Ω 6 nk − 2 n (2 k +1) Fiber integration C 2 n ( π ) ∗ : Ω 6 nk ( B ) dR yields a form C 2 n ( π ) ∗ Φ(Γ) ∈ Ω n (2 k − 2) ( B ). dR Let � [Γ] | Aut Γ | ∈ Ω n (2 k − 2) ζ 2 n ( π ; τ E ) := C 2 n ( π ) ∗ Φ(Γ) ( B ) ⊗ A 2 n . dR Γ ∈G 2 n 15
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.4. FROM GRAPHS TO DIFFERENTIAL FORMS We define a linear map Φ : G 2 n → Ω 6 nk dR ( EC 2 n ( π )) by � pr → EC 2 ( π )) ∗ ω. Φ(Γ) := ω e , ω e := ( EC 2 n ( π ) e dR ( EC 2 n ( π )) → Ω 6 nk − 2 n (2 k +1) Fiber integration C 2 n ( π ) ∗ : Ω 6 nk ( B ) dR yields a form C 2 n ( π ) ∗ Φ(Γ) ∈ Ω n (2 k − 2) ( B ). dR Let � [Γ] | Aut Γ | ∈ Ω n (2 k − 2) ζ 2 n ( π ; τ E ) := C 2 n ( π ) ∗ Φ(Γ) ( B ) ⊗ A 2 n . dR Γ 16
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.5. THEOREM (Kontsevich). ζ 2 n ( π ; τ E ) : characteristic class of framed ( D 2 k +1 , ∂ ) -bundles, i.e., 1. ζ 2 n ( π ; τ E ) is ( d ⊗ 1) -closed. 2. [ ζ 2 n ( π ; τ E )] ∈ H n (2 k − 2) ( B ; R ⊗ A 2 n ) does not depend on the closed extension ω chosen. 3. [ ζ 2 n ( π ; τ E )] is natural wrt maps between framed bundles. “Proof” By the generalized Stokes formula (for fiber integration) and vanishing of higher degenerations (Kontsevich’s lemma), � ( d ⊗ 1) ζ 2 n ( π ; τ E ) = (IHX + AS) = 0 . 17
2. KONTSEVICH’S CHARACTERISTIC CLASSES 2.5. THEOREM (Kontsevich). ζ 2 n ( π ; τ E ) : characteristic class of framed ( D 2 k +1 , ∂ ) -bundles, i.e., 1. ζ 2 n ( π ; τ E ) is ( d ⊗ 1) -closed. 2. [ ζ 2 n ( π ; τ E )] ∈ H n (2 k − 2) ( B ; R ⊗ A 2 n ) does not depend on the closed extension ω chosen. 3. [ ζ 2 n ( π ; τ E )] is natural wrt maps between framed bundles. “Proof” By the generalized Stokes formula (for fiber integration) and vanishing of higher degenerations (Kontsevich’s lemma), � ( d ⊗ 1) ζ 2 n ( π ; τ E ) = (IHX + AS) = 0 . 18
3. FEATURES OF THE SIMPLEST CLASS 3.0. CONTENT OF THIS SECTION - We define an unframed version ˆ Z 2 of the invariant of a ‘pointed’ framed ( D 2 k +1 , ∂ )-bundle π : E → D 2 k − 2 : Z 2 : π 2 k − 2 ( � B Diff( D 2 k +1 , ∂ )) → R � ω 3 ∈ R Z 2 ( π ; τ E ) = ζ 2 ( π ; τ E )[ D 2 k − 2 , ∂ ] | [Θ] �→ 12 = EC 2 ( π ) associated to the ‘Θ-graph’ by introducing a correction term. - Formula for ˆ Z 2 ⇒ ˆ Z 2 detects some exotic smooth structures on the total spaces. 19
3. FEATURES OF THE SIMPLEST CLASS 3.1. SIGNATURE DEFECT D 4 k - 1 W (correction term): = E cl( E ) * cl( E ) := E ∪ ∂ D 4 k − 1 W closing, canonical gluing. ∼ → R 2 k +1 × E * framing τ E : T fib E − extend (stable) framing τ ′ E on TW | ∂W =cl( E ) . � * L k ( TW ; τ ′ E )[ W, ∂W ]: relative L k -characteristic number. * (Signature defect) ∆ k ( π ; τ E ) := L k ( TW ; τ ′ E )[ W, ∂W ] − sign W gives a well-defined hom. π 2 k − 2 ( � B Diff( D 2 k +1 , ∂ )) → Q . 20
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