Trivial Cocycles, Casson invariant and a Conjecture of Perron Wolfgang Pitsch Departamento de matem´ aticas Universidad Aut´ onoma de Barcelona Topology and Geometry of Low-dimensional Manifolds Nara Womens University, October 25-28, 2016. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Setting the framework We start by a standard embedding of a genus g ≥ 3 surface Σ g , 1 with a disk embedded into the sphere S 3 . This decomposes S 3 into two handlebodies. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
� � � Mapping class group The diagram Σ g , 1 � � H g � � � � � ι g � S 3 = H g −H g � � ι g −H g is a Heegaard splitting of the 3-sphere and gives rise to a diagram of groups: Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
� � � � � � � � Mapping class group M g , 1 A g , 1 � � B g , 1 � � AB g , 1 = A g , 1 ∩ B g , 1 where ◮ M g , 1 = π 0 ( Diff (Σ g ; rel . D 2 )) is the ”mapping class group”. ◮ A g , 1 = subgroup of elements that extend over H g . ◮ B g , 1 = subgroup of elements that extend over −H g . ◮ AB g , 1 is the intersection: mapping classes that extend to the whole sphere. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Heegaard splittings of closed manifolds With this we can parametrize all manifolds. Definition Let V (3) be the set of oriented diffeomorphism classes of closed oriented 3-manifolds. Theorem (Singer, 1953) The map g →∞ B g , 1 \M g , 1 / A g , 1 lim − → V (3) � S 3 φ �− → φ = H g ι g φ −H g is a bijection. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
The limit is taken along the inclusion maps M g , 1 ֒ → M g +1 , 1 induced by extending a mapping class by the identity: This is why we need to have this fixed disc. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
The Johnson filtration The mapping class group has a very rich combinatorics: Theorem (Nielsen) Let π = π 1 (Σ g , 1 ) . The canonical action of diffeomorphisms on the surface induces an injection: M g , 1 ֒ → Aut ( π ) . Consider the lower central series of π : π ⊃ [ π, π ] ⊃ [ π, [ π, π ]] ⊃ · · · ⊃ Γ k ⊃ . . . Γ 0 = π Γ k +1 = [ π, Γ k ] Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
The Johnson filtration The action of M g , 1 on π respects this filtration π ⊃ [ π, π ] ⊃ [ π, [ π, π ]] ⊃ · · · ⊃ Γ k ⊃ . . . hence induces maps ∀ k ≥ 0 τ k � Aut ( π/ Γ k +1 ) M g , 1 For instance τ 0 = H 1 ( − ) Let M g , 1 ( k + 1) = ker τ k Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
The Johnson filtration This gives a descending and separated filtration of the mapping class group: ∞ � M g , 1 ⊃ M g , 1 (1) ⊃ M g , 1 (2) . . . M g ( k ) = { Id } k =1 This is the Johnsons filtration, and the quotients M g , 1 / M g , 1 ( k ) have been the object of much study (S. Morita, R. Hain, many others). Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Two questions with partial answers Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Recall Singer’s and Nielsen’s theorem: Theorem (Singer, 1953) The map g →∞ B g , 1 \M g , 1 / A g , 1 lim − → V (3) � S 3 φ �− → φ = H g ι g φ −H g is a bijection. Theorem (Nielsen) Let π = π 1 (Σ g , 1 ) . The canonical action of diffeomorphisms on the surface induces an injection: M g , 1 ֒ → Aut ( π ) . If you know the action of a mapping class on the fundamental group, then you ”know” the manifold it builds. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
� � � First question We have an increasingly accurate series of approximations of the action on the fundamental groups M g , 1 Aut π τ k Aut ( π/ Γ k +1 ) Question What can you say about the manifold S 3 φ if you know the action of φ on π up to k + 1 -commutators? i.e. you only know τ k ( φ ) ? Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Pointing towards an answer? Some easy cases: 1. By Mayer-Viettoris, if you know τ 0 ( φ ) = H 1 ( φ ; Z ) you know the cohomology of S 3 φ as a group. 2. To know the ring structure you only need τ 1 ( φ ) i.e the action on π/ [ π, [ π, π ]] (Stallings). Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Cochran,Gerges, Orr (2001) Definition Two closed 3-manifolds M 0 and M 1 are k -surgery equivalent if there exists a sequence M 0 = X 0 . . . X 2 . . . X m = X 1 such that ◮ X j +1 is obtained from X j by ± 1 q j surgery along a curve γ j ∈ Γ k ( π 1 ( X i )) Theorem (Cochran,Gerges,Orr (2001)) The following are equivalent: 1. M 0 and M 1 are k-equivalent ∼ 2. ∃ φ : π 1 ( M 0 ) / Γ k ( M 0 ) − → π 1 ( M 1 ) / Γ k ( M 1 ) such that φ ([ M 0 ]) = [ M 1 ] where [ M i ] is the image in H 3 ( π 1 ( M i ) / Γ k ( M i ); Z ) of the fundamental class of M along the canonical map f i : M i → K ( π 1 ( M i ) / Γ k ( M i ) , 1) . Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
1. Question: how is k -equvalence related to equality under τ k . ? S 3 φ ∼ k S 3 ⇐ ⇒ τ k ( φ ) = τ k ( ψ ) ψ 2. For k = 2 this is true (Cochran,Gerges,Orr) Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Second Question Question Assume know that φ ∈ M g , 1 ( k ) , i.e. the action on π up to k-commutators is trivial. What is S 3 φ ? By Mayer-Viettoris, S 3 φ is an integral homology sphere. Let S (3) = { M | H ∗ ( M ; Z ) = H ∗ ( S 3 ; Z ) } . and S (3) k = { S 3 φ | φ ∈ lim g M g , 1 ( k ) } Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Known cases 1. For k = 1 S (3) 1 = S (3) (exercise in Mayer-Viettoris) 2. For k = 2 S (3) 2 = S (3) (Prof. Morita) 3. For k = 3 S (3) 3 = S (3) (W. P. and Massuyeau-Meilhan) 4. For k ≥ 4, unknown. Maybe yes? Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Trivial cocycles and Casson invariant Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
One of the difficulites in the above question is to understand the restriction of the doble coset relation B g , 1 \M g , 1 / A g , 1 to the groups M g , 1 ( k ) . Denote by ≈ this equivalence relation. Proposition ∀ φ, ψ ∈ M g , 1 (1) � ∃ µ ∈ AB g , 1 such that φ ≈ ψ ⇔ φ = µψµ − 1 ∈ B g , 1 (1) \M g , 1 (1) / A g , 1 (1) where A g , 1 ( k ) = A g , 1 ∩ M g , 1 ( k ) and similarly for B g , 1 ( k ) . ≈ is double class in M g , 1 (1)+ coinvariants under the action of AB g , 1 . Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
� � From invariants to trivial cocycles Let F : S (3) → A be an invariant, where A is a group without 2-torsion. This is the same as a familly of functions lim g →∞ M g , 1 (1) / ≈ F � A M g , 1 (1) F g ∀ φ, ψ ∈ M g , 1 (1) let C g ( φ, ψ ) = F g ( φψ ) − F g ( φ ) − F g ( ψ ) This is a trivialized 2-cocycle on M g , 1 Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Properties of C g C g ( φ, ψ ) = F g ( φψ ) − F g ( φ ) − F g ( ψ ) Because F g is constant on the equivalence classes, C g has nice properties. 1. C g +1 restricted to M g , 1 (1) is C g . 2. C g is invariant under conjugation by AB g , 1 C g ( µφµ − 1 , µψµ − 1 ) = C g ( φ, ψ ) 3. C g = 0 on M g , 1 (1) × A g , 1 (1) ∪ B g , 1 (1) × M g , 1 (1) 4. C g � = 0, unless F = 0, equivalently C g is associated to a unique F . Observe that C g measures the defect to be a homomorphism. It can alos be seen as a kind of ”surgery instruction”. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
From cocycles to invariants Theorem (W.P.) Let A be an abelian group wihtout 2 -torsion. Let ( C g ) g ≥ 3 be a familiy of 2 -cocycles on M g , 1 (1) such that 1. C g +1 restricted to M g , 1 (1) is C g . 2. C g is invariant under conjugation by AB g , 1 C g ( µφµ − 1 , µψµ − 1 ) = C g ( φ, ψ ) . 3. C g = 0 on M g , 1 (1) × A g , 1 (1) ∪ B g , 1 (1) × M g , 1 (1) . 4. [ C g ] = 0 in H 2 ( M g , 1 (1); A ) . 5. The associated torsor ρ C g ∈ H 1 ( AB g , 1 ; Hom ( M g , 1 (1) , A )) is 0 . Then C g is the defect of a unique invariant F with values in A, where F g is the unique AB g , 1 -invariant trivialization of C g . Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
Algebraic construction of the cvAsson invariant ◮ The Casson invariant λ : S (3) → Z of M ∈ S (3) essentially counts the number of representations of π 1 ( M ) in SU (2). ◮ The Casson invariant is determined by surgery properties. Wolfgang Pitsch Trivial Cocycles, Casson invariant and a Conjecture of Perron
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