Homomorphisms between Diffeomorphism Groups Kathryn Mann University of Chicago May 5, 2012
A problem Given manifolds M 1 and M 2 , describe all homomorphisms Diff r c ( M 1 ) → Diff p c ( M 2 )
A problem Given manifolds M 1 and M 2 , describe all homomorphisms Diff r c ( M 1 ) → Diff p c ( M 2 ) ◮ Diff r c ( M ) = group of compactly supported C r diffeomorphisms isotopic to the identity.
A problem Given manifolds M 1 and M 2 , describe all homomorphisms Diff r c ( M 1 ) → Diff p c ( M 2 ) ◮ Diff r c ( M ) = group of compactly supported C r diffeomorphisms isotopic to the identity. ◮ This is a simple group [Mather, Thurston], so any nontrivial homomorphism is necessarily injective.
We understand isomorphisms completely Let M 1 and M 2 be smooth manifolds.
We understand isomorphisms completely Let M 1 and M 2 be smooth manifolds. Theorem (Filipkiewicz, 1982) c ( M 2 ) , then M 1 ∼ If ∃ an isomorphism Φ : Diff r c ( M 1 ) → Diff s = M 2 .
We understand isomorphisms completely Let M 1 and M 2 be smooth manifolds. Theorem (Filipkiewicz, 1982) c ( M 2 ) , then M 1 ∼ If ∃ an isomorphism Φ : Diff r c ( M 1 ) → Diff s = M 2 . Also, r = s and Φ is induced by a C r diffeomorphism f : M 1 → M 2 .
We understand isomorphisms completely Let M 1 and M 2 be smooth manifolds. Theorem (Filipkiewicz, 1982) c ( M 2 ) , then M 1 ∼ If ∃ an isomorphism Φ : Diff r c ( M 1 ) → Diff s = M 2 . Also, r = s and Φ is induced by a C r diffeomorphism f : M 1 → M 2 . “Induced” means Φ( g ) = fgf − 1
... but not homomorphisms! Question (Ghys, 1991) Let M 1 and M 2 be closed manifolds. ∃ (injective) homomorphism Diff ∞ ( M 1 ) 0 ֒ → Diff ∞ ( M 2 ) 0 ?? ⇒ dim( M 1 ) ≤ dim( M 2 )
... but not homomorphisms! Question (Ghys, 1991) Let M 1 and M 2 be closed manifolds. ∃ (injective) homomorphism Diff ∞ ( M 1 ) 0 ֒ → Diff ∞ ( M 2 ) 0 ?? ⇒ dim( M 1 ) ≤ dim( M 2 ) ◮ Diff ∞ ( M ) 0 = identity component of group of C ∞ diffeomorphisms on M .
... but not homomorphisms! Question (Ghys, 1991) Let M 1 and M 2 be closed manifolds. ∃ (injective) homomorphism Diff ∞ ( M 1 ) 0 ֒ → Diff ∞ ( M 2 ) 0 ?? ⇒ dim( M 1 ) ≤ dim( M 2 ) ◮ Diff ∞ ( M ) 0 = identity component of group of C ∞ diffeomorphisms on M . ◮ Can also ask this for general boundaryless M and Diff r c ( M ).
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 )
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms.
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms.
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms.
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms. ◮ Special cases: M 2 = M 1 × N ,
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms. ◮ Special cases: M 2 = M 1 × N ,
Examples of homomorphisms ◮ M 1 an open submanifold of M 2 , inclusion Diff r → Diff r c ( M 1 ) ֒ c ( M 2 ) ◮ Generalization: Topologically diagonal embedding U i ⊂ M 2 disjoint open sets, f i : M 1 → U i ⊂ M 2 diffeomorphisms. ◮ Special cases: M 2 = M 1 × N , unit tangent bundle of M 1 ...
(non)-Continuity
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ):
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ): ◮ α : R → R any non-continuous, injective, additive group homomorphism.
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ): ◮ α : R → R any non-continuous, injective, additive group homomorphism. ◮ ψ t compactly supported C r flow on M .
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ): ◮ α : R → R any non-continuous, injective, additive group homomorphism. ◮ ψ t compactly supported C r flow on M . t �→ ψ α ( t )
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ): ◮ α : R → R any non-continuous, injective, additive group homomorphism. ◮ ψ t compactly supported C r flow on M . t �→ ψ α ( t ) Can this happen with Diff instead of R ?
(non)-Continuity A non-continuous homomorphism R → Diff r c ( M ): ◮ α : R → R any non-continuous, injective, additive group homomorphism. ◮ ψ t compactly supported C r flow on M . t �→ ψ α ( t ) Can this happen with Diff instead of R ? How bad can injections Diff r c ( M 1 ) → Diff p c ( M 2 ) look?
Our results: small target
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds.
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds. Every homomorphism Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) is topologically diagonal.
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds. Every homomorphism Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) is topologically diagonal. (And if r ≤ p , the maps f i are C r )
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds. Every homomorphism Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) is topologically diagonal. (And if r ≤ p , the maps f i are C r ) Theorem 2 (-) Let M 1 be any manifold; r , p , M 2 as above.
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds. Every homomorphism Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) is topologically diagonal. (And if r ≤ p , the maps f i are C r ) Theorem 2 (-) Let M 1 be any manifold; r , p , M 2 as above. ∃ Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) nontrivial homomorphism ⇒ dim( M 1 ) = 1 and Φ is topologically diagonal
Our results: small target Theorem 1 (-) Let r ≥ 3 , p ≥ 2 ; M 1 and M 2 1-manifolds. Every homomorphism Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) is topologically diagonal. (And if r ≤ p , the maps f i are C r ) Theorem 2 (-) Let M 1 be any manifold; r , p , M 2 as above. ∃ Φ : Diff r c ( M 1 ) → Diff p c ( M 2 ) nontrivial homomorphism ⇒ dim( M 1 ) = 1 and Φ is topologically diagonal (This answers Ghys’ question in the dim( M 2 ) = 1 case)
Proof idea - theorem 1 ◮ Algebraic (group structure) data ↔ topological data
Proof idea - theorem 1 ◮ Algebraic (group structure) data ↔ topological data ◮ Continuity results
Proof idea - theorem 1 ◮ Algebraic (group structure) data ↔ topological data ◮ Continuity results ◮ Build f i
Topological data ↔ algebraic data Topology of the manifold ↔ group structure of Diff
Topological data ↔ algebraic data Topology of the manifold ↔ group structure of Diff ? Point x ↔
Topological data ↔ algebraic data Topology of the manifold ↔ group structure of Diff ? Point x ↔ G x point stablizer
Topological data ↔ algebraic data Topology of the manifold ↔ group structure of Diff ? Point x ↔ G x point stablizer Open set U ↔
Topological data ↔ algebraic data Topology of the manifold ↔ group structure of Diff ? Point x ↔ G x point stablizer Open set U ↔ G U group of diffeomorphisms fixing U pointwise.
G U is characterized by having large centralizer:
G U is characterized by having large centralizer: ( G U commutes with anything supported here)
G U is characterized by having large centralizer: ( G U commutes with anything supported here) Fact Let G ⊂ Diff r c ( R ) be nonabelian G has nonabelian centralizer ⇔ G pointwise fixes open U ⊂ R
G U is characterized by having large centralizer: ( G U commutes with anything supported here) Fact Let G ⊂ Diff r c ( R ) be nonabelian G has nonabelian centralizer ⇔ G pointwise fixes open U ⊂ R Proof techniques: ◮ H¨ older’s theorem (free actions on R ) ◮ Kopell’s lemma (centralizers of C 2 diffeomorphisms)
G U is characterized by having large centralizer: ( G U commutes with anything supported here) Fact Let G ⊂ Diff r c ( R ) be nonabelian G has nonabelian centralizer ⇔ G pointwise fixes open U ⊂ R Proof techniques: ◮ H¨ older’s theorem (free actions on R ) ◮ Kopell’s lemma (centralizers of C 2 diffeomorphisms) *These also [mostly] work for S 1 , but not for general M !*
Corollary For U, V open subsets of R U , V intersect ⇔ � G U , G V � pointwise fixes an open set ( U ∩ V ) ⇔ � G U , G V � has nonabelian centralizer in Diff
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