On the Van Est homomorphism for Lie groupoids Eckhard Meinrenken (based on joint work with David Li-Bland) Fields Institute, December 13, 2013 Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Overview Let G ⇒ M be a Lie groupoid, with Lie algebroid A = Lie( G ) Weinstein-Xu (1991) constructed a cochain map VE: C • ( G ) → C • ( A ) from smooth groupoid cochains to the Chevalley-Eilenberg complex of the Lie algebroid A of G . Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Overview Let G ⇒ M be a Lie groupoid, with Lie algebroid A = Lie( G ) Weinstein-Xu (1991) constructed a cochain map VE: C • ( G ) → C • ( A ) from smooth groupoid cochains to the Chevalley-Eilenberg complex of the Lie algebroid A of G . Crainic (2003) proved a Van Est Theorem for this map. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Overview Let G ⇒ M be a Lie groupoid, with Lie algebroid A = Lie( G ) Weinstein-Xu (1991) constructed a cochain map VE: C • ( G ) → C • ( A ) from smooth groupoid cochains to the Chevalley-Eilenberg complex of the Lie algebroid A of G . Crainic (2003) proved a Van Est Theorem for this map. Weinstein, Mehta (2006), and Abad-Crainic (2008, 2011) generalized to VE: W • , • ( G ) → W • , • ( A ) for suitably defined Weil algebras . Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Overview Applications Foliation theory Integration of (quasi-)Poisson manifolds and Dirac structures Multiplicative forms on groupoids (Mackenzie-Xu, Bursztyn-Cabrera-Ortiz) Index theory (Posthuma-Pflaum-Tang) Lie pseudogroups and Spencer operators (Crainic-Salazar-Struchiner), · · · Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Overview Applications Foliation theory Integration of (quasi-)Poisson manifolds and Dirac structures Multiplicative forms on groupoids (Mackenzie-Xu, Bursztyn-Cabrera-Ortiz) Index theory (Posthuma-Pflaum-Tang) Lie pseudogroups and Spencer operators (Crainic-Salazar-Struchiner), · · · Using the Fundamental Lemma of homological perturbation theory, we’ll give a simple construction of VE (and its properties). Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Let G ⇒ M be a Lie groupoid over M ⊆ G . g m 0 ← − m 1 . Multiplication ( g 1 , g 2 ) �→ g 1 g 2 defined for composable arrows : � � � � g 1 g 2 g 1 g 2 m 0 ← − m 1 ← − m 2 �→ m 0 ← − − m 2 . Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Let G ⇒ M be a Lie groupoid over M ⊆ G . g m 0 ← − m 1 . Multiplication ( g 1 , g 2 ) �→ g 1 g 2 defined for composable arrows : � � � � g 1 g 2 g 1 g 2 m 0 ← − m 1 ← − m 2 �→ m 0 ← − − m 2 . Examples Lie group G ⇒ pt Pair groupoid Pair( M ) = M × M ⇒ M Fundamental groupoid Π( M ) ⇒ M Foliation groupoid(s), e.g., Π F ( M ) ⇒ M Gauge groupoids of principal bundles Action groupoids K ⋉ M ⇒ M Groupoids associated with hypersurfaces Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Let B p G be the manifold of p -arrows ( g 1 , . . . , g p ): g p g 1 g 2 m 0 ← − m 1 ← − m 2 · · · ← − m p It is a simplicial manifold, with face maps ∂ i : B p G → B p − 1 G , i = 0 , . . . , p removing m i and degeneracies ǫ i : B p G → B p +1 G repeating m i . For example � � � � g p g p g 1 g 2 g 1 g 2 m 0 − m 1 − m 2 · · · − m p = m 0 − m 2 · · · − m p ∂ 1 ← ← ← ← − ← . Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Groupoid cochain complex: C • ( G ) := C ∞ ( B • G ) with differential p +1 � ( − 1) i ∂ ∗ i : C ∞ ( B p G ) → C ∞ ( B p +1 G ) δ = i =0 and algebra structure C p ( G ) ⊗ C p ′ ( G ) → C p + p ′ ( G ), f ∪ f ′ = pr ∗ f (pr ′ ) ∗ f ′ , where pr , pr ′ are the ‘front face’ and ‘back face’ projections. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Groupoid cochain complex: C • ( G ) := C ∞ ( B • G ) with differential p +1 � ( − 1) i ∂ ∗ i : C ∞ ( B p G ) → C ∞ ( B p +1 G ) δ = i =0 and algebra structure C p ( G ) ⊗ C p ′ ( G ) → C p + p ′ ( G ), f ∪ f ′ = pr ∗ f (pr ′ ) ∗ f ′ , where pr , pr ′ are the ‘front face’ and ‘back face’ projections. Variations: Normalized subcomplex � C • ( G ): kernel of degeneracy maps ǫ i . More generally, with coefficients in G -modules S → M . C • ( G ) M := C ∞ ( B • G ) M , the germs along M ⊆ B p G . Extends to double complex W • , • ( G ) := Ω • ( B • G ). Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie groupoid cohomology Example (Alexander-Spanier complex) C • (Pair( M )) M = C ∞ ( M p +1 ) M p +1 � ( − 1) i f ( m 0 , . . . , � ( δ f )( m 0 , . . . , m p +1 ) = m i , . . . , m p +1 ) i =0 Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie algebroid cohomology Let A → M be a Lie algebroid, with anchor a: A → TM and bracket [ · , · ] A on Γ( A ). Thus [ X , fY ] = f [ X , Y ] + (a( X ) f ) Y . Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie algebroid cohomology Let A → M be a Lie algebroid, with anchor a: A → TM and bracket [ · , · ] A on Γ( A ). Thus [ X , fY ] = f [ X , Y ] + (a( X ) f ) Y . Examples Lie algebra g Tangent bundle TM Tangent bundle to foliation T F M ⊂ TM Atiyah algebroid of principal bundle Cotangent Lie algebroid of Poisson manifold Action Lie algebroids k ⋉ M Lie algebroids associated with hypersurfaces ... Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie algebroid cohomology The Chevalley-Eilenberg complex is C • ( A ) = Γ( ∧ • A ∗ ) with differential ( d CE φ )( X 0 , . . . , X p ) p � ( − 1) i a( X i ) φ ( X 0 , . . . , � = X i , . . . , X p ) i =0 � ( − 1) i + j φ ([ X i , X j ] , X 0 , . . . , � X i , . . . , � + X j , . . . , X p ) i < j and with product the wedge product. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
Lie algebroid cohomology The Chevalley-Eilenberg complex is C • ( A ) = Γ( ∧ • A ∗ ) with differential ( d CE φ )( X 0 , . . . , X p ) p � ( − 1) i a( X i ) φ ( X 0 , . . . , � = X i , . . . , X p ) i =0 � ( − 1) i + j φ ([ X i , X j ] , X 0 , . . . , � X i , . . . , � + X j , . . . , X p ) i < j and with product the wedge product. More generally, with coefficients in A -modules S → M . Extends to double complex W • , • ( A ) For A = T F M , get foliated de Rham complex Ω F ( M ). Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
From Lie groupoids to Lie algebroids A Lie groupoid G ⇒ M has an associated Lie algebroid: Lie( G ) = ν ( M , G ) anchor a: Lie( G ) → TM induced from Tt − Ts : TG → TM , [ · , · ] from Γ(Lie( G )) = Lie(Γ( G )) where Γ( G ) is the group of bisections. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
From Lie groupoids to Lie algebroids A Lie groupoid G ⇒ M has an associated Lie algebroid: Lie( G ) = ν ( M , G ) anchor a: Lie( G ) → TM induced from Tt − Ts : TG → TM , [ · , · ] from Γ(Lie( G )) = Lie(Γ( G )) where Γ( G ) is the group of bisections. The Van Est map relates the corresponding cochain complexes. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
From Lie groupoids to Lie algebroids A Lie groupoid G ⇒ M has an associated Lie algebroid: Lie( G ) = ν ( M , G ) anchor a: Lie( G ) → TM induced from Tt − Ts : TG → TM , [ · , · ] from Γ(Lie( G )) = Lie(Γ( G )) where Γ( G ) is the group of bisections. The Van Est map relates the corresponding cochain complexes. We’ll explain this map using a double complex. Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
� � � Van Est double complex Define a principal G -bundle � B p G E p G κ p π p M where E p G ⊆ G p +1 consists of elements ( a 0 , . . . , a p ) with common source: m � � � � � �������������� � � � � � � a p a 0 � � � � � � � � a 1 � � � � � � � ... � � � � � � � � � � � � � � � � m 0 m 1 . . . � m p � � Eckhard Meinrenken (based on joint work with David Li-Bland) On the Van Est homomorphism for Lie groupoids
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