The cohomology of and the Eichler-Shimura Out( F r ) - PowerPoint PPT Presentation

The cohomology of and the Eichler-Shimura Out( F r ) isomorphism Jim Conant, University of Tennessee June 24, CIRM joint with Martin Kassabov and Karen Vogtmann

Happy Birthday, Karen!

Moving to the Lie category ∞ � PH ∗ ( ℓ + H ∗ ( Out ( F r ); Q ) ∞ ) sp r =2 sp-invariant theory ∼ ∼ = spectral sequence = (Kontsevich, on spine of outer Conant-Vogtmann) space Lie graph homology ∞ � PH ∗ ( ℓ + H ∗ ( Out ( F r ); Q ) ∞ ) sp ∼ = r =2

Moving to the Lie category ∞ � PH ∗ ( ℓ + H ∗ ( Out ( F r ); Q ) ∞ ) sp r =2 sp-invariant theory ∼ ∼ = spectral sequence = (Kontsevich, on spine of outer Conant-Vogtmann) space Lie graph homology ∞ � PH ∗ ( ℓ + H ∗ ( Out ( F r ); Q ) ∞ ) sp ∼ = r =2 So we study the homology of Out via the Lie algebra . ℓ + ∞

Lie Spiders Let be a vector space with a ( V, � , , � ) nondegenerate bilinear form. v 1 w 2 v 2 w 3 v 3 w 1 v 5 v 4 w 4 Modulo Jacobi (IHX) and antisymmetry.

v 1 w 2 v 2 w 3 [ , ] = v 3 w 1 v 5 v 4 w 4 w 2 v 1 v 2 w 3 � � v i , w j � i,j v 5 v 4 w 4 Antisymmetry: [x,y]=-[y,x] ⇒ � , � is symplectic. Jacobi Identity: ⇐ Generalized associativity (cyclic operad structure)

Let be a fixed standard 2n-dimensional V n symplectic vector space. is the Lie algebra of spiders labeled by , ℓ + V n n with at least 3 legs. ℓ + n →∞ ℓ + ∞ = lim n

Utility of the abelianization g → a H ∗ ( a ) → H ∗ ( g ) Λ ∗ ( a ) → H ∗ ( g ) Λ ∗ ( a ) sp → H ∗ ( g ) sp In some cases, the kernel is not too large.

Morita constructed a surjective Lie algebra map ∞ � ℓ + ∞ ։ Λ 3 V ⊕ S 2 k +1 V k =1 abelian Lie algebra He conjectured that this is precisely the abelianization.

v 1 v 2 v 2 tr � � v i , v j � v 3 v 3 M v 5 v 5 v 4 Not hard: tr vanishes on nontrivial brackets. S 2 k +1 V v 2 ... v 2 k +1 v 1

Idea: generalize Morita’s trace. Tr = exp( tr ): Λ ℓ + ∞ → HG v 1 v 2 v 2 Tr � � v i 1 , v j 1 � · · · � v i k , v j k � v 3 M v 5 v 4

Tr ∗ ( ℓ + ∞ ) ab ֒ → H 1 ( HG ) Theorem : where contains, for example, hairy graphs Im ( Tr ∗ ) labeled by . V + ⊂ V O v 3 v 1 v 2 δ + �→

is graded by loop degree. H 1 ( HG ) } H 0 = Λ 3 V 1 ( HG ) ∼ v 1 v 2 v 3 Morita ∞ � H 1 S 2 k +1 V 1 ( HG ) ∼ = k =1

L λ k, ℓ New: � H 2 1 ( HG ) ∼ ( F ( k, ℓ ) ) = k> ℓ ≥ 0 F ( k, ℓ ) = irrep of GL(V) is the dimension of the space of weight n cuspidal s n modular forms for . SL (2 , Z ) � if ℓ is even. s k − ℓ +2 λ k, ℓ = s k − ℓ +2 + 1 if ℓ is odd. New: H 3 1 ( HG ) � = 0

Example: � = 0 ∈ H 2 1 ( HG ) v 1 v 2 v 3 v 4 v i ∈ V + ⇒ this is in im(Tr), so represents a nonzero element of . ( ℓ + ∞ ) ab � v 3 , v 4 � � = 0 ⇒ this is not in im(Tr).

L λ k, ℓ Proof of � H 2 1 ( HG ) ∼ ( F ( k, ℓ ) ) = k> ℓ ≥ 0 Step 1: H r = H 2 r − 3 ( Out ( F r ); P ( V ⊕ r )) 1 ( HG ) ∼ Step 2: H 1 ( Out ( F 2 ); P ( V ⊕ V )) = H 1 ( GL (2 , Z ); P ( V ⊕ V )) Use existing results (Eichler-Shimura).

Proof of Step 1: Spine of Outer Space ρ ≃

∈ ˜ ρ C 2 r − 3 ( Out ( F r ) , P ( V ⊕ r )) ⊗ Φ ≃ ρ ∗ Φ ∈ P ( V ⊕ r ) = P ( V ⊗ H 1 ( R r , C )) ∼ = P ( V ⊗ H 1 ( G, C ))

∈ ˜ ρ C 2 r − 3 ( Out ( F r ) , P ( V ⊕ r )) ⊗ Φ ≃ ρ ∗ Φ ∈ P ( V ⊕ r ) = P ( V ⊗ H 1 ( R r , C )) ∼ = P ( V ⊗ H 1 ( G, C )) P ( V ⊗ H 1 ( G, C )) ↔ hairy graphs

P ( V ⊗ H 1 ( G, C )) ↔ hairy graphs ( e 1 ⊗ a ) 3 ( e 3 ⊗ b ) 2 ( e 6 ⊗ c ) 4 ( e 6 ⊗ d ) e 5 a e 3 e 1 b e 4 b a e 2 d e 6 c a c c c

Modulo the action of we are left with hairy Out ( F r ) graphs up to graph isomorphism. One verifies that in this top degree, the hairy graph boundary operator corresponds to the boundary operator for the spine (with local coefficients.) �

Step 2: H 1 ( Out ( F 2 ) , P ( V ⊗ C 2 )) =? Detour: modular forms.

H 1 ( Out ( F 2 ) , P ( V ⊗ C 2 )) =? Detour: modular forms. meromorphic. f : H → C

H 1 ( Out ( F 2 ) , P ( V ⊗ C 2 )) =? Detour: modular forms. meromorphic. f : H → C α z = az + b α ∈ SL (2 , Z ) cz + d

H 1 ( Out ( F 2 ) , P ( V ⊗ C 2 )) =? Detour: modular forms. meromorphic. f : H → C α z = az + b α ∈ SL (2 , Z ) cz + d Suppose f ( z ) = f ( α z )( cz + d ) − k

H 1 ( Out ( F 2 ) , P ( V ⊗ C 2 )) =? Detour: modular forms. meromorphic. f : H → C α z = az + b α ∈ SL (2 , Z ) cz + d Suppose f ( z ) = f ( α z )( cz + d ) − k f C H f(z)=f(z+1) so this ‘q- q ( z ) = e 2 π iz expansion’ exists. f ∞ C \ { 0 }

meromorphic on C ⇒ f ∞ f is a modular form of weight k. f ∞ (0) = 0 ⇒ f is cuspidal . Example: Eisenstein Series 1 � G k ( z ) = ( mz + n ) k k > 2 ( m,n ) � =(0 , 0)

The complex vector space of modular forms Theorem: is isomorphic to the polynomial ring . C [ G 4 , G 6 ] Exercise: � ⌊ k 12 ⌋ if k ≡ 2 mod 12 dim M k = ⌊ k 12 ⌋ + 1 if k �≡ 2 mod 12 M k ∼ = M 0 k ⊕ C

Eichler-Shimura isomorphism Let f be a cusp form of weight k.

Eichler-Shimura isomorphism Let f be a cusp form of weight k. ES f : PSL (2 , Z ) → R k − 1

Eichler-Shimura isomorphism Let f be a cusp form of weight k. ES f : PSL (2 , Z ) → R k − 1 f ( z ) z k − 2 dz   f ( z ) z k − 3 dz   ω ( f ) = .   .   .   f ( z ) z 0 dz

Eichler-Shimura isomorphism H 1 ( SL (2 , Z ); H k − 2 ) ∼ = M 0 k ⊕ M 0 k ⊕ E k

Let be the dimension of the space of weight n s n cuspidal modular forms for . SL (2 , Z ) Let be the irreducible representation of GL(V) F ( k, ℓ ) associated to the partition , . ( k, ℓ ) k ≥ ℓ Theorem: � H 1 ( Out ( F 2 ); P ( V ⊗ C 2 )) ∼ ( F ( k, ℓ ) ) ⊕ λ k, ℓ = k> ℓ ≥ 0 � if ℓ is even. s k − ℓ +2 where λ k, ℓ = s k − ℓ +2 + 1 if ℓ is odd.

Proof: Out ( F 2 ) = GL (2 , Z )

Proof: Out ( F 2 ) = GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ where is a Young diagram = M λ ⊗ N λ λ λ λ = ( m, n ) , m ≥ n C 2 irreps for V and .

Proof: Out ( F 2 ) = GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ where is a Young diagram = M λ ⊗ N λ λ λ λ = ( m, n ) , m ≥ n C 2 irreps for V and . N λ ∼ as modules. = C det n ⊗ H m − n GL (2 , Z )

Proof: Out ( F 2 ) = GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ where is a Young diagram = M λ ⊗ N λ λ λ λ = ( m, n ) , m ≥ n C 2 irreps for V and . N λ ∼ as modules. = C det n ⊗ H m − n GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ = H m − n ⊗ F ( m,n ) m ≥ n

Proof: Out ( F 2 ) = GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ where is a Young diagram = M λ ⊗ N λ λ λ λ = ( m, n ) , m ≥ n C 2 irreps for V and . N λ ∼ as modules. = C det n ⊗ H m − n GL (2 , Z ) � P [ V ⊗ C 2 ] ∼ = H m − n ⊗ F ( m,n ) m ≥ n � H 1 ( GL (2 , Z ); P [ V ⊗ C 2 ]) = H 1 ( GL (2 , Z ); H m − n ) ⊗ F ( m,n ) m ≥ n

H 1 ( GL (2 , Z ) , H k ) ∼ = H 1 ( SL (2 , Z ) , H k ) Z 2 H 1 ( SL (2 , Z ); H k ) ∼ = M 0 ( k + 2) ⊕ M 0 ( k + 2) ⊕ E k +2 � H 1 ( GL (2 , Z ); P [ V ⊗ C 2 ]) = H 1 ( GL (2 , Z ); H m − n ) ⊗ F ( m,n ) m ≥ n � M 0 ( m − n + 1) if n is even. H 1 ( SL (2 , Z ); H m − n ) Z 2 ∼ = M 0 ( m − n + 2) ⊕ E m − n +2 if n is odd. = C unless m = n E m − n +2 ∼ E 2 = 0 �

Fitting pieces of the abelianization together v 1 v 2 v 3 Λ 3 V can combine with themselves to detect H 0 ( Out ( F r ); Q ) = Q ∼ H 0 ( Out ( F 3 ); Q )

∞ can combine with themselves to � S 2 k +1 V create generalized Morita classes. k =1 H 4 ( Out ( F 4 ); Q ) H 8 ( Out ( F 6 ); Q ) H 12 ( Out ( F 8 ); Q ) (Vogtmann) (Conant,Vogtmann, (Gray) Ohashi)

H 7 ( Aut ( F 5 ); Q ) H 22 ( Out ( F 13 ); Q ) H 11 ( Aut ( F 7 ); Q ) nonzero?? (CKV, Gerlits)

Further Directions • Extend the 2-loop calculation to 3-loops and beyond. To generalize our argument, we need the cohomology of SL(n,Z) with coefficients in an irreducible representation (doable) as well as the cohomology of IA_n as a GL module, which is quite hard. • Show that classes produced from gluing together graphs in the abelianization give rise, in some large number of case, to nontrivial homology classes. Current methods require computer computations. The next Morita class is probably within reach, but essentially a new method will be needed for the general case. • All known classes for Aut and Out arise from this abelianization construction. Is this true in general?