BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS Index Theory and Singular Structures Toulouse, France Paul Baum Penn State 31 May, 2017 Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 1 / 50
BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLD K -homology is the dual theory to K -theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K -homology and K -cycle K -homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians. Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici — also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds. This is joint work with Erik van Erp. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 2 / 50
REFERENCE P. Baum and E. van Erp, K-homology and index theory on contact manifolds Acta. Math. 213 (2014) 1-48. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 3 / 50
FACT: If M is a closed odd-dimensional C ∞ manifold and D is any elliptic differential operator on M , then Index( D ) = 0. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 4 / 50
EXAMPLE: M = S 3 = { ( a 1 , a 2 , a 3 , a 4 ) ∈ R 4 | a 2 1 + a 2 2 + a 2 3 + a 2 4 = 1 } x 1 , x 2 , x 3 , x 4 are the usual co-ordinate functions on R 4 . x j ( a 1 , a 2 , a 3 , a 4 ) = a j j = 1 , 2 , 3 , 4 ∂ usual vector fields on R 4 j = 1 , 2 , 3 , 4 ∂x j Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 5 / 50
On S 3 consider the (tangent) vector fields V 1 , V 2 , V 3 ∂ ∂ ∂ ∂ V 1 = x 2 − x 1 + x 4 − x 3 ∂x 1 ∂x 2 ∂x 3 ∂x 4 ∂ ∂ ∂ ∂ V 2 = x 3 − x 4 − x 1 + x 2 ∂x 1 ∂x 2 ∂x 3 ∂x 4 ∂ ∂ ∂ ∂ V 3 = x 4 + x 3 − x 2 − x 1 ∂x 1 ∂x 2 ∂x 3 ∂x 4 Let r be any positive integer and let γ : S 3 − → M ( r, C ) be a C ∞ map. M ( r, C ) := { r × r matrices of complex numbers } . Form the operator P γ := iγ ( V 1 ⊗ I r ) − V 2 2 ⊗ I r − V 2 3 ⊗ I r . I r := r × r identity matrix. P γ : C ∞ ( S 3 , S 3 × C r ) − → C ∞ ( S 3 , S 3 × C r ) Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 6 / 50
P γ := iγ ( V 1 ⊗ I r ) − V 2 2 ⊗ I r − V 2 3 ⊗ I r i = √− 1 . I r := r × r identity matrix. P γ : C ∞ ( S 3 , S 3 × C r ) − → C ∞ ( S 3 , S 3 × C r ) LEMMA. Assume that for all p ∈ S 3 , γ ( p ) does not have any odd integers among its eigenvalues i.e. ∀ p ∈ S 3 , ∀ λ ∈ { . . . − 3 , − 1 , 1 , 3 , . . . } = ⇒ λI r − γ ( p ) ∈ GL ( r, C ) then dim C (Kernel P γ ) < ∞ and dim C (Cokernel P γ ) < ∞ . Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 7 / 50
With γ as in the above lemma, for each odd integer n , let γ n : S 3 − → GL ( r, C ) be p �− → nI r − γ ( p ) By Bott periodicity if r ≥ 2 , then π 3 GL ( r, C ) = Z . Hence for each odd integer n have the Bott number β ( γ n ) . PROPOSITION. With γ as above and r ≥ 2 � Index( P γ ) = β ( γ n ) n odd Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 8 / 50
S 2 n +1 = unit sphere of R 2 n +2 S 2 n +1 ⊂ R 2 n +2 n = 1 , 2 , 3 , . . . On S 2 n +1 there is the nowhere-vanishing vector field V V= ∂ ∂ ∂ ∂ ∂ ∂ x 2 − x 1 + x 4 − x 3 + · · · + x 2 n +2 − x 2 n +1 ∂x 1 ∂x 2 ∂x 3 ∂x 4 ∂x 2 n +1 ∂x 2 n +2 n +1 ∂ ∂ � V = x 2 i − x 2 i − 1 ∂x 2 i − 1 ∂x 2 i i =1 Let θ be the 1-form on S 2 n +1 n +1 � θ = x 2 i dx 2 i − 1 − x 2 i − 1 dx 2 i i =1 Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 9 / 50
Then: θ ( V ) = 1 θ ( dθ ) n is a volume form on S 2 n +1 i.e. θ ( dθ ) n is a nowhere-vanishing C ∞ 2 n + 1 form on S 2 n +1 . Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 10 / 50
Let H be the null-space of θ . H = { v ∈ TS 2 n +1 | θ ( v ) = 0 } H is a C ∞ sub vector bundle of TS 2 n +1 with For all x ∈ S 2 n +1 , dim R ( H x ) = 2 n The sub-Laplacian ∆ H : C ∞ ( S 2 n +1 ) → C ∞ ( S 2 n +1 ) is locally − W 2 1 − W 2 2 − · · · − W 2 2 n where W 1 , W 2 , . . . , W 2 n is a locally defined C ∞ orthonormal frame for H . These locally defined operators are then patched together using a C ∞ partition of unity to give the sub-Laplacian ∆ H . Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 11 / 50
Let r be a positive integer and let γ : S 2 n +1 − → M ( r, C ) be a C ∞ map. M ( r, C ) := { r × r matrices of complex numbers } . Assume: For each x ∈ S 2 n +1 { Eigenvalues of γ ( x ) } ∩ { . . . , − n − 4 , − n − 2 , − n, n, n + 2 , n + 4 , . . . } = ∅ i.e. ∀ x ∈ S 2 n +1 , λ ∈ { . . . − n − 4 , − n − 2 , − n, n, n +2 , n +4 , . . . } = ⇒ λI r − γ ( x ) ∈ GL ( r, C ) Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 12 / 50
Let γ : S 2 n +1 − → M ( r, C ) be as above, P γ : C ∞ ( S 2 n +1 , S 2 n +1 × C r ) → C ∞ ( S 2 n +1 , S 2 n +1 × C r ) is defined: √ P γ = iγ ( V ⊗ I r )+(∆ H ) ⊗ I r I r = r × r identity matrix i = − 1 P γ is a differential operator (of order 2) and is hypoelliptic but not elliptic. P γ is Fredholm. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 13 / 50
The formula for the index of P γ is Index P γ = � n + j − 1 N � � � β (( n + 2 j ) I r − γ ) + ( − 1) n +1 β (( n + 2 j ) I r ) + γ ) � j j =0 β (( n + 2 j ) I r − γ ) := the Bott number of ( n + 2 j ) I r − γ ( n + 2 j ) I r − γ : S 2 n +1 → GL ( r, C ) Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 14 / 50
Remark on the S 2 n +1 example n +1 ∂ ∂ � V = x 2 i − x 2 i − 1 ∂x 2 i − 1 ∂x 2 i i =1 θ is the 1-form on S 2 n +1 n +1 � x 2 i dx 2 i − 1 − x 2 i − 1 dx 2 i θ = i =1 θ ( V ) = 1 Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 15 / 50
V is the vector field along the orbits for the usual action of S 1 on S 2 n +1 . S 1 × S 2 n +1 − → S 2 n +1 The quotient space S 2 n +1 /S 1 is C P n . Denote the quotient map by π : S 2 n +1 → C P n . π : S 2 n +1 → C P n THEN H := null space of θ = π ∗ ( T C P n ) is a C vector bundle on S 2 n +1 . Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 16 / 50
Contact Manifolds A contact manifold is an odd dimensional C ∞ manifold X dimension( X ) = 2 n + 1 with a given C ∞ 1-form θ such that θ ( dθ ) n is non zero at every x ∈ X − i.e. θ ( dθ ) n is a volume form for X. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 17 / 50
Let X be a compact connected contact manifold without boundary ( ∂X = ∅ ) . Set dimension( X ) = 2 n + 1 . → M ( r, C ) be a C ∞ map. Let r be a positive integer and let γ : X − M ( r, C ) := { r × r matrices of complex numbers } . Assume: For each x ∈ X, { Eigenvalues of γ ( x ) } ∩ { . . . , − n − 4 , − n − 2 , − n, n, n + 2 , n + 4 , . . . } = ∅ i.e. ∀ x ∈ X, λ ∈ { . . . − n − 4 , − n − 2 , − n, n, n +2 , n +4 , . . . } = ⇒ λI r − γ ( x ) ∈ GL ( r, C ) Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 18 / 50
γ : X − → M ( r, C ) Are assuming : ∀ x ∈ X, λ ∈ { . . . − n − 4 , − n − 2 , − n, n, n +2 , n +4 , . . . } = ⇒ λI r − γ ( x ) ∈ GL ( r, C ) Associated to γ is a differential operator P γ which is hypoelliptic and Fredholm. P γ : C ∞ ( X, X × C r ) − → C ∞ ( X, X × C r ) P γ is constructed as follows. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 19 / 50
The sub-Laplacian ∆ H Let H be the null-space of θ . H = { v ∈ TX | θ ( v ) = 0 } H is a C ∞ sub vector bundle of TX with For all x ∈ X, dim R ( H x ) = 2 n The sub-Laplacian ∆ H : C ∞ ( X ) → C ∞ ( X ) is locally − W 2 1 − W 2 2 − · · · − W 2 2 n where W 1 , W 2 , . . . , W 2 n is a locally defined C ∞ orthonormal frame for H . These locally defined operators are then patched together using a C ∞ partition of unity to give the sub-Laplacian ∆ H . Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 20 / 50
The Reeb vector field The Reeb vector field is the unique C ∞ vector field W on X with : θ ( W ) = 1 and ∀ v ∈ TX, dθ ( W, v ) = 0 Let γ : X − → M ( r, C ) be as above, P γ : C ∞ ( X, X × C r ) → C ∞ ( X, X × C r ) is defined: √ P γ = iγ ( W ⊗ I r )+(∆ H ) ⊗ I r I r = r × r identity matrix i = − 1 P γ is a differential operator (of order 2) and is hypoelliptic but not elliptic. Paul Baum (Penn State) K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS 31 May, 2017 21 / 50
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