1 Drawing: Prof. Karl Heinrich Hofmann 2 Sylvesters Law of Inertia - PowerPoint PPT Presentation

1 Drawing: Prof. Karl Heinrich Hofmann

2 Sylvester’s Law of Inertia A demonstration of the theorem that every homogeneous quadratic ◮ polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. Philosophical Magazine IV, 138–142 (1852) ◮

3 Symmetric and symplectic forms over R ◮ Let ǫ = +1 or − 1. ◮ An ǫ -symmetric form ( K , φ ) is a finite dimensional real vector space K together with a bilinear pairing φ : K × K → R ; ( x , y ) �→ φ ( x , y ) such that φ ( x , y ) = ǫφ ( y , x ) ∈ R . ◮ The pairing φ can be identified with the adjoint linear map to the dual vector space φ : K → K ∗ = Hom R ( K , R ) ; x �→ ( y �→ φ ( x , y )) such that φ ∗ = ǫφ . ◮ The form ( K , φ ) is nonsingular if φ : K → K ∗ is an isomorphism. ◮ A 1-symmetric form is called symmetric . ◮ A ( − 1)-symmetric form is called symplectic .

4 Lagrangians and hyperbolic forms I. ◮ Definition A lagrangian of a nonsingular form ( K , φ ) is a subspace L ⊂ K such that L = L ⊥ , that is L = { x ∈ K | φ ( x , y ) = 0 for all y ∈ L } . ◮ Definition The hyperbolic ǫ -symmetric form is defined for any finite-dimensional real vector space L by � 0 � 1 H ǫ ( L ) = ( L ⊕ L ∗ , φ = ) , ǫ 0 φ : L ⊕ L ∗ × L ⊕ L ∗ → R ; (( x , f ) , ( y , g )) �→ g ( x ) + ǫ f ( y ) with lagrangian L . ◮ The graph of a ( − ǫ )-symmetric form ( L , λ ) is the lagrangian of H ǫ ( L ) Γ ( L ,λ ) = { ( x , λ ( x )) | x ∈ L } ⊂ L ⊕ L ∗ .

5 Lagrangians and hyperbolic forms II. ◮ Proposition The inclusion L → K of a lagrangian in a nonsingular ǫ -symmetric form ( K , φ ) extends to an isomorphism ∼ = � ( K , φ ) . H ǫ ( L ) ◮ Example For any nonsingular ǫ -symmetric form ( K , φ ) the inclusion of the diagonal lagrangian in ( K , φ ) ⊕ ( K , − φ ) ∆ : K → K ⊕ K ; x �→ ( x , x ) extends to the isomorphism − φ − 1   1 ∼ 2 = � ( K , φ ) ⊕ ( K , − φ ) .    : H ǫ ( K )   φ − 1  1 2

6 The classification of symmetric forms over R ◮ Proposition Every symmetric form ( K , φ ) is isomorphic to � � � ( R , 1) ⊕ ( R , − 1) ⊕ ( R , 0) p q r with p + q + r = dim R ( K ). Nonsingular if and only if r = 0. ◮ Two forms are isomorphic if and only if they have the same p , q , r . ◮ Definition The signature (or the index of inertia ) of ( K , φ ) is σ ( K , φ ) = p − q ∈ Z . ◮ Proposition The following conditions on a nonsingular form ( K , φ ) are equivalent: ◮ σ ( K , φ ) = 0, that is p = q , ◮ ( K , φ ) admits a lagrangian L , ( R , − 1) ∼ ◮ ( K , φ ) is isomorphic to � = H + ( R p ). ( R , 1) ⊕ � p p

7 The classification of symplectic forms over R ◮ Theorem Every symplectic form ( K , φ ) is isomorphic to � H − ( R p ) ⊕ ( R , 0) r with 2 p + r = dim R ( K ). Nonsingular if and only if r = 0. ◮ Two forms are isomorphic if and only if they have the same p , r . ◮ Proposition Every nonsingular symplectic form ( K , φ ) admits a lagrangian. ◮ Proof By induction on dim R ( K ). For every x ∈ K have φ ( x , x ) = 0. If x � = 0 ∈ K the linear map K → R ; y �→ φ ( x , y ) is onto, so there exists y ∈ K with φ ( x , y ) = 1 ∈ R . The subform ( R x ⊕ R y , φ | ) is isomorphic to H − ( R ), and ( K , φ ) ∼ = H − ( R ) ⊕ ( K ′ , φ ′ ) with dim R ( K ′ ) = dim R ( K ) − 2.

8 Poincar´ e duality ◮ H.P. Analysis Situs and its Five Supplements (1892–1904) (English translation by John Stillwell, 2009) ◮

9 The ( − ) n -symmetric form of a 2 n -dimensional manifold ◮ Manifolds will be oriented. ◮ Homology and cohomology will be with R -coefficients. ◮ The intersection form of a 2 n -dimensional manifold with boundary ( M , ∂ M ) is the ( − ) n -symmetric form given by the evaluation of the cup product on the fundamental class [ M ] ∈ H 2 n ( M , ∂ M ) ( H n ( M , ∂ M ) , φ M : ( x , y ) �→ � x ∪ y , [ M ] � ) . ◮ By Poincar´ e duality and universal coefficient isomorphisms H n ( M , ∂ M ) ∼ = H n ( M ) , H n ( M , ∂ M ) ∼ = H n ( M , ∂ M ) ∗ the adjoint linear map φ M fits into an exact sequence φ M � H n ( M , ∂ M ) . . . � H n ( ∂ M ) � H n ( M ) � H n − 1 ( ∂ M ) � . . . . ◮ The isomorphism class of the form is a homotopy invariant of ( M , ∂ M ). ◮ If M is closed, ∂ M = ∅ , then ( H n ( M , ∂ M ) , φ M ) is nonsingular. ◮ The intersection form of S n × S n is H ( − ) n ( R ).

� � � � 10 The lagrangian of a (2 n + 1) -dimensional manifold with boundary ◮ Proposition If ( N 2 n +1 , M 2 n ) is a (2 n + 1)-dimensional manifold with boundary then L = ker( H n ( M ) → H n ( N )) = im( H n ( N ) → H n ( M )) ⊂ H n ( M ) is a lagrangian of the ( − ) n -symmetric intersection form ( H n ( M ) , φ M ). ◮ Proof Consider the commutative diagram H n ( N ) H n ( M ) H n +1 ( N , M ) ∼ ∼ ∼ φ M � = = = � H n ( M ) � H n ( N ) H n +1 ( N , M ) with H n ( N ) ∼ = H n +1 ( N , M ), H n +1 ( N , M ) ∼ = H n ( N ) the Poincar´ e-Lefschetz duality isomorphisms.

11 The signature of a manifold I. Analisis situs combinatorio ◮ H.Weyl, Rev. Mat. Hispano-Americana 5, 390–432 (1923) ◮ ◮ Published in Spanish in South America to spare the author the shame of being regarded as a topologist.

12 The signature of a manifold II. ◮ The signature of a 4 k -dimensional manifold with boundary ( M , ∂ M ) is σ ( M ) = σ ( H 2 k ( M , ∂ M ) , φ M ) ∈ Z . ◮ Theorem (Thom, 1954) If a 4 k -dimensional manifold M is the boundary M = ∂ N of a (4 k + 1)-dimensional manifold N then σ ( M ) = σ ( H 2 k ( M ) , φ M ) = 0 ∈ Z . Cobordant manifolds have the same signature. ◮ The signature map σ : Ω 4 k → Z is onto for k � 1, with σ ( C P 2 k ) = 1 ∈ Z . Isomorphism for k = 1.

13 Novikov additivity of the signature ◮ Let M 4 k be a closed 4 k -dimensional manifold which is a union of 4 k -dimensional manifolds with boundary M 1 , M 2 M 4 k = M 1 ∪ M 2 with intersection a separating hypersurface ( M 1 ∩ M 2 ) 4 k − 1 = ∂ M 1 = ∂ M 2 ⊂ M . ▼ ✶ ▼ ✷ ▼ ✶ ❭ ▼ ✷ ◮ Theorem (N., 1967) The union has signature σ ( M ) = σ ( M 1 ) + σ ( M 2 ) ∈ Z . ✶

14 Formations ◮ Definition An ǫ -symmetric formation ( K , φ ; L 1 , L 2 ) is a nonsingular ǫ -symmetric form ( K , φ ) with an ordered pair of lagrangians L 1 , L 2 . ◮ Example The boundary of a ( − ǫ )-symmetric form ( L , λ ) is the ǫ -symmetric formation ∂ ( L , λ ) = ( H ǫ ( L ); L , Γ ( L ,λ ) ) with Γ ( L ,λ ) = { ( x , λ ( x )) | x ∈ L } the graph lagrangian of H ǫ ( L ). ◮ Definition (i) An isomorphism of ǫ -symmetric formations f : ( K , φ ; L 1 , L 2 ) → ( K ′ , φ ′ ; L ′ 1 , L ′ 2 ) is an isomorphism of forms f : ( K , φ ) → ( K ′ , φ ′ ) such that f ( L 1 ) = L ′ 1 , f ( L 2 ) = L ′ 2 . ◮ (ii) A stable isomorphism of ǫ -symmetric formations [ f ] : ( K , φ ; L 1 , L 2 ) → ( K ′ , φ ′ ; L ′ 1 , L ′ 2 ) is an isomorphism of the type 2 ) ⊕ ( H ǫ ( L ′ ); L ′ , L ′∗ ) . : ( K , φ ; L 1 , L 2 ) ⊕ ( H ǫ ( L ); L , L ∗ ) → ( K ′ , φ ′ ; L ′ 1 , L ′ f ◮ Two formations are stably isomorphic if and only if dim R ( L 1 ∩ L 2 ) = dim R ( L ′ 1 ∩ L ′ 2 ) .

15 Formations and automorphisms of forms ◮ Proposition Given a nonsingular ǫ -symmetric form ( K , φ ), a lagrangian L , and an automorphism α : ( K , φ ) → ( K , φ ) there is defined an ǫ -symmetric formation ( K , φ ; L , α ( L )). ◮ Proposition For any formation ( K , φ ; L 1 , L 2 ) there exists an automorphism α : ( K , φ ) → ( K , φ ) such that α ( L 1 ) = L 2 . ◮ Proof The inclusions ( L i , 0) → ( K , φ ) ( i = 1 , 2) extend to isomorphisms f i : H ǫ ( L i ) ∼ = ( K , φ ). Since dim R ( L 1 ) = dim R ( H ) / 2 = dim R ( L 2 ) there exists an isomorphism g : L 1 ∼ = L 2 . The composite automorphism f − 1 f 2 h 1 � H ǫ ( L 1 ) � H ǫ ( L 2 ) � ( K , φ ) α : ( K , φ ) ∼ ∼ ∼ = = = is such that α ( L 1 ) = L 2 , where ∼ � g � 0 = � H ǫ ( L 2 ) . h = : H ǫ ( L 1 ) ( g ∗ ) − 1 0

16 The ( − ) n -symmetric formation of a (2 n + 1) -dimensional manifold ◮ Proposition Let N 2 n +1 be a closed (2 n + 1)-dimensional manifold. ◆ ✶ ◆ ✷ ▼ A separating hypersurface M 2 n ⊂ N = N 1 ∪ M N 2 determines a ( − ) n -symmetric formation ( K , φ ; L 1 , L 2 )=( H n ( M ) , φ M ; im( H n ( N 1 ) → H n ( M )) , im( H n ( N 2 ) → H n ( M ))) If H r ( M ) → H r ( N 1 ) ⊕ H r ( N 2 ) is onto for r = n + 1 and one-one for r = n − 1 then L 1 ∩ L 2 = H n ( N ) = H n +1 ( N ) , H / ( L 1 + L 2 ) = H n +1 ( N ) = H n ( N ) . ◮ The stable isomorphism class of the formation is a homotopy invariant of N . If N = ∂ P for some P 2 n +2 the class includes ∂ ( H n +1 ( P ) , φ P ). ✶

17 The triple signature ◮ Definition (Wall 1969) The triple signature of lagrangians L 1 , L 2 , L 3 in a nonsingular symplectic form ( K , φ ) is σ ( L 1 , L 2 , L 3 ) = σ ( L 123 , λ 123 ) ∈ Z with ( L 123 , λ 123 ) the symmetric form defined by � K ) , L 123 = ker( L 1 ⊕ L 2 ⊕ L 3  0 λ 12 λ 13  � K ∗ , λ 123 = λ ∗  : K 123 = λ 21 0 λ 23  λ 31 λ 32 0 φ � K � K ∗ � L ∗ λ ij = λ ∗ ji : L j . i ◮ Motivation A stable isomorphism of formations [ f ] : ( K , φ ; L 1 , L 2 ) ⊕ ( K , φ ; L 2 , L 3 ) ⊕ ( K , φ ; L 3 , L 1 ) → ∂ ( L 123 , λ 123 )