Drawing: Prof. Karl Heinrich Hofmann

Drawing: Prof. Karl Heinrich Hofmann 1

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)

Symmetric and symplectic forms over R 3 Let ɛ = +1 or 1. An ɛ-symmetric form (K, φ) is a finite dimensional real vector space K together with a bilinear pairing such that φ : K K R ; (x, y) φ(x, y) φ(x, y) = ɛφ(y, x) R. The pairing φ can be identified with the adjoint linear map to the dual vector space φ : K K such that φ = ɛφ. = Hom R (K, R) ; x (y φ(x, y)) 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 ɛ-symmetric form is defined for any finite-dimensional real vector space L by ( ) 0 1 H ɛ (L) = (L L, φ = ), ɛ 0 φ : L L L L R ; ((x, f ), (y, g)) g(x) + ɛf (y) with lagrangian L. The graph of a ( ɛ)-symmetric form (L, λ) is the lagrangian of H ɛ (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 ɛ-symmetric form (K, φ) extends to an isomorphism H ɛ (L) = (K, φ). Example For any nonsingular ɛ-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 φ 1 : H ɛ(k) 1 2 = (K, φ) (K, φ).

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, (K, φ) is isomorphic to (R, 1) (R, 1) = H + (R p ). p p

The classification of symplectic forms over R 7 Theorem Every symplectic form (K, φ) is isomorphic to H (R p ) r (R, 0) with 2p + 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 (Rx Ry, φ ) is isomorphic to H (R), and with dim R (K ) = dim R (K) 2. (K, φ) = H (R) (K, φ )

8 Poincaré duality H.P. Analysis Situs and its Five Supplements (1892 1904) (English translation by John Stillwell, 2009)

The ( ) n -symmetric form of a 2n-dimensional manifold Manifolds will be oriented. Homology and cohomology will be with R-coefficients. The intersection form of a 2n-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 2n (M, M) ( H n (M, M), φ M : (x, y) x y, [M] ). By Poincaré 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... H n ( M) H n (M) φ M H n (M, 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). 9

The lagrangian of a (2n + 1)-dimensional manifold with boundary Proposition If (N 2n+1, M 2n ) is a (2n + 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 10 H n (N) H n (M) H n+1 (N, M) = φ M = = H n+1 (N, M) H n (M) H n (N) with H n (N) = H n+1 (N, M), H n+1 (N, M) = H n (N) the Poincaré-Lefschetz duality isomorphisms.

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

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

Novikov additivity of the signature 13 Let M 4k be a closed 4k-dimensional manifold which is a union of 4k-dimensional manifolds with boundary M 1, M 2 M 4k = M 1 M 2 with intersection a separating hypersurface (M 1 M 2 ) 4k 1 = M 1 = M 2 M. M 1 M 2 M 1 \ M 2 Theorem (N., 1967) The union has signature σ(m) = σ(m 1 ) + σ(m 2 ) Z.

Formations Definition An ɛ-symmetric formation (K, φ; L 1, L 2 ) is a nonsingular ɛ-symmetric form (K, φ) with an ordered pair of lagrangians L 1, L 2. Example The boundary of a ( ɛ)-symmetric form (L, λ) is the ɛ-symmetric formation (L, λ) = (H ɛ (L); L, Γ (L,λ) ) with Γ (L,λ) = {(x, λ(x)) x L} the graph lagrangian of H ɛ (L). Definition (i) An isomorphism of ɛ-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 ɛ-symmetric formations [f ] : (K, φ; L 1, L 2 ) (K, φ ; L 1, L 2 ) is an isomorphism of the type f : (K, φ; L 1, L 2 ) (H ɛ (L); L, L ) (K, φ ; L 1, L 2) (H ɛ (L ); L, L ). Two formations are stably isomorphic if and only if dim R (L 1 L 2 ) = dim R (L 1 L 2). 14

Formations and automorphisms of forms Proposition Given a nonsingular ɛ-symmetric form (K, φ), a lagrangian L, and an automorphism α : (K, φ) (K, φ) there is defined an ɛ-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 ɛ (L i ) = (K, φ). Since dim R (L 1 ) = dim R (H)/2 = dim R (L 2 ) there exists an isomorphism g : L 1 = L2. The composite automorphism f1 1 α : (K, φ) h H ɛ (L 1 ) f 2 = H ɛ (L 2 ) = (K, φ) = 15 is such that α(l 1 ) = L 2, where ( ) g 0 h = 0 (g ) 1 : H ɛ (L 1 ) = H ɛ (L 2 ).

The ( ) n -symmetric formation of a (2n + 1)-dimensional manifold Proposition Let N 2n+1 be a closed (2n + 1)-dimensional manifold. 16 N 1 N 2 M A separating hypersurface M 2n 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 2n+2 the class includes (H (P), φ ).

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 L 123 = ker(l 1 L 2 L 3 K), 0 λ 12 λ 13 λ 123 = λ 123 = λ 21 0 λ 23 : K K, λ 31 λ 32 0 λ ij = λ ji : L j K φ K L 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 )

M = M1 [ M2 [ M3 Wall non-additivity for M 4k = M 1 M 2 M 3 I. Let M 4k be a closed 4k-dimensional manifold which is a triple union M 4k = M 1 M 2 M 3 of 4k-dimensional manifolds with boundary M 1, M 2, M 3 such that the double intersections M 4k 1 ij = M i M j (1 i < j 3) are codimension 1 submanifolds of M. The triple intersection M123 4k 2 = M 1 M 2 M 3 is required to be a codimension 2 submanifold of M, with M 1 = (M 2 M23 M 3 ) = M 12 M123 M 13 etc. ; 18 M12 M1 M2 M123 M13 M23 ; M3 ;

19 Wall non-additivity for M 4k = M 1 M 2 M 3 II. Theorem (W. Non-additivity of the signature, Invent. Math. 7, 269 274 (1969)) The signature of a triple union M = M 1 M 2 M 3 of 4k-dimensional manifolds with boundary is σ(m) = σ(m 1 ) + σ(m 2 ) + σ(m 3 ) + σ(l 1, L 2, L 3 ) Z with σ(l 1, L 2, L 3 ) the triple signature of the three lagrangians L i = im(h 2k (M jk, M 123 ) K) K = H 2k 1 (M 123 ) in the symplectic intersection form of M 123 (K, φ) = (H 2k 1 (M 123 ), φ M123 ).

Wall non-additivity for M 4k = M 1 M 2 M 3 III. 20 Idea of proof σ(l 1, L 2, L 3 ) = σ(n) Z is the signature of a manifold neighbourhood (N 4k, N) of M 12 M 13 M 13 M N = (M 12 M 23 M 13 ) D 1 (M 123 D 2 ). M 12 D 1 M 0 1 M 0 2 M 123 D 2 M 13 D 1 M 23 D 1 M 0 3

The space of lagrangians Λ(n) 21 Definition For n 1 let Λ(n) be the spaces of lagrangians L H (R n ). Use the complex structure on H (R n ) J : R n R n R n R n ; (x, y) ( y, x) to associate to every lagrangian L Λ(n) a canonical complement JL Λ(n) with L JL = R n R n. For every L Λ(n) there exists a unitary matrix A U(n) such that A(R n {0}) = L Λ(n). If A U(n) is another such unitary matrix then ( ) (A ) 1 B 0 A = 0 B t (B O(n)) with (b jk ) t = (b kj ) the transpose.

Maslov index : π 1 (Λ(n)) = Z Proposition (Arnold, 1967) (i) The function U(n)/O(n) Λ(n) : A A(R n {0}) is a diffeomorphism. Λ(n) is a compact manifold of dimension dim Λ(n) = dim U(n) dim O(n) = n 2 n(n 1) n(n + 1) =. 2 2 The graphs {Γ (R n,φ) φ = φ} Λ(n) define a chart at R n Λ(n). (ii) The square of the determinant function det 2 : Λ(n) S 1 ; L = A(R n {0}) det(a) 2 induces the Maslov index isomorphism det 2 = : π 1 (Λ(n)) π 1 (S 1 ) = Z. Proposition (Kashiwara and Schapira, 1992) The triple signature σ(l 1, L 2, L 3 ) Z of L 1, L 2, L 3 Λ(n) is the Maslov index of a loop S 1 Λ(n) passing through L 1, L 2, L 3. 22

23 The algebraic η-invariant Definition/Proposition (Atiyah-Patodi-Singer 1974, Cappell-Lee-Miller 1994, Bunke 1995) (i) The algebraic η-invariant of L 1, L 2 Λ(n) is η(l 1, L 2 ) = n j=1,θ j 0 (1 2θ j /π) R with θ 1, θ 2,..., θ n [0, π) such that ±e iθ 1, ±e iθ 2,..., ±e iθn are the eigenvalues of any A U(n) with A(L 1 ) = L 2. (ii) The algebraic η-invariant is a cocycle for the triple index of L 1, L 2, L 3 Λ(n) σ(l 1, L 2, L 3 ) = η(l 1, L 2 ) + η(l 2, L 3 ) + η(l 3, L 1 ) Z R.

The real signature I. Let M be a 4k-dimensional manifold with a decomposed boundary M = N 1 P N 2, where P M is a separating codimension 1 submanifold. Let (H 2k 1 (P), φ P ) be the nonsingular symplectic intersection form, and n = dim R (H 2k 1 (P))/2. Given a choice of isomorphism J : (H 2k 1 (P), φ P ) = H (R n ) (or just a complex structure on (H 2k 1 (P), φ P )) define the real signature using the lagrangians σ J (M, N 1, N 2, P) = σ(m) + η(l 1, L 2 ) R L j = ker(h 2k 1 (P) H 2k 1 (N j )) (H 2k 1 (P), φ P ). Proposition The real signature is additive σ J (M N2 M ; N 1, N 3, P) = σ J (M; N 1, N 2, P) + σ J (M ; N 2, N 3, P) R. 24

The real signature II. Proof Apply the Wall non-additivity formula to the union M M (M N2 M ) = ((M N2 M ) I ), which is an (un)twisted double with signature σ = 0. ; 25 N2 M M 0 P N1 N3 ; M [ N2 M 0 ; Note Analogue of the additivity of M L-genus = σ(m) + η( M) R in the Atiyah-Patodi-Singer signature theorem. Note In general σ J (M; N 1, N 2, P) R depends on the choice of complex structure J on (H 2k 1 (P), φ P ).

Real and complex vector bundles 26 In view of the fibration Λ(n) = U(n)/O(n) BO(n) BU(n) Λ(n) classifies real n-plane bundles β with a trivialisation δβ : C β = ɛ n of the complex n-plane bundle C β. The canonical real n-plane bundle γ over Λ(n) is E(γ) = {(L, x) L Λ(n), x L}. The complex n-plane bundle C γ E(C γ) = {(L, z) L Λ(n), z C R L} is equipped with the canonical trivialisation δγ : C γ = ɛ n defined by δγ : E(C γ) = E(ɛ n ) = Λ(n) C n ; (L, z) (L, (x, y)) if z = (x, y) = x + iy C R L = L JL = C n.

The Maslov index, whichever way you slice it! I. The lagrangians L Λ(1) are parametrized by θ R L(θ) = {(rcos θ, rsin θ) r R} R R with indeterminacy L(θ) = L(θ + π). The map det 2 is a diffeomorphism. The canonical R-bundle γ over Λ(1) : Λ(1) = U(1)/O(1) S 1 ; L(θ) e 2iθ E(γ) = {(L, x) L Λ(1), x L} is nontrivial = infinite Möbius band. The induced C-bundle over Λ(1) E(C R γ) = {(L, z) L Λ(1), z C R L} is equipped with the canonical trivialisation δγ : C R γ = ɛ defined by = δγ : E(C R γ) E(ɛ) = Λ(1) C ; (L, z) = (L(θ), (x + iy)(cos θ, sin θ)) (L(θ), (x + iy)e iθ ). 27

The Maslov index, whichever way you slice it! II. Given a bagel B = S 1 D 2 R 3 and a map λ : S 1 Λ(1) = S 1 slice B along C = {(x, y) B y λ(x)}. The slicing line (x, λ(x)) B is the fibre over x S 1 of the pullback [ 1, 1]-bundle [ 1, 1] C = D(λ γ) S 1 with boundary (where the knife goes in and out of the bagel) C = {(x, y) C y λ(x)} a double cover of S 1. There are two cases: C is a trivial [ 1, 1]-bundle over S 1 (i.e. an annulus), with C two disjoint circles, which are linked in R 3. The complement B\C has two components, with the same linking number. C is a non trivial [ 1, 1]-bundle over S 1 (i.e. a Möbius band), with C a single circle, which is self-linked in R 3. The complement B\C is connected, with the same self-linking number (= linking of C and S 1 {(0, 0)} C R 3 ). 28

The Maslov index, whichever way you slice it! III. 29 By definition, Maslov index(λ) = degree(λ) Z. degree : π 1 (S 1 ) Z is an isomorphism, so it may be assumed that λ : S 1 Λ(1) ; e 2iθ L(nθ) with Maslov index = n 0. The knife is turned through a total angle nπ as it goes round B. It may also be assumed that the bagel B is horizontal. The projection of C onto the horizontal cross-section of B consists of n = λ 1 (L(0)) points. For n > 0 this corresponds to the angles θ = jπ/n [0, π) (0 j n 1) where L(nθ) = L(0), i.e. sin nθ = 0. The two cases are distinguished by: If n = 2k then C is a union of two disjoint linked circles in R 3. Each successive pair of points in the projection contributes 1 to the linking number n/2 = k. If n = 2k + 1 then C is a single self-linked circle in R 3. Each point in the projection contributes 1 to the self-linking number n = 2k + 1. (Thanks to Laurent Bartholdi for explaining this case to me.)

Maslov index = 0, C = annulus, linking number = 0 30 λ : S 1 S 1 ; z 1.

31 Maslov index = 1, C = Möbius band, self-linking number = 1 (Clara Löh (05/2009) clara.loeh@uni-muenster.de) λ : S 1 S 1 ; z z. Thanks to Clara Löh for this picture.

32 Maslov index = 2, C = annulus, linking number = 1 λ : S 1 S 1 ; z z 2. http://www.georgehart.com/bagel/bagel.html