COBORDISM IN ALGEBRA AND TOPOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ e aar Dedicated to Robert Switzer and Desmond Sheiham GÄottingen,13th May, 2005 1 Cobordism ² There is a cobordism equivalence relation on each of the following 6 classes of mathe- matical structures, which come in 3 match- ing pairs of topological and algebraic types: { (manifolds, quadratic forms) { (knots, Seifert forms) { (boundary links, partitioned Seifert forms) ² The cobordism groups are the abelian groups of equivalence classes, with forgetful mor- phisms ftopological cobordismg ! falgebraic cobordismg ² How large are these groups? To what ex- tent are these morphisms isomorphisms? 2 Matrices and forms ² An r£r matrix A = (aij) has entries aij 2 Z with 1 6 i; j 6 r. ² The direct sum of A and an s £ s matrix B = (bk`) is the (r + s) £ (r + s) matrix à ! A 0 A © B = : 0 B ² The transpose of A is the r £ r matrix T A = (aji) : ² A quadratic form is an r£r matrix A which is symmetric and invertible AT = A; det(A) = §1 : A symplectic form is an r£r matrix A which is (¡1)-symmetric and invertible AT = ¡ A; det(A) = §1 : 3 Cobordism of quadratic forms ² Quadratic forms A; A0 are congruent if A0 = UT AU for an invertible matrix U. ² A quadratic formà A is! null-cobordant if it is 0 P congruent to with P an invertible P T Q s£s matrix, and Q a symmetric s£s matrix. à ! 0 1 ² Example H = is null-cobordant. 1 0 ² Quadratic forms A; A0 (which may be of dif- ferent sizes) are cobordant if A © B is con- gruent to A0 © B0 for null-cobordant B; B0. ² Similarly for symplectic forms. 4 Calculation of the cobordism group of quadratic forms ² The Witt group W (Z) is the abelian group of cobordism classes of quadratic forms, with addition by direct sum A © A0. ² De¯nition (Sylvester, 1852) The signature of a quadratic form A is σ(A) = r+ ¡ r¡ 2 Z with r+ the number of positive eigenvalues of A, r¡ the number of negative eigenval- ues of A. ² σ(1) = 1, σ(¡1) = ¡1, σ(H) = 0. ² Theorem Signature de¯nes isomorphism σ : W (Z) ! Z ; A 7! σ(A) : ² The Witt group of symplectic forms = 0. 5 Manifolds ² An n-manifold M is a topological space such that each x 2 M has a neighbourhood U ½ M which is homeomorphic to Euclid- ean n-space Rn. Will assume di®erentiable structure. ² The solution set M = f ¡1(0) of equation f(x) = 0 2 Rm for function f : Rm+n ! Rm is generically an n-manifold. ² The n-sphere Sn = fx 2 Rn+1 j kxk = 1g is an n-manifold. ² A surface is a 2-manifold, e.g. sphere S2, torus S1 £ S1. ² Will only consider oriented manifolds: no MÄobiusbands, Klein bottles etc. 6 Cobordism of manifolds ² An (n + 1)-manifold with boundary (W; @W ½ W ) has W n@W an (n+1)-manifold and @W an n-manifold. ² Will only consider compact oriented mani- folds with boundary (which may be empty). ² Example (Dn+1;Sn) is a compact oriented (n + 1)-manifold with boundary, where Dn+1 = fx 2 Rn+1 j kxk 6 1g. ² Two n-manifolds M0;M1 are cobordant if the disjoint union M0t¡M1 is the boundary @W of an (n + 1)-manifold W , where ¡M1 is M1 with reverse orientation. ² Every surface M is the boundary M = @W of a 3-manifold W , so any two surfaces M; M0 are cobordant. 7 The cobordism groups of manifolds ² The cobordism group ­n of cobordism classes of n-manifolds, with addition by disjoint union M t M0. L The cobordism ring ­¤ = ­n with mul- n tiplication by cartesian product M £ N. ² Theorem (Thom, 1952) Each cobordism group ­n is ¯nitely generated with 2-torsion only. The cobordism ring is M ­¤ = Z[x4; x8;::: ] © Z2 : 1 Z[x4; x8;::: ] is the polynomial algebra with one generator x4k in each dimension 4k. Note that ­n grows in size as n increases. ² Nice account of manifold cobordism in Switzer's book Algebraic Topology { Homotopy and Homology (Springer, 1975) 8 The signature of a 4k-manifold ² (Poincar¶e,1895) The intersection matrix A = (aij) of a 2q-manifold M de¯ned by intersection numbers aij = zi \ zj 2 Z for a basis z1; z2; : : : ; zr of the homology group r Hq(M) = Z © torsion, with AT = (¡1)qA; det(A) = § 1 : A is a quadratic form if q is even. A is a symplectic form if q is odd. à ! 0 1 ² If M = Sq £ Sq then A = . (¡1)q 0 ² The signature of a 4k-manifold M4k is σ(M) = σ(A) 2 Z : 4k 2k 2k ² σ(S ) = σ(S £ S ) = 0, σ(x4k) = 1. 9 The signature morphism σ : ­4k ! W (Z) ² Let M; M0 be 4k-manifolds with intersec- tion matrices A; A0. If M and M0 are cobor- dant then A and A0 are cobordant, and σ(M) = σ(A) = σ(A0) = σ(M0) 2 Z : However, a cobordism of A and A0 may not come from a cobordism of M and M0. ² Signature de¯nes surjective ring morphism σ : ­4k ! W (Z) = Z ; M 7! σ(M) with x4k 7! 1. Isomorphism for k = 1. 2 ² Example The 8-manifolds (x4) , x8 have same signature σ = 1, but are not cobor- 2 dant, (x4) ¡ x8 6= 0 2 ker(σ : ­8 ! Z). ² Can determine class of 4k-manifold M in ­4k=torsion = Z[x4; x8;::: ] from signatures σ(N) of submanifolds N4` ⊆ M (` 6 k). 10 Cobordism of knots ² A n-knot is an embedding K : Sn ½ Sn+2 : Traditional knots are 1-knots. n n+2 ² Two n-knots K0; K1 : S ½ S are cobordant if there exists an embedding J : Sn £ [0; 1] ½ Sn+2 £ [0; 1] n such that J (x; i) = Ki(x)(x 2 S ; i = 0; 1). ² The n-knot cobordism group Cn is the abelian group of cobordism classes of n-knots, with addition by connected sum. First de¯ned for n = 1 by Fox and Milnor (1966). 11 Cobordism of Seifert surfaces ² A Seifert surface for n-knot K : Sn ½ Sn+2 is a submanifold V n+1 ½ Sn+2 with bound- ary @V = K(Sn) ½ Sn+2. ² Every n-knot K has Seifert surfaces V { highly non-unique! n n+2 ² If K0; K1 : S ½ S are cobordant n- knots, then for any Seifert surfaces V0;V1 ½ Sn+2 there exists a Seifert surface cobor- dism W n+2 ½ Sn+2 £ [0; 1] such that n+2 W \ (S £ fig) = Vi (i = 0; 1). ² Theorem (Kervaire 1965) C2q = 0 (q > 1) Proof: for every K : S2q ½ S2q+2 and Seifert surface V 2q+1 ½ S2q+2 can construct null- cobordism by `killing H¤(V ) by ambient surgery'. 12 The trefoil knot, with a Seifert surface J.B. 13 Seifert forms ² A Seifert (¡1)q-form is an r £ r matrix B such that the (¡1)q-symmetric matrix A = B + (¡1)qBT is invertible. ² A (2q ¡ 1)-knot K : S2q¡1 ½ S2q+1 with a Seifert surface V 2q ½ S2q+1 determine a Seifert (¡1)q-form B. ² B is the r £ r matrix of linking numbers 0 bij = `(zi; zj) 2 Z, for any basis z1; z2; : : : ; zr 2 0 0 0 2q+1 Hq(V ), with z1; z2; : : : ; zr 2 Hq(S nV ) the images of the zi's under a map V ! S2q+1nV pushing V o® itself in S2q+1. A = B +(¡1)qBT is the intersection matrix of V . 14 Cobordism of Seifert forms ² The cobordism of Seifert (¡1)q-forms de- ¯ned as for quadratic forms, with cobordism group G(¡1)q(Z). ² Depends only on q(mod 2). ² Theorem (Levine, 1969) The morphism C2q¡1 ! G(¡1)q(Z); K 7! B (any V ) is an isomorphism for q > 2 and surjective for q = 1. Thus for q > 2 knot cobordism C2q¡1 = algebraic cobordism G(¡1)q(Z). ² For q > 2 can realize Seifert (¡1)q-form cobordisms by Seifert surface and (2q ¡ 1)- knot cobordisms! 15 The calculation of the knot cobordism group C2q¡1 ² Theorem (Levine 1969) For q > 2 M M M C2q¡1 = G(¡1)q(Z) = Z © Z2 © Z4 : 1 1 1 Countably in¯nitely generated. ² The Z's are signatures, one for each alge- braic integer s 2 C (= root of monic in- tegral polynomial) with Re(s) = 1=2 and Im(s) > 0, so that s + ¹s = 1. ² The Z2's and Z4's are Hasse-Minkowski in- variants, as in the Witt group of rational quadratic forms L L W (Q) = Z © Z2 © Z4 : 1 1 ² Corollary For q > 2 an algorithm for decid- ing if two (2q ¡ 1)-knots are cobordant. 16 The Milnor-Levine knot signatures ² For an r£r Seifert (¡1)q-form B de¯ne the complex vector space K = Cr and the linear map J = A¡1B : K ! K with A = B + (¡1)qBT . The eigenvalues of J are algebraic integers, the roots s 2 C of the characteristic monic integral polyno- mial det(sI ¡ J) of J. K and A split as M M K = Ks ;A = As s s S1 n with Ks = n=0 ker(sI ¡ J) the general- ized eigenspace. For each s with s + ¹s = 1 (Ks;As) has signature σs(B) = σ¹s(B) 2 Z. ² The morphism M M G(¡1)q(Z) ! Z ; B 7! σs(B) s s is an isomorphism modulo 4-torsion, with s running over all the algebraic integers s 2 C with Re(s) = 1=2 and Im(s) > 0.