THE NUMBER EIGHT IN TOPOLOGY Andrew Ranicki (Edinburgh) - PowerPoint PPT Presentation

1 THE NUMBER EIGHT IN TOPOLOGY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ ∼ aar/eight.htm Drawings by Carmen Rovi ECSTATIC Imperial College, London 11th June, 2015

2 Sociology and topology ◮ It is a fact of sociology that topologists are interested in quadratic forms (Serge Lang) ◮ The 8 in the title refers to the applications in topology of the mod 8 properties of the signatures of integral symmetric matrices, such as the celebrated 8 × 8 matrix E 8 with signature( E 8 ) = 8 ∈ Z . ◮ A compact oriented 4 k -manifold with boundary has an integral symmetric matrix of intersection numbers. The signature of the manifold is defined by signature(manifold) = signature(matrix) ∈ Z . ◮ Manifolds with intersection matrix E 8 have been used to distinguish the categories of differentiable, PL and topological manifolds, and so are of particular interest to topologists!

3 Quadratic forms and manifolds ◮ The algebraic properties of quadratic forms were already studied in the 19th century: Sylvester, H.J.S. Smith, . . . ◮ Similarly, the study of the topological properties of manifolds reaches back to the 19th century: Riemann, Poincar´ e, . . . ◮ The combination of algebra and topology is very much a 20th century story. But in 1923 when Weyl first proposed the definition of the signature of a manifold, topology was so dangerous that he thought it wiser to write the paper in Spanish and publish it in Mexico. And this is his signature :

4 Symmetric matrices ◮ R = commutative ring. Main examples today: Z , R , Z 4 , Z 2 . ◮ The transpose of an m × n matrix Φ = (Φ ij ) with Φ ij ∈ R is the n × m matrix Φ T with (Φ T ) ji = Φ ij (1 � i � m , 1 � j � n ) . ◮ Let Sym n ( R ) be the set of n × n matrices Φ which are symmetric Φ T = Φ. ◮ Φ , Φ ′ ∈ Sym n ( R ) are conjugate if Φ ′ = A T Φ A for an invertible n × n matrix A ∈ GL n ( R ). ◮ Can also view Φ as a symmetric bilinear pairing on the n -dimensional f.g. free R -module R n n n Φ : R n × R n → R ; (( x 1 , . . . , x n ) , ( y 1 , . . . , y n )) �→ ∑ ∑ Φ ij x i y j . i =1 j =1 ◮ Φ ∈ Sym n ( R ) is unimodular if it is invertible, or equivalently if det(Φ) ∈ R is a unit.

5 The signature ◮ The signature of Φ ∈ Sym n ( R ) is σ (Φ) = p + − p − ∈ Z with p + the number of eigenvalues > 0 and p − the number of eigenvalues < 0. ◮ Law of Inertia (Sylvester 1853) Symmetric matrices Φ , Φ ′ ∈ Sym n ( R ) are conjugate if and only if p + = p ′ + , p − = p ′ − . ◮ The signature of Φ ∈ Sym n ( Z ) σ (Φ) = σ ( R ⊗ Z Φ) ∈ Z . is an integral conjugacy invariant. ◮ The conjugacy classification of symmetric matrices is much harder for Z than R . For example, can diagonalize over R but not over Z .

6 Type I and type II ◮ Φ ∈ Sym n ( Z ) is of type I if at least one of the diagonal entries Φ ii ∈ Z is odd. ◮ Φ is of type II if each Φ ii ∈ Z is even. ◮ Type I cannot be conjugate to type II. So unimodular type II cannot be diagonalized, i.e. not conjugate to ⊕ ± 1. n ◮ Φ is positive definite if n = p + , or equivalently if σ (Φ) = n . Choosing an orthonormal basis for R ⊗ Z ( Z n , Φ) defines an embedding as a lattice ( Z n , Φ) ⊂ ( R n , dot product). Lattices (including E 8 ) much used in coding theory. ◮ Examples (i) Φ = (1) ∈ Sym 1 ( Z ) is unimodular, positive definite, type I, signature 1. (ii) Φ = (2) ∈ Sym 1 ( Z ) is positive definite, type II, signature 1. ( 0 ) 1 (iii) Φ = ∈ Sym 2 ( Z ) is unimodular, type II, signature 0. 1 1

7 Characteristic elements and the signature mod 8 ◮ An element u ∈ R n is characteristic for Φ ∈ Sym n ( R ) if Φ( x , u ) − Φ( x , x ) ∈ 2 R ⊆ R for all x ∈ R n . ◮ Every unimodular Φ admits characteristic elements u ∈ R n which constitute a coset of 2 R n ⊆ R n . ◮ Theorem (van der Blij, 1958) The mod 8 signature of a unimodular Φ ∈ Sym n ( Z ) is such that σ (Φ) ≡ Φ( u , u ) mod 8 for any characteristic element u ∈ Z n . ◮ Corollary A unimodular Φ ∈ Sym n ( Z ) is of type II if and only if u = 0 ∈ Z n is characteristic, in which case σ (Φ) ≡ 0 mod 8 .

8 The E 8 -form I. ◮ Theorem (H.J.S. Smith 1867, Korkine and Zolotareff 1873) There exists an 8-dimensional unimodular positive definite type II symmetric matrix 2 1 0 0 0 0 0 0   1 2 1 0 0 0 0 0     0 1 2 1 0 0 0 0     0 0 1 2 1 0 0 0   ∈ Sym 8 ( Z ) . E 8 =   0 0 0 1 2 1 0 1     0 0 0 0 1 2 1 0     0 0 0 0 0 1 2 1   0 0 0 0 1 0 1 2 ◮ E 8 has signature σ ( E 8 ) = 8 ∈ Z .

9 The E 8 -form II. ◮ E 8 ∈ Sym 8 ( Z ) is determined by the Dynkin diagram of the simple Lie algebra E 8 2 2 2 2 2 2 2 2 weighted by χ ( S 2 ) = 2 at each vertex, with  1 if i th vertex is adjacent to j th vertex   Φ ij = 2 if i = j  0 otherwise .  ◮ Theorem (Mordell, 1938) Any unimodular positive definite type II symmetric matrix Φ ∈ Sym 8 ( Z ) is conjugate to E 8 .

10 The intersection matrix of a 4 k -manifold ◮ The intersection matrix of a 4 k -manifold with boundary ( M , ∂ M ) with respect to a basis ( b 1 , b 2 , . . . , b n ) for H 2 k ( M ) / torsion ∼ = Z n is the symmetric matrix Φ( M ) = ( b i ∩ b j ) 1 � i , j � n ∈ Sym n ( Z ) with b i ∩ b j ∈ Z the homological intersection number. ◮ If b i , b j are represented by disjoint closed 2 k -submanifolds N i , N j ⊂ M which intersect transversely then b i ∩ b j ∈ Z is the number of points in the actual intersection N i ∩ N j ⊂ M , counted algebraically. N i N j ◮ A different basis gives a conjugate intersection matrix.

11 (2 k − 1) -connected 4 k -manifolds ◮ For j � 0 a space M is j -connected if it is connected and H i ( M ) = 0 for i = 1 , 2 , . . . , j . ◮ An m -manifold with boundary ( M , ∂ M ) is j -connected if M is j -connected and ∂ M is connected. ◮ Proposition If ( M , ∂ M ) is a (2 k − 1)-connected 4 k -manifold with boundary then ◮ H 2 k ( M ) is f.g. free, ◮ ∂ M is (2 k − 2)-connected, ◮ there is an exact sequence 0 → H 2 k ( ∂ M ) → H 2 k ( M ) Φ( M ) �� H 2 k ( M ) ∗ → H 2 k − 1 ( ∂ M ) → 0 with H 2 k ( M ) ∗ = Hom Z ( H 2 k ( M ) , Z ).

12 Homology spheres ◮ A homology ℓ -sphere Σ is a closed ℓ -manifold such that H ∗ (Σ) = H ∗ ( S ℓ ) . ◮ An m -manifold with boundary ( M , ∂ M ) is almost closed if either M is closed, i.e. ∂ M = ∅ , or ∂ M is a homology ( m − 1)-sphere H ∗ ( ∂ M ) = H ∗ ( S m − 1 ) . ◮ Proposition The intersection matrix Φ( M ) ∈ Sym n ( Z ) of a (2 k − 1)-connected 4 k -dimensional manifold with boundary ( M , ∂ M ) with H 2 k ( M ) = Z n is unimodular if and only if ( M , ∂ M ) is almost closed.

13 The 2 k th Wu class of an almost closed ( M 4 k , ∂ M ) ◮ Proposition For an almost closed (2 k − 1)-connected 4 k -manifold with boundary ( M 4 k , ∂ M ) and intersection e dual of the 2 k th Wu matrix Φ( M ) ∈ Sym n ( Z ) the Poincar´ characteristic class of the normal bundle ν M v 2 k ( ν M ) ∈ H 2 k ( M ; Z 2 ) ∼ = H 2 k ( M ; Z 2 ) is characteristic for 1 ⊗ Φ( M ) ∈ Sym n ( Z 2 ). An element u ∈ H 2 k ( M ) is characteristic for Φ( M ) if and only if [ u ] = v 2 k ( ν M ) ∈ H 2 k ( M ) / 2 H 2 k ( M ) = H 2 k ( M ; Z 2 ) . . ◮ Φ( M ) is of type II if and only if v 2 k ( ν M ) = 0. ◮ By van der Blij’s theorem, for any lift u ∈ H 2 k ( M ) of v 2 k ( ν M ). σ ( M ) ≡ Φ( u , u ) mod 8 . ◮ If ( M 4 k , ∂ M ) is framed, i.e. ν M is trivial, then v 2 k ( ν M ) = 0 , u = 0 and σ ( M ) ≡ 0 (mod8) .

14 The Poincar´ e homology 3-sphere and E 8 ◮ Poincar´ e (1904) constructed a differentiable homology 3-sphere Σ 3 = dodecahedron / opposite faces with π 1 (Σ 3 ) = binary icosahedral group of order 120 ̸ = { 1 } . This disproved the naive Poincar´ e conjecture that every homology 3-sphere is homeomorphic to S 3 . ◮ Modern construction: Σ 3 = ∂ M is the boundary of a framed ( M 4 , ∂ M ) with intersection matrix E 8 obtained by the “geometric plumbing” of 8 copies of τ S 2 according to the E 8 graph.

15 Exotic spheres and E 8 ◮ An exotic ℓ -sphere Σ ℓ is a differentiable ℓ -manifold which is homeomorphic but not diffeomorphic to S ℓ . ◮ Milnor (1956) constructed the first exotic spheres, Σ 7 , using the Hirzebruch signature theorem (1953) to detect non-standard differentiable structure. ◮ Kervaire and Milnor (1963) classified exotic ℓ -spheres Σ ℓ for all ℓ � 7, involving the finite abelian groups Θ ℓ of differentiable structures on S ℓ . ◮ The subgroup bP 4 k ⊆ Θ 4 k − 1 consists of the exotic (4 k − 1)-spheres Σ 4 k − 1 = ∂ M which are the boundary of a framed (2 k − 1)-connected 4 k -manifold ( M 4 k , ∂ M ) obtained by geometric plumbing, with Φ( M ) = ⊕ E 8 . ◮ In particular, the Brieskorn (1965) exotic spheres arising in algebraic geometry are such boundaries, including the e homology 3-sphere Σ 3 as a special case. Poincar´

16 bP 4 k ◮ The subgroup bP 4 k ⊆ Θ 4 k − 1 of diffeomorphism classes of the bounding exotic spheres Σ 4 k − 1 = ∂ M is a finite cyclic group Z bp 4 k , with an isomorphism ∼ � Z bp 4 k ; Σ 4 k − 1 = ∂ M �→ σ ( M ) / 8 . = bP 4 k ◮ The order | bP 4 k | = bp 4 k is related to the numerators of the Bernoulli numbers. ◮ The group bP 8 = Θ 7 = Z 28 of 28 differentiable structures on S 7 is generated by Σ 7 = ∂ M with Φ( M ) = E 8 .