ALGEBRAIC TRANSVERSALITY ANDREW RANICKI (Edinburgh) - PDF document

ALGEBRAIC TRANSVERSALITY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ � aa r • Ho w do es one build ( n + 1)-dimensional manifolds from n -dimensional manifolds? 1 Fib re bundles over S . { Op en b o oks. { • Ho w do es one �nd n -dimensional submani- folds inside ( n + 1)-dimensional manifolds? Geometric transversalit y . { • What is algeb raic transversalit y? 1

Time scale • 2- and 3-dimensional manifolds : 1900 { • Algeb raic va rieties : 1920 { • Algeb raic K - and L -theo ry : 1940 { • High-dimensional manifolds : 1960 { • 3- and 4-dimensional manifolds, TQFT : 1980 { 2

Aims • 1 : Give homological criterion on a high- dimensional manifold M which is neces- sa ry and su�cient to decomp ose M as a 1 �b re bundle over S . Algeb raic K -theo ry of chain complexes. { • 2 : Lik ewise fo r op en b o ok decomp osition. Algeb raic L -theo ry of chain complexes { with P oinca r � e dualit y . 3

1 over S Fib re bundles • The mapping to rus of a map h : F → F is the identi�cation space T ( h ) = ( F × [0 , 1]) / ∼ with ( x, 0) ∼ ( h ( x ) , 1). • If F is a closed n -dimensional manifold and is an automo rphism then T ( h ) is a closed h ( n + 1)-dimensional manifold which is a 1 �b re bundle over S , with p rojection 1 T ( h ) → [0 , 1] / (0 ∼ 1) = S ; [ x, t ] → [ t ] . 4

1 Brief histo ry of �b re bundles over S • Stallings (1961) : a su�cient group- and homotop y-theo retic criterion fo r a 1 3-dimensional manifold M to �b re over S . • Bro wder and Levine (1964) ( π ( M ) = Z ) 1 and F a rrell (1970) (any π ( M )) : necessa ry 1 and su�cient conditions fo r an n -dimensional 1 manifold M to �b re over S , fo r n ≥ 6 : a �nitely dominated in�nite cyclic cover M , { the vanishing of the Whitehead to rsion { τ = τ ( M → T ( ζ )) ∈ Wh ( π ( M )) 1 with ζ : M → M generating covering translation. • Novik ov, F a rb er and P azhitnov (1981{ ) : 1 -valued Mo rse theo ry . S 5

F redholm lo calization • A = ring. • A [ z, z − 1 ] = Laurent p olynomial extension. in A [ z, z − 1 • De�nition : A squa re matrix ω ] is F redholm if cok er ( ω ) is a f.g. p rojective A -mo dule. • 1 − z Example : ω = is F redholm. in A [ z, z − 1 • De�nition : � = F redholm matrices ]. � − 1 A [ z, z − 1 ] = the noncommutative lo cal- of A [ z, z − 1 ization ] inverting each ω ∈ �. • Example : if A = K is a �eld then � − 1 A [ z, z − 1 ] = K ( z ) is the function �eld. (� − 1 A [ z, z − ( A [ z, z − 1 1 • K ]) = K ]) ⊕ Aut ( A ) . 1 1 0 6

Recognizing �b re bundles homologically • M = compact n -dimensional manifold with in�nite cyclic cover M , such that ( M ) = π × Z , π ( M ) = π . π 1 1 , A [ z, z − 1 • A = Z [ π ] ] = Z [ π × Z ] . • Theo rem 1 M is �nitely dominated if � − 1 A [ z, z − 1 and only if H ∗ ( M ; ]) = 0. • Theo rem 2 If n ≥ 6 then M is a �b re 1 bundle over S if and only if M is �nitely � − 1 A [ z, z − 1 dominated with ]-co e�cient Reidemeister-Whitehead to rsion (= F a rrell obstruction) is 0, that is � − 1 A [ z, z − 1 τ ( M ; ]) = 0 (� − 1 A [ z, z − 1 ∈ K ]) / ( {± ( π × Z ) } ⊕ Aut ( A )) 1 0 = Wh ( π × Z ) . 7

Op en b o oks • The relative mapping to rus of an automo rphism of an n -dimensional manifold with b ounda ry h : ( F, ∂F ) → ( F, ∂F ) with h | ∂F = id . is the closed ( n + 1)-dimensional manifold 2 . t ( h ) = T ( h ) ∪ ∂F × S 1 ∂F × D • A closed ( n + 1)-dimensional manifold has an op en b o ok decomp osition if M M = t ( h ) fo r some h . 1 ⊂ S n : S n − +1 • Example : fo r a �b red knot k S n +1 = t ( h ), with h the mono dromy of the surface F n ⊂ S n ( S n − +1 1 Seifert , ∂F = k ). 1 • Note : op en b o ok = �b re bundle over S if ∂F = ∅ . 8

Brief histo ry of op en b o oks • Alexander (1923) : every 3-dimensional manifold has an op en b o ok decomp osition. • r n ≥ Wink elnk emp er (1972) : fo 7 a simply- connected n -dimensional manifold has an op en b o ok decomp osition if and only if 0 ∈ Z . signature( M ) = • Quinn (1979) : non-simply-connected obstruction theo ry in dimensions ≥ 5 to the existence and uniqueness of op en b o ok decomp ositions : asymmetric Witt obstruction in { even dimensions, no obstruction in o dd dimensions. { 9

Recognizing op en b o oks homologically • M = compact n -dimensional manifold. , A [ z, z − 1 1 • A ( M × S = Z [ π ( M )] ] = Z [ π )] . 1 1 • Theo rem 3 If n ≥ 6 then M has an op en b o ok decomp osition if and only � − 1 A [ z, z − 1 if the ]-co e�cient symmetric signature (= Quinn op en b o ok obstruction) is 0 σ ∗ � − 1 A [ z, z − 0 ∈ L n (� − 1 A [ z, z − 1 1 ( M ; ]) = ]) . • dimension ≥ A manifold of 6 has an op en b o ok decomp osition if and only if it is π - 1 1 b o rdant to a �b re bundle over S . 10

Algeb raic K -theo ry transversalit y • Co dimension 1 geometric transversalit y : every in�nite cyclic cover of a compact { manifold has a compact fundamental domain. • Co dimension 1 algeb raic transversalit y : free A [ z, z − 1 every �nite f.g. ]-mo dule { chain complex C has an algeb raic fun- damental domain, with a chain equiva- lence [ z, z − [ z, z − 1 1 ( f − zg : D ] → E ]) C fo r A -mo dule chain maps f, g : D → E . (Higman (1940), W aldhausen (1972)) ( A [ z, z − 2 � 1 • K ( A ) ⊕ K ( A ) ⊕ ]) = K Nil ( A ) . 1 1 0 0 (Bass, Heller & Sw an (1965)) 11

Algeb raic P oinca r � e complexes • A = ring with involution a → a . • An n -dimensional algeb raic P oinca r � e complex over A is an A -mo dule chain complex C with a symmetric chain equivalence φ ≃ φ ∗ : C n −∗ = Hom A ( C, A ) n −∗ → C ) ∼ y H n −∗ inducing dualit ( C = H ∗ ( C ). • Simila rly fo r pairs, cob o rdism. • L n ( A ) = cob o rdism group of n -dimensional algeb raic P oinca r � e complexes over A . • Simila r description of W all surgery obstruc- tion groups L n ( A ), using extra quadratic structure. Di�erence in 2-p rima ry to rsion only . 12

Symmetric signature • The symmetric signature of an n -dimensional P oinca r � e space M is σ ∗ ) ∈ L n ( � ( M ) = ( C M ) , φ ( Z [ π ( M )]) 1 with � the universal cover of M and M = [ M ] ∩ − (Mishchenk o (1974)). φ • Homotop y inva riant. • Bo rdism inva riant σ ∗ ) → L n ; M → σ ∗ : � n ( K ( Z [ π ( K )]) ( M ) . 1 • Example : fo r n = 4 k σ ∗ 4 k ( M ) = signature( M ) ∈ L ( Z ) = Z . 13

raic L Algeb -theo ry transversalit y • The asymmetric L -groups of a ring with involution A a re the cob o rdism groups of pairs ( C, λ ) with C an n -dimensional f.g. free A -mo dule chain complex and : C n −∗ → C λ a chain equivalence. • The asymmetric L -groups a re 0 fo r o dd n . • Theo rem 4 The symmetric L -groups L n (� − 1 A [ z, z − 1 ]) of F redholm lo calization = z − 1 (with z ) a re isomo rphic to the asymmetric L -groups of A . 14

Manifold transversalit y • T rue. • The b o rdism group � n ( X ) of maps M n manifold → X = n -dimensional is a generalized homology theo ry . • K� unneth fo rmula 1 � n ( K × S ) = � n ( K ) ⊕ � n − ( K ) 1 • Co dimension 1 transversalit y: every map : M n → K × S 1 f homotopic to one with N n − = f − 1 1 pt . ) ⊂ M n ( K × a co dimension 1 submanifold. • An in�nite cyclic cover of a compact man- ifold has a compact manifold fundamental domain. 15

P oinca r � e space transversalit y • F alse in general { same obstructions as fo r algeb raic P oinca r � e complex transversalit y . • A �nite n -dimensional P oinca r � e space P is a �nite CW complex with ) ∼ H n −∗ ( P = H ∗ ( P ) . � h • The b o rdism group ( X ) of maps P → X n is not a generalized homology theo ry . � p � h � h 1 • ( K × S ) = ( K ) ⊕ ( K ) n n n − 1 � p with ( K ) the b o rdism group of �nitely ∗ dominated P oinca r � e spaces with map to K . • An in�nite cyclic cover of a �nite P oinca r � e space has a �nitely dominated fundamental domain. 16