COHN LOCALIZATION, GENERALIZED FREE PRODUCTS AND BOUNDARY LINKS - PDF document

COHN LOCALIZATION, GENERALIZED FREE PRODUCTS AND BOUNDARY LINKS ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ � aar • Given a ring A and a set Σ of elements, ma- trices, morphisms, . . . , it is possible to con- struct a new ring Σ − 1 A , the Cohn localization of A inverting all the elements in Σ. In gen- eral, A and Σ − 1 A are noncommutative. • The Cohn localization of triangular matrix rings gives a new construction of gener- alized free products G (= amalgamated free product G 1 ∗ H G 2 and HNN extension G ∗ H { z } ) and a new way of relating mod- ules, chain complexes and quadratic forms over Z [ G ] to the components. For the ap- plication to µ -component boundary links G = F µ = { z 1 , z 2 , . . . , z µ } . 1

Ore localization • The Ore localization of a ring A Σ − 1 A = ( A × Σ) / ∼ is defined for a multiplicatively closed sub- set Σ ⊂ A of elements s ∈ A satisfying: • Ore condition for all a ∈ A , s ∈ Σ there exists b ∈ A , t ∈ Σ such that at = sb ∈ A (e.g. central, as = sa for all a ∈ A , s ∈ Σ) • The Ore localization is the ring of fractions Σ − 1 A = ( A × Σ) / ∼ with a s ∈ Σ − 1 A the equivalence class ( a, s ) ∼ ( b, t ) iff atu = bsu ∈ A for some u ∈ Σ • Σ − 1 A is a flat A -module, with H ∗ (Σ − 1 C ) = Σ − 1 H ∗ ( C ) for any A -module chain complex C . 2

Cohn localization • A = ring, Σ = a set of morphisms s : P → Q of f.g. projective A -modules. • A ring morphism A → B is Σ-inverting if each 1 ⊗ s : B ⊗ A P → B ⊗ A Q ( s ∈ Σ) is a B -module isomorphism. • The Cohn localization Σ − 1 A is a ring with a Σ-inverting morphism A → Σ − 1 A such that any Σ-inverting morphism A → B has a unique factorization A → Σ − 1 A → B . • Σ − 1 A exists, but could be 0. Σ − 1 A need not be a flat A -module, H ∗ (Σ − 1 C ) � = Σ − 1 H ∗ ( C ). • An element fs − 1 g ∈ Σ − 1 A is an equiva- lence class of generalized fractions, triples ( s : P → Q, f : P → A, g : A → Q ) with s ∈ Σ (Malcolmson). 3

The lifting problem for chain complexes free Σ − 1 A -module chain • A lift of a f.g. complex C is a f.g. projective A -module chain complex B with Σ − 1 B ≃ C . • Every n -dimensional f.g. free Σ − 1 A -module chain complex C can be lifted if n � 2, or if Σ − 1 A is an Ore localization. • For n � 3 there are lifting obstructions in Tor A i (Σ − 1 A, Σ − 1 A ) for i � 1. • Chain complex lifting = algebraic analogue of transversality. e-print AT.0304362 4

Stable flatness • Definition A localization Σ − 1 A of a ring A inverting a set Σ of morphisms of f.g. projective A -modules is stably flat if i (Σ − 1 A, Σ − 1 A ) = 0 ( i � 1) . Tor A • For stably flat Σ − 1 A have stable exactness: H ∗ (Σ − 1 C ) = Σ − 1 H ∗ ( D ) lim − → D with C → D such that Σ − 1 C ≃ Σ − 1 D . • (Neeman, R. and Schofield) Examples of Σ − 1 A which are not stably flat, and Σ − 1 A -module chain complexes which cannot be lifted. Math. Proc. Camb. Phil. Soc. 2004, e-print RA.0205034 5

Theorem of Neeman + R. If A → Σ − 1 A is injective and stably flat then : • have ’fibration sequence of exact categories’ T ( A, Σ) → P ( A ) → P (Σ − 1 A ) with P ( A ) the category of f.g. projec- tive A -modules and T ( A, Σ) the category of h.d. 1 Σ-torsion A -modules, and free Σ − 1 A -module chain • every finite f.g. complex can be lifted • there are long exact sequences · · · → K n ( A ) → K n (Σ − 1 A ) → K n − 1 ( T ( A, Σ)) → K n − 1 ( A ) → . . . · · · → L n ( A ) → L n (Σ − 1 A ) → L n ( T ( A, Σ)) → L n − 1 ( A ) → . . . e-print RA.0109118 6

Group rings and Cohn localization • Given a group G consider (commutative or Ore) localization of the integral group ring Z [ G ], e.g. Q [ G ] = ( Z − { 0 } ) − 1 Z [ G ]. Local- ization is a ”better” ring than Z [ G ], e.g. Q [ G ] is semisimple for finite G . • The ‘augmentation localization’ Π − 1 Z [ F µ ] inverts the set Π of square matrices in Z [ F µ ] which become invertible over Z . • If G is a generalized free product the matrix ring M k ( Z [ G ]) for some k � 1 is a Cohn lo- calization Π − 1 A of a k × k triangular matrix ring A . The localization map A → Π − 1 A is an ‘assembly’ map. In the ‘injective case’ it is possible to describe the homological algebra of Z [ G ]-modules and the algebraic K - and L -theory of Z [ G ] in terms of A and Π. In particular, this is the case for G = F µ with k = µ + 1. 7

Triangular matrix rings Given rings A 1 , A 2 and an ( A 2 , A 1 )-bimodule B define the triangular matrix ring � � A 1 0 A = B A 2 � � � � A 1 0 with P 1 = , P 2 = f.g. projective B A 2 A -modules such that A = P 1 ⊕ P 2 . Proposition (i) The category of A -modules is equivalent to the category of triples = ( M 1 , M 2 , µ : B ⊗ A 1 M 1 → M 2 ) M with M 1 an A 1 -module, M 2 an A 2 -module and µ an A 2 -module morphism. (ii) If A → C is a ring morphism such that there is a C -module isomorphism C ⊗ A P 1 ∼ = C ⊗ A P 2 then C = M 2 ( D ) with D = End C ( C ⊗ A P 1 ), { A -modules } → { C -modules } ≈ { D -modules } ; M �→ ( D D ) ⊗ A M = coker( D ⊗ A 2 B ⊗ A 1 M 1 → D ⊗ A 1 M 1 ⊕ D ⊗ A 2 M 2 ) 8

Generalized free products • Theorem (Schofield, R.) Given group mor- phisms H → G 1 , H → G 2 define   Z [ H ] 0 0   A = Z [ G 1 ] Z [ G 1 ] 0   Z [ G 2 ] 0 Z [ G 2 ] and let Π = { P 2 ⊂ P 1 , P 3 ⊂ P 1 } with P i the i th column of A . Then Π − 1 A = M 3 ( Z [ G 1 ∗ H G 2 ]) . Stably flat for injective H → G 1 , H → G 2 . • Similarly for HNN extensions. • See survey article Noncommutative localization in topology, e-print AT.0303046, for the connection with the Bass-Serre theory of groups acting on trees, and the algebraic K - and L -theory splitting theorems of Wald- hausen and Cappell. 9

The codimension 2 placement problem • For a connected space X with universal cover � X and a Z [ π 1 ( X )]-module A H ∗ ( X ; A ) := H ∗ ( A ⊗ Z [ π 1 ( X )] C ( � X )) • Let X = M \ N be the complement of a codimension 2 embedding N n ⊂ M n +2 . By Alexander duality H ∗ ( X ) = H n +2 −∗ ( M, N ) ( ∗ � = 0 , n + 2) depends only on the homotopy class of the inclusion N ⊂ M . However, H ∗ ( � X ) depends on the knotting of N ⊂ M . • The applications of Cohn localization to boundary links ( M, N ) = ( S n +2 , � µ S n ) are a joint project with Des Sheiham. 10

Boundary links • An ( n + 2)-dimensional µ -component boundary link is a locally flat embedding � µ S n ⊂ S n +2 with a µ -component Seifert surface µ � ( M n +1 , ∂M ) = ( M i , S n ) ⊂ S n +2 i =1 The Z -homology equivalence to the trivial link complement � � � f : X = S n +2 \ ( S n +1 ∨ S 1 S n ) → Y = µ µ µ − 1 induces a surjection π 1 ( X ) → π 1 ( Y ) = F µ . • Can construct a Seifert surface M by tak- ing f to be transverse at ∗ 1 ∪ · · · ∪ ∗ µ ⊂ Y and setting M i = f − 1 ( ∗ i ). 11

The augmentation localization • The augmentation Z [ F µ ] → Z factors through the Cohn localization Σ − 1 Z [ F µ ] inverting the set Σ of square matrices in Z [ F µ ] which augment to invertible matrices in Z . Stably flat (Farber and Vogel, 1992) • A finite f.g. free Z [ F µ ]-module chain com- plex C is such that H ∗ (Σ − 1 Z [ F µ ] ⊗ Z [ F µ ] C ) = 0 if and only if H ∗ ( Z ⊗ Z [ F µ ] C ) = 0. • The localization map Z [ F µ ] → Σ − 1 Z [ F µ ] detects knotting of a boundary link � µ S n ⊂ S n +2 , in the sense that X ) , H ∗ ( X ; Σ − 1 Z [ F µ ]) = 0 H ∗ ( X ; Z [ F µ ]) = H ∗ ( � for ∗ � = 0 , 1 , n +1, with X the boundary link complement and � X the cover of X induced from the universal cover � Y of Y . 12

� � � Γ -groups • Theorem (Cappell-Shaneson, 1980) For n � 4 the concordance group C n ( F µ ) of µ -component ( n +2)-dimensional boundary links (with F µ -structure) is the relative Γ- group � Z [ F µ ] Z [ F µ ] � C n ( F µ ) = Γ n +3 Φ Z [ F µ ] � Z in the exact sequence · · · → L n +3 ( Z [ F µ ]) → Γ n +3 ( Z [ F µ ] → Z ) → Γ n +3 (Φ) → L n +2 ( Z [ F µ ]) → . . . . • In particular, C 2 q ( F µ ) = 0 for q � 2. 13

Seifert, Blanchfield, computation • (Levine for µ = 1 1969, Ko, Mio, 1987) The expression of C 2 q − 1 ( F µ ) in terms of Seifert matrices. • (Kearton for µ = 1 1973, Duval, 1986) The expression of C 2 q − 1 ( F µ ) for q � 2 in terms of Blanchfield forms. • (Levine for µ = 1 1970, Sheiham, 2002) The computation of C 2 q − 1 ( F µ ) (infinitely generated) for q � 2, using Seifert forms. 14

The L -theory localization sequence • Theorem (R., 2003) The Cappell-Shaneson exact sequence is the noncommutative L - theory localization exact sequence · · · → L n +3 ( Z [ F µ ]) → L n +3 (Σ − 1 Z [ F µ ]) → L n +3 ( T ( Z [ F µ ] , Σ)) → L n +2 ( Z [ F µ ]) → . . . with Γ n +3 (Φ) = L n +3 ( T ( Z [ F µ ] , Σ)) the cobor- dism group of ( n +2)-dimensional Z -contractible quadratic Poincar´ e complexes over Z [ F µ ]. The F µ -link concordance class of a bound- ary link � µ S n ⊂ S n +2 is the cobordism class of the complex ( C ( � f ) ∗ +1 , ψ ) with � f : C ( � X ) → C ( � Y ) the canonical Z -coefficient chain equiv- alence. • Can recover the middle dimensional Blanchfield- Duval form for n = 2 q − 1. 15