EXACT BRAIDS AND OCTAGONS Andrew Ranicki (Edinburgh) - PowerPoint PPT Presentation

1 EXACT BRAIDS AND OCTAGONS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar Lewisfest, Dublin, 23 July 2009

2 Organized by Eva Bayer-Fluckiger, David Lewis and Andrew Ranicki.

3 Exact braids ◮ An exact braid is a commutative diagram of 4 exact sequences W ... ... A n B n C n D n E n F n G n ... ... H n I n J n W ◮ The 4 exact sequences are ... ... A n E n I n J n G n ... ... D n A n B n F n J n ... ... D n H n I n F n C n ... ... H n E n B n C n G n

4 Brief history of exact braids ◮ Eilenberg and Steenrod (1952) Axiomatic treatment of Mayer-Vietoris exact sequences, with commutative diagrams. ◮ Kervaire-Milnor (1963), Levine (1965/1984). Application of braids to the classification of exotic spheres. ◮ Wall (1966) On the exactness of interlocking sequences . General theory: exactness of three sequences implies exactness of fourth. Applications in homology theory, simplifying the Eilenberg-Steenrod treatment of triples and the Mayer-Vietoris sequence. ◮ 1966 – . . . Many applications in the surgery theory of high-dimensional manifolds (Wall, R., Hambleton-Taylor-Williams, Harsiladze . . . ) ◮ Hardie and Kamps (1985) Homotopy theory application. ◮ Iversen (1986) Triangulated category application. ◮ 1983 – . . . Many applications in quadratic form theory of equivariant forms and Clifford algebras, via the exact octagons of Lewis et al.

5 The first exact braid ◮ In a letter from Milnor to Kervaire, 29 June, 1961: with Θ n = π n ( PL / O ) the group of n -dimensional exotic spheres, F Θ n = π n ( PL ) the group of framed n -dimensional exotic spheres, P n = L n ( Z ) = π n ( G / PL ) the simply-connected surgery obstruction group, π n = Ω fr n = π n ( G ) the stable homotopy groups of spheres = the framed cobordism group, A n = π n ( G / O ) the almost framed cobordism group, and π n ( SO ) → π n the J -homomorphism. ◮ Exact braids are sometimes called Kervaire diagrams .

� � � 6 Homotopy and homology groups ◮ The homotopy groups of a space X are the groups of homotopy classes of maps S n → X π n ( X ) = [ S n , X ] ( n � 1) . ◮ The relative homotopy groups π n ( X , Y ) of a map of spaces Y → X are the homotopy classes of commutative squares S n − 1 Y � X D n with an exact sequence · · · → π n ( Y ) → π n ( X ) → π n ( X , Y ) → π n − 1 ( Y ) → . . . . ◮ Similarly for homology H ∗ ( X ), H ∗ ( X , Y ).

� � � 7 Fibre squares ◮ A commutative square of spaces and maps X + Y � X X − is a fibre square if the natural maps of relative homotopy groups π ∗ ( X + , Y ) → π ∗ ( X , X − ) are isomorphisms, or equivalently if the natural maps π ∗ ( X − , Y ) → π ∗ ( X , X + ) are isomorphisms.

� � � � � � � � � � � � � � � � � � � � � � � � � � � 8 The exact braid of homotopy groups of a fibre square ◮ Proposition The homotopy groups of a fibre square X + Y � X X − fit into an exact braid . . . . . . π n +1 ( X , X + ) π n ( X + ) π n ( X , X − ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � π n +1 ( X ) π n ( Y ) π n ( X ) π n − 1 ( Y ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � . . . � . . . π n ( X , X + ) π n +1 ( X , X − ) π n ( X − )

9 The Mayer-Vietoris sequence of an exact braid ◮ Proposition An exact braid W ... ... A + B + A − n n n − 1 B n +1 A n − 1 A n B n ... A + ... A − B − n n n − 1 W determines an exact sequence ... ... B + B n +1 n ⊕ B − A n − 1 A n B n n ◮ Exactness proved by diagram chasing.

� � � � � � � � � � � � � � � � � � 10 The Mayer-Vietoris exact sequence of a union ◮ Let X be a topological space with a decomposition X = X + ∪ Y X − with X + , X − , Y ⊆ X closed subspaces, Y = X + ∩ X − . ◮ Proposition The excision isomorphisms H ∗ ( X + , Y ) ∼ = H ∗ ( X , X − ) , H ∗ ( X − , Y ) ∼ = H ∗ ( X , X + ) determine an exact braid of homology sequences . . . . . . H n +1 ( X , X + ) H n ( X + ) � � � � � � � � � � � � � � � � � � � H n +1 ( X ) H n ( Y ) H n ( X ) � � � � � � � � � � � � � � � � � � . . . � . . . H n +1 ( X , X − ) H n ( X − ) and hence the Mayer-Vietoris exact sequence � H n +1 ( X ) � H n ( Y ) � H n ( X + ) ⊕ H n ( X − ) � H n ( X ) � . . . . . .

11 Almost an exact braid ◮ From Eilenberg and Steenrod, Foundations of algebraic topology (1952)

12 The homology isomorphisms ◮ Proposition The top and bottom rows of an exact braid W ... ... A + B + A − n n n − 1 B n +1 A n − 1 A n B n ... A + ... A − B − n n n − 1 W are chain complexes with isomorphic homology n → A + ker( B + n → A − n − 1 ) ker( B − n − 1 ) ∼ . = im( A + n → B + im( A − n → B − n ) n ) ◮ The elements b + ∈ ker( B + n − 1 ), b − ∈ ker( B − n → A + n → A − n − 1 ) match up if and only if they have the same image in B n .

13 4-periodicity ◮ An exact braid is 4-periodic if X n = X n +4 for X ∈ { A , B , A + , B + , A − , B − } . ◮ Proposition For a 4-periodic exact braid with bottom row 0 W ... ... A + B + A − 2 n 2 n 2 n − 1 B 2 n +1 A 2 n B 2 n A 2 n − 1 ... ... A + 2 n = 0 2 n = 0 2 n − 1 = 0 A − B − W the top row is an exact sequence ... ... A + B + A − B − A − 2 n 2 n 2 n − 1 2 n − 1 2 n − 2 defining . . .

14 The exact octagon of a 4-periodic exact braid with bottom row 0 A + B + 0 0 B − A − 1 3 A − B − 1 3 B + A + 2 2

15 The coat of arms of the Isle of Man

� � � � � � � � � � � � 16 The surgery exact braid ◮ Given an m -dimensional manifold M and x : S n × D m − n ⊂ M define the m -dimensional manifold M ′ obtained from M by surgery M ′ = M 0 ∪ D n +1 × S m − n − 1 with M 0 = cl.( M \ S n × D m − n ) . ◮ The homology groups of the trace cobordism ( W ; M , M ′ ) = ( M × I ∪ D n +1 × D m − n ; M , M ′ ) fit into an exact braid x x ! H i +1 ( W , M ) H i ( M ) H i ( W , M ′ ) � � � � � � � � � � � � � � � � H i ( M 0 ) H i ( W ) � � � � � � � � � � � � � � � � H i +1 ( W , M ′ ) H i ( M ′ ) H i ( W , M ) x ′ ! x ′ with H n +1 ( W , M ) = Z , H m − n ( W , M ′ ) = Z , = 0 otherwise.

17 Algebraic L -theory via forms and automorphisms ◮ Wall (1970) defined the 4-periodic algebraic L -groups L n ( A ) = L n +4 ( A ) of a ring with involution A . Applications to surgery theory of n -dimensional manifolds with n � 5. ◮ L 2 k ( A ) is the Witt group of nonsingular ( − ) k -quadratic forms on f.g. free A -modules. ◮ L 2 k +1 ( A ) is the commutator quotient of the stable unitary group of automorphisms of the hyperbolic ( − ) k -quadratic forms on f.g. free A -modules. ◮ If X is an n -dimensional space with Poincar´ e duality and a normal vector bundle there is an obstruction in L n ( Z [ π 1 ( X )]) to X being homotopy equivalent to an n -dimensional manifold. ◮ If f : M → X is a normal homotopy equivalence of n -dimensional manifolds there is an obstruction in L n +1 ( Z [ π 1 ( X )]) to f being homotopic to a diffeomorphism.

18 Algebraic L -theory via Poincar´ e chain complexes ◮ (R., 1980) Expression of L n ( A ) as the cobordism group of n -dimensional f.g. free A -module chain complexes C : C n → C n − 1 → · · · → C 1 → C 0 with an n -dimensional quadratic Poincar´ e duality H n −∗ ( C ) ∼ = H ∗ ( C ) . e complexes C , C ′ are cobordant if there exists an ◮ Quadratic Poincar´ ( n + 1)-dimensional f.g. free A -module chain complex D with chain maps C → D , C ′ → D and an ( n + 1)-dimensional quadratic Poincar´ e-Lefschetz duality H n +1 −∗ ( D , C ) ∼ = H ∗ ( D , C ′ ) . ◮ The 4-periodicity isomorphisms are defined by double suspension L n ( A ) → L n +4 ( A ) ; C �→ S 2 C with ( S 2 C ) r = C r − 2 .

19 Induction in L -theory ◮ A morphism of rings with involution f : A → B determines an induction functor of additive categories with duality involution f ! : { f.g. free A -modules } → { f.g. free B -modules } ; M �→ B ⊗ A M ◮ (R., 1980) The relative L -group L n ( f ! ) in the exact sequence f ! � L n ( A ) � L n ( B ) � L n ( f ! ) � L n − 1 ( A ) � . . . . . . is the cobordism group of pairs ( D , C ) with C an ( n − 1)-dimensional quadratic Poincar´ e complex over A and D a null-cobordism of f ! C over B L n ( f ! ) → L n − 1 ( A ) ; ( D , C ) �→ C .