CUTTING AND PASTING MANIFOLDS FROM THE ALGEBRAIC POINT OF VIEW - PDF document

CUTTING AND PASTING MANIFOLDS FROM THE ALGEBRAIC POINT OF VIEW ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ � aa r 1

Cutting and P asting closed n -dimensional manifold M • Cut a M = M 1 ∪ M 2 with M 1 , M manifolds with b ounda ry 2 ∂M = ∂M 1 2 together M and M • P aste using an iso- 1 2 mo rphism h : ∂M 1 → ∂M 2 closed n -dimensional to obtain a new man- ifold M ′ = M 1 ∪ h M 2 • What a re the inva riants of manifolds which do not change under cutting and pasting? 2

Schneiden und Kleb en • J � anich (1968) cha racterized signature and Euler cha racteristic as cut and paste inva ri- ants. • Ka rras, Kreck, Neumann and Ossa (1973) de�ned SK -groups, universal groups of cut and paste inva riants. • Applications to index of elliptic op erato rs. • Some recent applications of cut and paste signatures, L 2 metho ds to higher -cohomology erger, . . . { Leichtnam, Lott, L� uck, W einb 3

rdism SK The b o -groups ( X closed n - • � n ) = b o rdism of maps from dimensional manifolds : M n → X f De�nition SK n ( X ( X ) / ∼ • � n ) = with M 1 ∪ g M 2 ∼ M 1 ∪ h M 2 rphisms g, h : ∂M 1 → ∂M fo r any isomo 2 4

Twisted doubles closed n -dimensional manifold M • A is a t wisted double if M = N ∪ h N r n -dimensional fo manifold with b ounda ry ( N, ∂N rphism h : ∂N → ∂N ) and automo . map f : M → X • Lemma A from a closed n -dimensional manifold M rep resents 0 in SK n ( X if f : M → X ) if and only is b o rdant to a t wisted double. • Pro of Whitehead identit y in b o rdism M 1 ∪ g M + M 2 ∪ h M = M 1 ∪ hg M ( X 3 ∈ � n ) 2 3 with M = M . 3 1 5

Main result of SK n ( X r n ≥ • The identi�cation ) fo 6 with the image of the assembly map in the asymmetric L [ π ( X of Z -theo ry )]. 1 • Geometric realization of algeb raic result: { A symmetric P oinca r � e complex is an al- geb raic t wisted double if and only if it is null-cob o rdant as an asymmetric P oinca r � e complex. • Identi�cation almost p roved in High dimensional knot theo ry (Sp ringer, 1998) 6

Symmetric L -theo ry (I.) • A = ring with involution An n -dimensional • symmetric P oinca r � e ( C, φ an n -dimensional complex ) is f.g. free A -mo dule chain complex C 0 → C n → · · · → C 1 → C : · · · → 0 with a chain equivalence : C n −∗ φ ( C, A ) ∗− n → C Hom A = that φ ≃ φ ∗ such , and higher symmetry conditions. • Cob o rdism of symmetric P oinca r � e complexes • L n ( A ) = cob o rdism group (Mishchenk o, R.) 7

Symmetric L -theo ry (I I.) Symmetric L quadratic L • -groups = W all - groups mo dulo 2-p rima ry to rsion L n [1 / 2] ∼ ( A ) ⊗ Z = L n ( A ) ⊗ Z [1 / 2] 4 ∗ • L ( Z = Z ) (signature) an n -dimensional • The symmetric signature of manifold M ) ∈ L n σ ∗ ( � ( M ( C M ) , φ [ π ( M ( Z ) = )]) 1 • Symmetric signature map on b o rdism σ ∗ ) → L ∗ ( X [ π ( X ( Z � ∗ : )]) 1 8

Asymmetric L -theo ry (I.) An n -dimensional • asymmetric P oinca r � e ( C, φ an n -dimensional complex ) is f.g. free A -mo dule chain complex C 0 → C n → · · · → C : · · · → 0 with a chain equivalence : C n −∗ φ ( C, A ) ∗− n → C Hom A = (no symmetry condition) • Cob o rdism of asymmetric P oinca r � e com- plexes • LAsy n ( A ) = cob o rdism group maps L n ( A ) → LAsy n ( A ) • F o rgetful 9

Asymmetric L -theo ry (I I.) • 2-p erio dic LAsy n ( A ) ∼ = LAsy n ( A ) +2 asymmetric L • Odd-dimensional -groups van- ish 2 ∗ LAsy ( A ) +1 = 0 asymmetric L • Even-dimensional -groups a re la rge, e.g. � � � LAsy ( Z Z ⊕ Z Z 2 ⊕ 0 ) = 4 ∞ ∞ ∞ of n -dimensional • Asymmetric signature man- ifold M ) ∈ LAsy n Asyσ ∗ ( � ( M ( C M ) , φ [ π ( M ( Z ) = )]) 1 10

Algeb raic t wisted doubles An n -dimensional • Theo rem symmetric P oinca r e � ( C, φ over A complex ) is an algeb raic t wisted symmetric P oinca r � e double if and only if 0 ∈ LAsy n ( C, φ ( A ) ) = • Pro of Chapter 30 of High dimensional knot theo ry If M = N ∪ h N • Example is a t wisted dou- then C ( M ) → C ( N, ∂N ble manifold ) deter- mines an asymmetric P oinca r � e null-cob o rdism of M of the symmetric P oinca r � e complex , so that ( L n )]) → LAsy n σ ∗ ( M [ π ( M [ π ( M ( Z ( Z ) ∈ k er )])) 1 1 11

Recognizing t wisted doubles r n ≥ an n -dimensional • Theo rem F o 6 man- ifold M is a t wisted double if and only if 0 ∈ LAsy n Asy ([ M [ π ( M ( Z ] L ) = )]) 1 • Pro of The asymmetric signature is the Quinn (1979) obstruction to the existence of op en on M b o ok structure M = T ( h : F → F ) ∪ ∂F × D 2 ( h, id . ) rel ∂ ( n − • = automo rphism of 1)- ( F, ∂F dimensional manifold with b ounda ry ). r n ≥ • F o 6 op en b o ok if and only if t wisted double. 12

Assembly symmetric L • Assembly map in -theo ry )) → L n A : H n ( X [ π ( X ; L ( Z ( Z )]) 1 = L ∗ space X with π ∗ ( L ( Z ( Z fo r any , )) ). Every n -dimensional manifold M an L • has - theo ry o rientation [ M ] L ∈ H n ( M ; L ( Z )) ) ∈ L n = σ ∗ with A ([ M ( M [ π ( M ( Z ] L ) )]) 1 • Symmetric signature facto rs through as- sembly )) A → L n σ ∗ ( X ) → H n ( X [ π ( X ; L ( Z ( Z � n : )]) 1 asymmetric L • Assembly map in -theo ry A → L n Asy : H n ( X [ π ( X ; L ( Z ( Z )) )]) 1 → LAsy n [ π ( X ( Z )]) 1 13

The identi�cation of the b o rdism SK -groups space X and n ≥ • Co rolla ry F o r any 6 the asymmetric signature de�nes an isomo r- phism ) ∼ SK n ( X = )) → LAsy n ( Asy : H n ( X [ π ( X ; L ( Z ( Z im )])) 1 • Pro of Theo rem gives that ) ∼ ) → LAsy n ( Asy σ ∗ SK n ( X ( X [ π ( X ( Z � n = im : )])) 1 with σ ∗ ( X ) → H ∗ ( X ( f : M → X ) �→ f ∗ [ M ; L ( Z � ∗ : )) ; ] of L ( Z • Computation of homotop y t yp e ) that σ ∗ (T a ylo r and Williams, 1979) sho ws is onto, so ) ⊆ LAsy n ( Asy σ ∗ ( Asy [ π ( X ( Z im ) = im )]) 1 14