MANIFOLDS AND DUALITY ANDREW RANICKI Classi�cation of manifolds • • Uniqueness Problem • Existence Problem • Quadratic algeb ra • Applications 1
Manifolds manifold M n • An n -dimensional is a top ological space which is lo cally to R n homeomo rphic . compact, o riented, connected. { • Classi�cation of manifolds up to homeomo rphism. F o r n = 1: circle { F o r n = 2: { sphere, to rus, . . . , handleb o dy . F o r n ≥ 3: in general imp ossible. { 2
The Uniqueness Problem • Is every homotop y equivalence of n -dimen- sional manifolds : M n → N n f homotopic to a homeomo rphism? F o r n = 1 , 2: Y es. { F o r n ≥ 3: in general No. { 3
The P oinca r � e conjecture 3 → S 3 • Every homotop y equivalence f : M is homotopic to a homeomo rphism. Stated in 1904 and still unsolved! { • Theo rem ( n ≥ 5: Smale, 1960, n = 4: F reedman, 1983) : M n → S n Every homotop y equivalence f is homotopic to a homeomo rphism. 4
Old solution of the Uniqueness Problem • Surgery theo ry w o rks b est fo r n ≥ 5. F rom no w on let n ≥ 5. { • Theo rem (Bro wder, Novik ov, Sullivan, W all, 1970) : M n → N n A homotop y equivalence f is homotopic to a homeomo rphism if and only if t w o obstructions vanish. • The 2 obstructions of surgery theo ry: 1. In the top ological K -theo ry of vecto r bundles over N . 2. In the algeb raic L -theo ry of quadratic fo rms over the fundamental group ring [ π ( N )]. Z 1 5
T raditional surgery theo ry • Advantage: Suitable fo r computations . { • Disadvantages: Inaccessible. { A complicated mix of top ology and al- { geb ra. P assage from a homotop y equivalence { to the obstructions is indirect . Obstructions a re not indep endent . { 6
W all's p rogramme • \The theo ry of quadratic structures on chain complexes should p rovide a simple and sat- isfacto ry algeb raic version of the whole setup." C.T.C.W all, Surgery on compact mani- { folds , 1970 • Such a theo ry is no w available. Ranicki, Algeb raic L -theo ry and top o- { logical manifolds , 1992 7
Sieb enmann's theo rem The k ernel groups of a map f : M → N a re • the relative homology groups ( f − 1 ( x ) = H r ( x ) → { x } ) ( x ∈ N ) . K r +1 • Exact sequence ( f − 1 ( x ) → H r ( x )) → H r ( { x } ) · · · → K r → K r − ( x ) → . . . . 1 ( x ) = 0 fo r a homeomo rphism f . • K ∗ • (Sieb enmann, 1972) Theo rem : M n → N n A homotop y equivalence f with ( x ) = 0 ( x ∈ N ) K ∗ is homotopic to a homeomo rphism. 8
New solution of the Uniqueness Problem • The total surgery obstruction s ( f ) of a : M n → N n homotop y equivalence f is the cob o rdism class of the sheaf of Z -mo dule chain complexes { with n -dimensional P oinca r � e dualit y { over N { with stalk homology K ∗ ( x ) ( x ∈ N ). { • Cob o rdism and P oinca r � e dualit y a re algeb raic. A homotop y equivalence f is ho- • Theo rem motopic to a homeomo rphism if and only if s ( f ) = 0 . 9
P oinca r � e dualit y • (P oinca r e, � 1895) Theo rem The homology and cohomology of a com- pact o riented n -dimensional manifold M a re isomo rphic: ) ∼ H n − r ( M = H r ( M ) ( r = 0 , 1 , 2 , . . . ) . (Bro wder, 1962) • De�nition An n -dimensional dualit y space X is a space with isomo rphisms: ) ∼ H n − r ( X = H r ( X ) ( r = 0 , 1 , 2 , . . . ) . 10
The Existence Problem • Is an n -dimensional dualit y space X homotop y equivalent to an n -dimensional manifold? F o r n = 1 , 2: Y es. { F o r n ≥ 3: in general No. { 11
Old solution of the Existence Problem • Theo rem (Bro wder, Novik ov, Sullivan, W all, 1970) An n -dimensional dualit y space X is homo- top y equivalent to an n -dimensional mani- fold if and only if 2 obstructions vanish. The 2 obstructions (as fo r Uniqueness): • 1. In the top ological K -theo ry of vecto r bundles over X . 2. In the algeb raic L -theo ry of quadratic fo rms over the fundamental group ring [ π ( X )]. Z 1 • Same (dis)advantages as fo r the old solu- tion of the Uniqueness Problem. 12
The Theo rem of Galewski and Stern • The k ernel groups K r ( x ) of an n -dimensional dualit y space X �t into the exact sequence ) → H n − r · · · → K r ( x ( { x } ) → H r ( X, X \{ x } ) → K r − ( x ) → . . . . 1 ( x ) = 0 fo r a manifold. • K ∗ • (Galewski and Stern, 1977) Theo rem A p olyhedral dualit y space X with ( x ) = 0 ( x ∈ X ) (a homology manifold) K ∗ is homotop y equivalent to a manifold. 13
New solution of the Existence Problem • The total surgery obstruction s ( X ) of and n -dimensional dualit y space X is the cob o r- dism class of the sheaf of Z -mo dule chain complexes { with ( n − 1)-dimensional P oinca r � e dual- { it y over X { with stalk homology K ∗ ( x ) ( x ∈ X ). { • Cob o rdism and P oinca r � e dualit y a re algeb raic. • A dualit y space X is homotop y Theo rem equivalent to a manifold if and only if s ( X ) = 0 . 14
Quadratic algeb ra • Chain complexes with the homological p rop- erties of manifolds and dualit y spaces. • An n -dimensional dualit y complex is a chain complex d d 2 2 → . . . → C ( d = 0) C n → C n − → C n − 1 0 with isomo rphisms ) ∼ H n − r ( C = H r ( C ) ( r = 0 , 1 , 2 , . . . ) . generalized quadratic fo rms { • cob o rdism of dualit y complexes 15
Lo cal and global dualit y complexes = connected space X • The global surgery group L n ( Z [ π ( X )]) of 1 W all is the cob o rdism group of n -dimensional dualit y complexes of Z [ π ( X )]-mo dules. 1 Generalized Witt groups. { The lo cal surgery group H n ( X ; L ( Z )) is the • cob o rdism group of n -dimensional dualit y complexes of Z -mo dule sheaves over X . Generalized homology with co e�cients { ( Z ). L ∗ 16
The surgery exact sequence • The lo cal and global surgery groups Theo rem a re related b y the exact sequence A . . . → H n ( X ; L ( Z )) ( Z [ π ( X )]) → L n 1 → S n ( X ) → H n − ( X ; L ( Z )) → . . . . 1 • The assembly map A is the passage from lo cal to global dualit y . The structure group S n ( X ) is the cob o r- • dism group of ( n − 1)-dimensional lo cal du- alit y complexes over X which a re globally underlinetrivial. 17
The total surgery obstructions Uniqueness : the total surgery obstruction • : M n → N n of a homotop y equivalence f ( f ) ∈ S n ( N ) . s +1 ( f ) is the cob o rdism class of the n - { s dimensional globally trivial lo cal dualit y complex with stalk homology the k er- nels K ∗ ( x ) ( x ∈ N ). • Existence : the total surgery obstruction of an n -dimensional dualit y space X ( X ) ∈ S n ( X ) . s ( X ) is the cob o rdism class of the ( n − { s 1)-dimensional globally trivial lo cal du- alit y complex with stalk homology the k ernels K ∗ ( x ) ( x ∈ X ). 18
T op ology and homotop y theo ry The di�erence b et w een the top ology of man- • ifolds and the homotop y theo ry of dualit y spaces = the di�erence b et w een the cob o r- dism theo ries of the lo cal and global dualit y complexes. manifolds lo cal dualit y − → A � � dualit y spaces − → global dualit y Converse of P oinca r � e dualit y : • A dualit y space with su�cient lo cal dualit y is homotop y equivalent to a manifold. 19
The Novik ov and Bo rel conjectures • The Novik ov conjecture on the homotop y inva riance of the higher signatures is algeb raic: : H ∗ ( Bπ ; L ( Z )) → L ∗ ( Z [ π ]) is ratio- { A nally injective, fo r every group π . • The Bo rel conjecture on the existence and uniqueness of aspherical manifolds is algeb raic : : H ∗ ( Bπ ; L ( Z )) → L ∗ ( Z [ π ]) is an iso- { A mo rphism if Bπ is a dualit y space. • The va rious solution metho ds can no w b e turned into algeb ra : ( C ∗ top ology , geometry , analysis -algeb ra), { index theo rems, . . . . 20
Applications algeb raic computations of the L -groups • numb er theo ry { • singula r spaces algeb raic va rieties { di�erential geometry • hyp erb olic geometry { • non-compact manifolds controlled top ology { • 3- and 4-dimensional manifolds 21
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