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Manifolds and Duality

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Manifolds and Duality MANIFOLDS AND DUALITY ANDREW RANICKI • Classi cation of manifolds • Uniqueness Problem • Existence Problem • Quadratic algebra • Applications 1 Manifolds n -dimensional manifold is a • An n M top ological space which is lo cally n R . homeomorphic to { compact, oriented, connected. • Classi cation of manifolds up to homeomorphism. { For 1: circle n = { For 2: n = , handleb o dy. torus, sphere, ... { For in general imp ossible. n ≥ 3: 2 The Uniqueness Problem every homotopy equivalence of -dimen- • Is n sional manifolds n → n f : M N homotopic to a homeomorphism? = 1 2: Yes. For { n , ≥ 3: in general No. For { n 3 The Poincar e conjecture 3 3 homotopy equivalence : → • Every f M S is homotopic to a homeomorphism. { Stated in 1904 and still unsolved! • Theorem ≥ 5: Smale, 1960, = 4: Freedman, n n 1983 n n homotopy equivalence : → Every f M S is homotopic to a homeomorphism. 4 Old solution of the Uniqueness Problem theory works b est for ≥ 5. • Surgery n ≥ 5. From now on let { n • Theorem Browder, Novikov, Sullivan, Wall, 1970 n n : → is homotopy equivalence A f M N homotopic to a homeomorphism if and only if two obstructions vanish. • The 2 obstructions of surgery theory: -theory of vector In the top ological 1. K bundles over N . -theory of quadratic In the algebraic 2. L forms over the fundamental group ring Z[ ]. π1 N 5 Traditional surgery theory • Advantage: { Suitable for computations . • Disadvantages: . { Inaccessible { A complicated mix of top ology and al- gebra. { Passage from a homotopy equivalence . to the obstructions is indirect { Obstructions are not indep endent . 6 Wall's programme • \The theory of quadratic structures on chain complexes should provide a simple and sat- isfactory algebraic version of the whole setup." { C.T.C.Wall, Surgery on compact mani- folds, 1970 • Such a theory is now available. Ranicki, Algebraic -theory and top o- { L logical manifolds, 1992 7 Sieb enmann's theorem kernel groups of a map : → are • The f M N the relative homology groups −1 = →{ } ∈ K x H f x x x N . r r +1 • Exact sequence −1 → → { } ···→Kr x Hr f x Hr x → → Kr−1 x ... • = 0 for a homeomorphism . K∗ x f • Theorem Sieb enmann, 1972 n n A homotopy equivalence → with f : M N = 0 ∈ K∗ x x N is homotopic to a homeomorphism. 8 New solution of the Uniqueness Problem total surgery obstruction of a • The s f n n y equivalence : → is the homotop f M N cob ordism class of { the sheaf of Z-mo dule chain complexes with -dimensional Poincar e duality { n over { N with stalk homology ∈ . { K∗ x x N • Cob ordism and Poincar e duality are algebraic. rem A homotopy equivalence is ho- • Theo f motopic to a homeomorphism if and only = 0. if s f 9 Poincar e duality • Theorem Poincar e, 1895 The homology and cohomology of a com- oriented -dimensional manifold are pact n M isomorphic: n−r ∼ = 0 1 2 H M = Hr M r , , ,... • De nition Browder, 1962 An duality space is a space n-dimensional X with isomorphisms: n−r ∼ = 0 1 2 H X = Hr X r , , ,... 10 The Existence Problem an -dimensional duality space • Is n X y equivalent to an -dimensional homotop n manifold? { For 1 2: Yes. n = , ≥ 3: in general No. For { n 11 Old solution of the Existence Problem • Theorem Browder, Novikov, Sullivan, Wall, 1970 -dimensional duality space is homo- An n X topy equivalent to an mani- n-dimensional fold if and only if 2 obstructions vanish. • The 2 obstructions as for Uniqueness: 1. In the top ological ry of vector K -theo over . bundles X -theory of quadratic In the algebraic 2. L forms over the fundamental group ring Z[ ]. π1 X • Same disadvantages as for the old solu- tion of the Uniqueness Problem. 12 The Theorem of Galewski and Stern kernel groups of an -dimensional • The Kr x n t into the exact sequence y space dualit X n−r → { } → \{ } ···→Kr x H x Hr X, X x → → Kr−1 x ... • = 0 for a manifold. K∗ x • Theorem Galewski and Stern, 1977 A p olyhedral duality space X with = 0 ∈ a homology manifold K∗ x x X is homotopy equivalent to a manifold. 13 New solution of the Existence Problem total surgery obstruction of and • The s X duality space is the cob or- n-dimensional X dism class of { the sheaf of Z-mo dule chain complexes { with Poincar e dual- n − 1-dimensional ity over { X with stalk homology ∈ . { K∗ x x X • Cob ordism and Poincar e duality are algebraic. rem A duality space is homotopy • Theo X to a manifold if and only if = equivalent s X 0. 14 Quadratic algebra • Chain complexes with the homological prop- erties of manifolds and duality spaces. -dimensional duality complex is a chain • An n complex d d → → 2 → → = 0 −2 0 Cn Cn−1 Cn ... C d with isomorphisms n−r ∼ = 0 1 2 H C = Hr C r , , ,... { generalized quadratic forms • cob ordism of duality complexes 15 Lo cal and global duality complexes connected space X = • The global surgery group Z[ ] of Ln π1 X Wall is the cob ordism group of n-dimensional duality complexes of Z[ ]-mo dules. π1 X { Generalized Witt groups. lo cal surgery group ; LZ is the • The Hn X rdism group of -dimensional duality cob o n Z-mo dule sheaves over . of complexes X { Generalized homology with co ecients Z. L∗ 16 The surgery exact sequence • Theorem The lo cal and global surgery groups are related by the exact sequence A → → ; LZ Z[ ] ... Hn X Ln π1 X → S → ; LZ → n X Hn−1 X ... • assembly map is the passage from The A lo cal to global duality. structure group S is the cob or- • The n X − 1-dimensional lo cal du- group of dism n which are globally y complexes over alit X underlinetrivial. 17 The total surgery obstructions • Uniqueness: the total surgery obstruction n n a homotopy equivalence : → of f M N ∈ S s f N . n+1 is the cob ordism class of the - { s f n dimensional globally trivial lo cal duality complex with stalk homology the ker- nels ∈ . K∗ x x N • Existence: the total surgery obstruction an -dimensional duality space of n X ∈ S sX n X . is the cob ordism class of the − { s X n 1-dimensional globally trivial lo cal du- ality complex with stalk homology the kernels ∈ . K∗ x x X 18 Top ology and homotopy theory • The di erence between the top ology of man- ifolds and the homotopy theory of duality spaces = the di erence between the cob or- dism theories of the lo cal and global duality complexes. manifolds duality −→ lo cal y yA duality spaces −→ global duality • Converse of Poincar e duality: A duality space with sucient lo cal duality is homotopy equivalent to a manifold. 19 The Novikov and Borel conjectures • The Novikov conjecture on the homotopy : invariance of the higher signatures is algebraic { ; LZ → Z[ ] is ratio- A : H∗ Bπ L∗ π . injective, for every group nally π • The Borel conjecture on the existence and uniqueness of aspherical manifolds is algebraic : { ; LZ → Z[ ] is an iso- A : H∗ Bπ L∗ π rphism if is a duality space. mo Bπ • The various solution metho ds can now be turned into algebra : ∗ -algebra, top ology, geometry, analysis { C theorems, . index ... 20 Applications raic computations of the -groups • algeb L { numb er theory • singular spaces { algebraic varieties • di erential geometry { hyp erb olic geometry • non-compact manifolds { controlled top ology • 3- and 4-dimensional manifolds 21.
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