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 dis advantages 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 ; L Z 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
→
→ ; L Z Z[ ]
... Hn X Ln π1 X
→ S → ; L Z →
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
s X 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
{ ; L Z → 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 :
{ ; L Z → 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