Algebraic and geometric surgery
Lille 23.3, 24.3, 1.4, 2.4, 1999
by Andrew Ranicki
In these lectures I shall try to outline the surgery classification
of high-dimensional differentiable and topological manifolds,
using the algebraic theory of quadratic structures on chain complexes.
The titles of the individual lectures will be
1. "The surgery exact sequence"
2. "Algebraic and geometric cobordism"
3. "Algebraic and geometric transversality"
4. "Localization in surgery theory"
The two classic references for surgery theory are the books
"Surgery on compact manifolds" by C.T.C. Wall, Academic Press (1970)
"Surgery on simply-connected manifolds" by W. Browder, Springer (1972)
The material in the lectures will be based on these, and also on my
own books
"Exact sequences in the algebraic theory of surgery", Princeton (1982)
"Algebraic L-theory and topological manifolds", Cambridge (1992)
"High dimensional knot theory", Springer (1998)
However, I shall not presume prior knowledge of surgery theory,
only standard homotopy theory and cobordism. I hope to cover
the following topics, at various levels of detail.
1. "The surgery exact sequence"
The first lecture will state the basic definitions of surgery,
and outline the classification theory of differentiable manifolds
from the surgery point of view.
There are two basic problems:
(a) when is a space with Poincare duality homotopy equivalent
to a manifold?, and
(b) when is a homotopy equivalence of manifolds homotopic to
a diffeomorphism?
In fact (b) is just the relative version of (a).
Surfaces.
h- and s-cobordism theorems.
Poincare complexes and Spivak normal fibrations, homotopy-theoretic
analogues of manifolds and vector bundles.
In dimensions >4 Browder-Novikov-Sullivan-Wall provided a 2-stage
obstruction theory for both (a) and (b), with a primary obstruction
in the topological K-theory of vector bundles and a secondary obstruction
in the algebraic L-theory of quadratic forms over the fundamental group
ring Z[\pi_1].
The surgery exact sequence for the set of manifold structures within a
homotopy type.
The L-groups of Z.
The value of the sequence will be illustrated by means of three classics
of simply-connected surgery theory
(i) The surgery classification of the Milnor exotic spheres
(ii) Browder's converse of the Hirzebruch signature theorem
(iii) Novikov's systematic construction of homotopy equivalences
of manifolds which are not homotopic to diffeomorphisms, using
products of spheres
2. "Algebraic and geometric cobordism"
Morse functions and surgery.
Cobordism and surgery, for manifolds and normal maps.
Whitney trick in dimension >4.
Effect of surgery in homotopy and homology.
Manifolds and quadratic forms.
Symmetric L-groups L^*(Z[\pi_1]), the algebraic cobordism groups
of chain complexes with symmetric Poincare duality.
Symmetric signature, L^*(Z).
Quadratic L-groups L_*(Z[\pi_1]), identified with the Witt groups of
quadratic forms and formations.
The surgery obstruction of a normal map is identified with the
algebraic cobordism class of a chain complex over Z[\pi_1] with
Poincare duality.
In high dimensions there is a one-one correspondence between geometric
surgeries on spheres in the Hurewicz dimension of the normal map and
the algebraic surgeries on the quadratic Poincare chain complex.
The surgery exact sequence of Lecture 1 is extended to the PL and topological
categories. The extensions will be illustrated by three classics of
non-simply-connected surgery theory:
(i) Novikov's theorem on the topological invariance of the rational
Pontrjagin classes
(ii) Casson and Sullivan's disproof of the manifold Hauptvermutung
(iii) Wall's surgery classification of tori
3. "Algebraic and geometric transversality"
The traditional correspondence between the cobordism groups of manifolds
and generalized homology theory.
Identification of topological normal map set with the generalized homology
theory of sheaves of quadratic Poincare complexes.
The assembly map in the topological surgery exact sequence.
Algebraic transversality for the L-groups of group ring Z[\pi] when \pi has a
geometric structure.
The Borel and Novikov conjectures on rigidity and homotopy invariance
of the higher signatures.
Surgery on submanifolds.
Codimension q splitting obstruction theory.
L-groups of Laurent polynomial extensions via algebraic Seifert surface.
Mayer-Vietoris sequences for L-groups of HNN extensions
and amalgamated free products.
The Cappell Unil obstruction for splitting a homotopy equivalence of manifolds
along a codimension 1 submanifold.
4. "Localization in surgery theory"
Linking forms on torsion homology.
The localization exact sequence for localization of rings.
Connection with Hasse-Witt invariants.
Computation of the L-groups of finite groups.
High-dimensional knot theory and localization of commutative
polynomial extensions.
Links and noncommutative polynomial extensions.
Circle-valued Morse theory and noncommutative localization.
The topological manifold structure set and its deloop as `total surgery
obstruction' groups.
The total surgery obstruction of a Poincare complex unites the two stages
of the Browder-Novikov-Sullivan-Wall surgery theory.