In the mathematical surgery theory, the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold. The surgery structure set is a pointed set.

it is enough to understand the other terms in the exact sequence, which are usually easier to determine. These are on one hand the normal invariants which form generalized cohomology groups, and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand, there are the L-groups which are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the surgery obstruction maps from normal invariants to the L-groups. For these maps there are certain characteristic classes formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems).

it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion).

The original 1962 work of Browder and Novikov on the existence and uniqueness of manifolds within a simply-connected homotopy type was reformulated by Sullivan in 1966 as a surgery exact sequence. In 1970 Wall developed non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group.

Definition

The surgery exact sequence is defined as

where:

are the abelian groups of normal invariants,

are the L-groups associated to the group ring,

are the surgery obstruction maps,

will be explained below.

Versions

.

The entries

Normal invariants

-dimensional oriented closed manifold from the stable tangent bundle. Two such maps are equivalent if there exists a normal bordism between them (that means a bordism of the sources covered by suitable bundle data). The equivalence classes of degree one normal maps are called normal invariants.

. However the Pontrjagin-Thom a structure of an abelian group. In fact we have a non-natural bijection

, which is an infinite loop space when working with topological manifolds.

L-groups

-groups are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.

Surgery obstruction maps

:

is normally cobordant to a homotopy equivalence.

\eta \colon \mathcal{S}(X) \to \mathcal{N} (X)

defines a degree one normal map.

\partial \colon L_{n+1} (\pi_1 (X)) \to \mathcal{S}(X)

-groups which reads as follows:

. Then there exists a degree one normal map of manifolds with boundary

with the following properties:

1. is a diffeomorphism

2. is a homotopy equivalence of closed manifolds

.

The exactness

might not be a homomorphism. Hence it is necessary to explain what is meant when talking about the "exact sequence". So the surgery exact sequence is an exact sequence in the following sense:

.

Versions revisited

In the topological category the surgery obstruction map can be made into a homomorphism. This is achieved by putting an alternative abelian group structure on the normal invariants as described here. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki the structure of an abelian group. Note, however, that there is to this date no satisfactory geometric description of this abelian group structure.

Classification of manifolds

The answer to the organizing questions of the surgery theory can be formulated in terms of the surgery exact sequence. In both cases the answer is given in the form of a two-stage obstruction theory.

must have a vector bundle.

represent two elements in the surgery structure set to make this normal cobordism to an h-cobordism (or s-cobordism in fact represent the same element in the surgery structure set.

Quinn's surgery fibration

In his thesis written under the guidance of Browder, Frank Quinn introduced a fiber sequence so that the surgery long exact sequence is the induced sequence on homotopy groups.

Examples

1. Homotopy sphereHomotopy spheres

.

The idea of the surgery exact sequence is implicitly present already in the original article of Kervaire and Milnor on the groups of homotopy spheres. In the present terminology we have

-periodicity of the L-groups)

The surgery exact sequence in this case is an exact sequence of abelian groups. In addition to the above identifications we have

Because the odd-dimensional L-groups are trivial one obtains these exact sequences:

to stable homotopy theory.

2. Topological spheres

The generalized Poincaré conjecture in the topological category we see that

by some ad-hoc methods.)

3. Complex projective spaceprojective spaces in the topological category

is always surjective. Hence we have

From the work of Sullivan one can calculate

4. Aspherical spaceAspherical manifolds in the topological category

. Hence the only non-trivial homotopy group

One way to state the Borel conjecture we have that the Whitehead group is trivial and that

, when it is the fundamental group of a negatively curved manifold or when it is a word-hyperbolic group or a CAT(0)-group.

The statement is equivalent to showing that the surgery obstruction map to the right of the surgery structure set is injective and the surgery obstruction map to the left of the surgery structure set is surjective. Most of the proofs of the above-mentioned results are done by studying these maps or by studying the assembly maps with which they can be identified. See more details in Borel conjecture, Farrell-Jones Conjecture.

References