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.
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 into account or not (decorations
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.
The surgery exact sequence is defined as where: the entries
One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological.
, which is an infinite loop space and hence maps into it define a generalized cohomology theory.
is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property (when
is normally cobordant to a homotopy equivalence if and only if the image
This arrow describes in fact an action of the group
Then there exists a degree one normal map of manifolds with boundary with the following properties: 1.
Hence it is necessary to explain what is meant when talking about the "exact sequence".
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.
Note, however, that there is to this date no satisfactory geometric description of this abelian group structure.
In both cases the answer is given in the form of a two-stage obstruction theory.
must have a vector bundle reduction of its Spivak normal fibration.
This condition can be also formulated as saying that the set of normal invariants
represent two elements in the surgery structure set
The question whether they represent the same element can be answered in two stages as follows.
to make this normal cobordism to an h-cobordism (or s-cobordism) relative to the boundary vanishes then
in fact represent the same element in the surgery structure set.
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.
Because the odd-dimensional L-groups are trivial one obtains these exact sequences: The results of Kervaire and Milnor are obtained by studying the middle map in the first two sequences and by relating the groups
in the topological category the surgery obstruction map
is trivial and that This conjecture was proven in many special cases - for example when