Surgery exact sequence

In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension formula_1. The surgery structure set formula_2 of a compact formula_3-dimensional manifold formula_4 is a pointed set which classifies formula_3-dimensional manifolds within the homotopy type of formula_4.
The basic idea is that in order to calculate formula_2 it is enough to understand the other terms in the 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).
In practice one has to proceed case by case, for each manifold formula_8 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 formula_9 or formula_10).
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:
the entries formula_12 and formula_13 are the abelian groups of normal invariants,
the entries formula_14 and formula_15 are the L-groups associated to the group ring formula_16,
the maps formula_17 and formula_18 are the surgery obstruction maps,
the arrows formula_19 and formula_20 will be explained below.
Versions.
There are various versions of the surgery exact sequence. One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological. Another possibility is to work with the decorations formula_9 or formula_10.
The entries.
Normal invariants.
A degree one normal map formula_23 consists of the following data: an formula_3-dimensional oriented closed manifold formula_25, a map formula_26 which is of degree one (that means formula_27), and a bundle map formula_28 from the stable tangent bundle of formula_25 to some bundle formula_30 over formula_4. 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.
When defined like this the normal invariants formula_13 are just a pointed set, with the base point given by formula_33. However the Pontrjagin-Thom construction gives formula_13 a structure of an abelian group. In fact we have a non-natural bijection
where formula_36 denotes the homotopy fiber of the map formula_37, which is an infinite loop space and hence maps into it define a generalized cohomology theory. There are corresponding identifications of the normal invariants with formula_38 when working with PL-manifolds and with formula_39 when working with topological manifolds.
L-groups.
The formula_40-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.
The map formula_18 is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property (when formula_42:
A degree one normal map formula_23 is normally cobordant to a homotopy equivalence if and only if the image formula_44 in formula_45.
The normal invariants arrow formula_20.
Any homotopy equivalence formula_47 defines a degree one normal map.
The surgery obstruction arrow formula_19.
This arrow describes in fact an action of the group formula_49 on the set formula_50 rather than just a map. The definition is based on the realization theorem for the elements of the formula_40-groups which reads as follows:
Let formula_25 be an formula_3-dimensional manifold with formula_54 and let formula_55. Then there exists a degree one normal map of manifolds with boundary
with the following properties:
1. formula_57
2. formula_58 is a diffeomorphism
3. formula_59 is a homotopy equivalence of closed manifolds
Let formula_47 represent an element in formula_2 and let formula_55. Then formula_63 is defined as formula_64.
The exactness.
Recall that the surgery structure set is only a pointed set and that the surgery obstruction map formula_65 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:
For a normal invariant formula_66 we have formula_67 if and only if formula_68. For two manifold structures formula_69 we have formula_70 if and only if there exists formula_71 such that formula_72. For an element formula_71 we have formula_74 if and only if formula_75.
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 which is an exact sequence of abelian groups by definition. This gives the structure set formula_2 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.
The existence question. Let formula_4 be a finite Poincaré complex. It is homotopy equivalent to a manifold if and only if the following two conditions are satisfied. Firstly, formula_4 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 formula_13 is non-empty. Secondly, there must be a normal invariant formula_80 such that formula_81. Equivalently, the surgery obstruction map formula_82 hits formula_83.
The uniqueness question. Let formula_47 and formula_85 represent two elements in the surgery structure set formula_2. The question whether they represent the same element can be answered in two stages as follows. First there must be a normal cobordism between the degree one normal maps induced by formula_87 and formula_88, this means formula_89 in formula_13. Denote the normal cobordism formula_91. If the surgery obstruction formula_92 in formula_93 to make this normal cobordism to an h-cobordism (or s-cobordism) relative to the boundary vanishes then formula_87 and formula_88 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 spheres.
This is an example in the smooth category, formula_42.
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
formula_98 the cobordism group of almost framed formula_3 manifolds, formula_100
formula_101 where formula_102 mod formula_103 (recall the formula_103-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
formula_105
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 formula_109 to stable homotopy theory.
2. Topological spheres.
The generalized Poincaré conjecture in dimension formula_3 can be phrased as saying that formula_111. It has been proved for any formula_3 by the work of Smale, Freedman and Perelman. From the surgery exact sequence for formula_113 for formula_42 in the topological category we see that
is an isomorphism. (In fact this can be extended to formula_116 by some ad-hoc methods.)
3. Complex projective spaces in the topological category.
The complex projective space formula_117 is a formula_118-dimensional topological manifold with formula_119. In addition it is known that in the case formula_120 in the topological category the surgery obstruction map formula_65 is always surjective. Hence we have
From the work of Sullivan one can calculate
4. Aspherical manifolds in the topological category.
An aspherical formula_3-dimensional manifold formula_4 is an formula_3-manifold such that formula_128 for formula_129. Hence the only non-trivial homotopy group is formula_130
One way to state the Borel conjecture is to say that for such formula_4 we have that the Whitehead group formula_132 is trivial and that
This conjecture was proven in many special cases - for example when formula_130 is formula_135, 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.