Surgery structure set
Encyclopedia
In mathematics, the structure set 
is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion
is taken into account or not.
from closed manifolds 
of dimension 
to 
(
) equivalent if there exists a cobordism
together with a map 
such that 
, 
and 
are homotopy equivalences.
The structure set
is the set of equivalence classes of homotopy equivalences 
from closed manifolds of dimension n to X.
This set has a preferred base point:
.
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F,
and 
to be simple homotopy equivalences then we obtain the simple structure set 
.
in the definition of 
resp. 
is an h-cobordism
resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set
, provided that n>4: The simple structure set 
is the set of equivalence classes of homotopy equivalences 
from closed manifolds 
of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences 
(i=0,1) are equivalent if there exists a
diffeomorphism (or PL-homeomorphism or homeomorphism)
such that 
is homotopic to 
.
As long as we are dealing with differential manifolds, there is in general no canonical group structure on
. If we deal with topological manifolds, it is possible to endow 
with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).
Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence
whose equivalence class is the base point in 
. Some care is necessary because it may be possible that a given simple homotopy equivalence 
is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on 
.
The basic tool to compute the simple structure set is the surgery exact sequence.
in the topological category says that
only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).
Exotic Spheres: The classification of exotic spheres
by Kervaire and Milnor gives
for n > 4 (smooth category).
is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsionWhitehead torsion
In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
is taken into account or not.
Definition
Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences from closed manifolds of dimension to () equivalent if there exists a cobordismCobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
together with a map such that , and are homotopy equivalences.The structure set
is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X.This set has a preferred base point:
.There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F,
and to be simple homotopy equivalences then we obtain the simple structure set .Remarks
Notice that in the definition of resp. is an h-cobordismH-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...
resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set
, provided that n>4: The simple structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences (i=0,1) are equivalent if there exists adiffeomorphism (or PL-homeomorphism or homeomorphism)
such that is homotopic to .As long as we are dealing with differential manifolds, there is in general no canonical group structure on
. If we deal with topological manifolds, it is possible to endow with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence
whose equivalence class is the base point in . Some care is necessary because it may be possible that a given simple homotopy equivalence is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on .The basic tool to compute the simple structure set is the surgery exact sequence.
Examples
Topological Spheres: The generalized Poincaré conjectureGeneralized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...
in the topological category says that
only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).Exotic Spheres: The classification of exotic spheres
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
by Kervaire and Milnor gives
for n > 4 (smooth category).