Surgery structure set explained
In mathematics, the surgery 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.
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
cobordism
together with a map
(F;f0,f1):(W;M0,M1)\to(X x [0,1];X x \{0\},X x \{1\})
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-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 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 conjecture 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
l{S}s(Sn)=\thetan=\pin(PL/O)
for n > 4 (smooth category).
External links
