In mathematics, the **surgery structure set**
S
(
X
)
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.

Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences
f
i
:
M
i
→
X
from closed manifolds
M
i
of dimension
n
to
X
(
i
=
0
,
1
) equivalent if there exists a cobordism
(
W
;
M
0
,
M
1
)
together with a map
(
F
;
f
0
,
f
1
)
:
(
W
;
M
0
,
M
1
)
→
(
X
×
[
0
,
1
]
;
X
×
{
0
}
,
X
×
{
1
}
)
such that
F
,
f
0
and
f
1
are homotopy equivalences. The **structure set**
S
h
(
X
)
is the set of equivalence classes of homotopy equivalences
f
:
M
→
X
from closed manifolds of dimension n to X. This set has a preferred base point:
i
d
:
X
→
X
.

There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F,
f
0
and
f
1
to be simple homotopy equivalences then we obtain the **simple structure set**
S
s
(
X
)
.

Notice that
(
W
;
M
0
,
M
1
)
in the definition of
S
h
(
X
)
resp.
S
s
(
X
)
is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set
S
s
(
X
)
, provided that n>4: The simple structure set
S
s
(
X
)
is the set of equivalence classes of homotopy equivalences
f
:
M
→
X
from closed manifolds
M
of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences
f
i
:
M
i
→
X
(i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism)
g
:
M
0
→
M
1
such that
f
1
∘
g
is homotopic to
f
0
.

As long as we are dealing with differential manifolds, there is in general no canonical group structure on
S
s
(
X
)
. If we deal with topological manifolds, it is possible to endow
S
s
(
X
)
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
ϕ
:
M
→
X
whose equivalence class is the base point in
S
s
(
X
)
. Some care is necessary because it may be possible that a given simple homotopy equivalence
ϕ
:
M
→
X
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
S
s
(
X
)
.

The basic tool to compute the simple structure set is the surgery exact sequence.

**Topological Spheres:** The generalized Poincaré conjecture in the topological category says that
S
s
(
S
n
)
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
S
s
(
S
n
)
=
θ
n
=
π
n
(
P
L
/
O
)
for n > 4 (smooth category).