In mathematics, specifically in surgery theory, the surgery obstructions define a map
θ
:
N
(
X
)
→
L
n
(
π
1
(
X
)
)
from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when
n
≥
5
:
A degree-one normal map
(
f
,
b
)
:
M
→
X
is normally cobordant to a homotopy equivalence if and only if the image
θ
(
f
,
b
)
=
0
in
L
n
(
Z
[
π
1
(
X
)
]
)
.
The surgery obstruction of a degree-one normal map has a relatively complicated definition.
Consider a degree-one normal map
(
f
,
b
)
:
M
→
X
. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve
(
f
,
b
)
so that the map
f
becomes
m
-connected (that means the homotopy groups
π
∗
(
f
)
=
0
for
∗
≤
m
) for high
m
. It is a consequence of Poincaré duality that if we can achieve this for
m
>
⌊
n
/
2
⌋
then the map
f
already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on
M
to kill elements of
π
i
(
f
)
. In fact it is more convenient to use homology of the universal covers to observe how connected the map
f
is. More precisely, one works with the surgery kernels
K
i
(
M
~
)
:=
k
e
r
{
f
∗
:
H
i
(
M
~
)
→
H
i
(
X
~
)
}
which one views as
Z
[
π
1
(
X
)
]
-modules. If all these vanish, then the map
f
is a homotopy equivalence. As a consequence of Poincaré duality on
M
and
X
there is a
Z
[
π
1
(
X
)
]
-modules Poincaré duality
K
n
−
i
(
M
~
)
≅
K
i
(
M
~
)
, so one only has to watch half of them, that means those for which
i
≤
⌊
n
/
2
⌋
.
Any degree-one normal map can be made
⌊
n
/
2
⌋
-connected by the process called surgery below the middle dimension. This is the process of killing elements of
K
i
(
M
~
)
for
i
<
⌊
n
/
2
⌋
described here when we have
p
+
q
=
n
such that
i
=
p
<
⌊
n
/
2
⌋
. After this is done there are two cases.
1. If
n
=
2
k
then the only nontrivial homology group is the kernel
K
k
(
M
~
)
:=
k
e
r
{
f
∗
:
H
k
(
M
~
)
→
H
k
(
X
~
)
}
. It turns out that the cup-product pairings on
M
and
X
induce a cup-product pairing on
K
k
(
M
~
)
. This defines a symmetric bilinear form in case
k
=
2
l
and a skew-symmetric bilinear form in case
k
=
2
l
+
1
. It turns out that these forms can be refined to
ε
-quadratic forms, where
ε
=
(
−
1
)
k
. These
ε
-quadratic forms define elements in the L-groups
L
n
(
π
1
(
X
)
)
.
2. If
n
=
2
k
+
1
the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group
L
n
(
π
1
(
X
)
)
.
If the element
θ
(
f
,
b
)
is zero in the L-group surgery can be done on
M
to modify
f
to a homotopy equivalence.
Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in
K
k
(
M
~
)
possibly creates an element in
K
k
−
1
(
M
~
)
when
n
=
2
k
or in
K
k
(
M
~
)
when
n
=
2
k
+
1
. So this possibly destroys what has already been achieved. However, if
θ
(
f
,
b
)
is zero, surgeries can be arranged in such a way that this does not happen.
In the simply connected case the following happens.
If
n
=
2
k
+
1
there is no obstruction.
If
n
=
4
l
then the surgery obstruction can be calculated as the difference of the signatures of M and X.
If
n
=
4
l
+
2
then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over
Z
2
.
