In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds.
A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to Z satisfying the following properties:
λ(S3) = 0.
Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference
is independent of
n. Here
Σ
+
1
m
⋅
K
denotes
1
m
Dehn surgery on Σ by
K.
For any boundary link K ∪ L in Σ the following expression is zero:
The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.
If K is the trefoil then
The Casson invariant is 1 (or −1) for the Poincaré homology sphere.
The Casson invariant changes sign if the orientation of M is reversed.
The Rokhlin invariant of M is equal to the Casson invariant mod 2.
The Casson invariant is additive with respect to connected summing of homology 3-spheres.
The Casson invariant is a sort of Euler characteristic for Floer homology.
For any integer n
where
ϕ
1
(
K
)
is the coefficient of
z
2
in the Alexander-Conway polynomial
∇
K
(
z
)
, and is congruent (mod 2) to the Arf invariant of
K.
The Casson invariant is the degree 1 part of the LMO invariant.
The Casson invariant for the Seifert manifold
Σ
(
p
,
q
,
r
)
is given by the formula:
where
Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology 3-sphere M into the group SU(2). This can be made precise as follows.
The representation space of a compact oriented 3-manifold M is defined as
R
(
M
)
=
R
i
r
r
(
M
)
/
S
O
(
3
)
where
R
i
r
r
(
M
)
denotes the space of irreducible SU(2) representations of
π
1
(
M
)
. For a Heegaard splitting
Σ
=
M
1
∪
F
M
2
of
M
, the Casson invariant equals
(
−
1
)
g
2
times the algebraic intersection of
R
(
M
1
)
with
R
(
M
2
)
.
Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres. A Casson-Walker invariant is a surjective map λCW from oriented rational homology 3-spheres to Q satisfying the following properties:
1. λ(S3) = 0.
2. For every 1-component Dehn surgery presentation (K, μ) of an oriented rational homology sphere M′ in an oriented rational homology sphere M:
λ
C
W
(
M
′
)
=
λ
C
W
(
M
)
+
⟨
m
,
μ
⟩
⟨
m
,
ν
⟩
⟨
μ
,
ν
⟩
Δ
W
′
′
(
M
−
K
)
(
1
)
+
τ
W
(
m
,
μ
;
ν
)
where:
m is an oriented meridian of a knot K and μ is the characteristic curve of the surgery.
ν is a generator the kernel of the natural map H1(∂N(K), Z) → H1(M−K, Z).
⟨
⋅
,
⋅
⟩
is the intersection form on the tubular neighbourhood of the knot, N(K).
Δ is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of
H
1
(
M
−
K
)
/
Torsion
in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
τ
W
(
m
,
μ
;
ν
)
=
−
s
g
n
⟨
y
,
m
⟩
s
(
⟨
x
,
m
⟩
,
⟨
y
,
m
⟩
)
+
s
g
n
⟨
y
,
μ
⟩
s
(
⟨
x
,
μ
⟩
,
⟨
y
,
μ
⟩
)
+
(
δ
2
−
1
)
⟨
m
,
μ
⟩
12
⟨
m
,
ν
⟩
⟨
μ
,
ν
⟩
where
x,
y are generators of
H1(∂
N(
K),
Z) such that
⟨
x
,
y
⟩
=
1
,
v = δ
y for an integer δ and
s(
p,
q) is the Dedekind sum.
Note that for integer homology spheres, the Walker's normalization is twice that of Casson's:
λ
C
W
(
M
)
=
2
λ
(
M
)
.
Christine Lescop defined an extension λCWL of the Casson-Walker invariant to oriented compact 3-manifolds. It is uniquely characterized by the following properties:
If the first Betti number of M is zero,
If the first Betti number of M is one,
where Δ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.
If the first Betti number of M is two,
where γ is the oriented curve given by the intersection of two generators
S
1
,
S
2
of
H
2
(
M
;
Z
)
and
γ
′
is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by
S
1
,
S
2
.
If the first Betti number of M is three, then for a,b,c a basis for
H
1
(
M
;
Z
)
, then
If the first Betti number of M is greater than three,
λ
C
W
L
(
M
)
=
0
.
The Casson–Walker–Lescop invariant has the following properties:
If the orientation of M, then if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, otherwise it changes sign.
For connect-sums of manifolds
In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has gauge theoretic interpretation as the Euler characteristic of
A
/
G
, where
A
is the space of SU(2) connections on M and
G
is the group of gauge transformations. He led Chern–Simons invariant as a
S
1
-valued Morse function on
A
/
G
and pointed out that the SU(3) Casson invariant is important to make the invariants independent on perturbations. (Taubes (1990))
Boden and Herald (1998) defined an SU(3) Casson invariant.
