Rational homology 3-spheres

Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres. A Casson-Walker invariant is a surjective map λ_CW 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:

${\displaystyle \lambda _{CW}(M^{\prime })=\lambda _{CW}(M)+{\frac {\langle m,\mu \rangle }{\langle m,\nu \rangle \langle \mu ,\nu \rangle }}\Delta _{W}^{\prime \prime }(M-K)(1)+\tau _{W}(m,\mu$ ;\nu )}

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).
• ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 ${\displaystyle H_{1}(M-K)/{\text{Torsion}}}$ in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
• ${\displaystyle \tau _{W}(m,\mu$ ;\nu )=-\mathrm {sgn} \langle y,m\rangle s(\langle x,m\rangle ,\langle y,m\rangle )+\mathrm {sgn} \langle y,\mu \rangle s(\langle x,\mu \rangle ,\langle y,\mu \rangle )+{\frac {(\delta ^{2}-1)\langle m,\mu \rangle }{12\langle m,\nu \rangle \langle \mu ,\nu \rangle }}}

where x, y are generators of H₁(∂N(K), Z) such that ${\displaystyle \langle x,y\rangle =1}$, v = δy 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: ${\displaystyle \lambda _{CW}(M)=2\lambda (M)}$.

Compact oriented 3-manifolds

Christine Lescop defined an extension λ_CWL 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,

${\displaystyle \lambda _{CWL}(M)={\tfrac {1}{2}}\left\vert H_{1}(M)\right\vert \lambda _{CW}(M)}$.

• If the first Betti number of M is one,

${\displaystyle \lambda _{CWL}(M)={\frac {\Delta _{M}^{\prime \prime }(1)}{2}}-{\frac {\mathrm {torsion} (H_{1}(M,\mathbb {Z} ))}{12}}}$

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,

${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M))\right\vert \mathrm {Link} _{M}(\gamma ,\gamma ^{\prime })}$

where γ is the oriented curve given by the intersection of two generators ${\displaystyle S_{1},S_{2}}$ of ${\displaystyle H_{2}(M;\mathbb {Z} )}$ and ${\displaystyle \gamma ^{\prime }}$ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by ${\displaystyle S_{1},S_{2}}$.

• If the first Betti number of M is three, then for a,b,c a basis for ${\displaystyle H_{1}(M;\mathbb {Z} )}$, then

${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M;\mathbb {Z} ))\right\vert \left((a\cup b\cup c)([M])\right)^{2}}$.

• If the first Betti number of M is greater than three, ${\displaystyle \lambda _{CWL}(M)=0}$.

The Casson–Walker–Lescop invariant has the following properties:

• When the orientation of M changes the behavior of ${\displaystyle \lambda _{CWL}(M)}$ depends on the first Betti number ${\displaystyle b_{1}(M)=\operatorname {rank} H_{1}(M;\mathbb {Z} )}$of M: if ${\displaystyle {\overline {M}}}$ is M with the opposite orientation, then

${\displaystyle \lambda _{CWL}({\overline {M}})=(-1)^{b_{1}(M)+1}\lambda _{CWL}(M).}$

That is, if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, while if it is even it changes sign.

• For connect-sums of manifolds

${\displaystyle \lambda _{CWL}(M_{1}\#M_{2})=\left\vert H_{1}(M_{2})\right\vert \lambda _{CWL}(M_{1})+\left\vert H_{1}(M_{1})\right\vert \lambda _{CWL}(M_{2})}$

SU(N)

In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has a gauge theoretic interpretation as the Euler characteristic of ${\displaystyle {\mathcal {A}}/{\mathcal {G}}}$, where ${\displaystyle {\mathcal {A}}}$ is the space of SU(2) connections on M and ${\displaystyle {\mathcal {G}}}$ is the group of gauge transformations. He regarded the Chern–Simons invariant as a ${\displaystyle S^{1}}$-valued Morse function on ${\displaystyle {\mathcal {A}}/{\mathcal {G}}}$ and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. (Taubes (1990))

H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.