Casson invariant
A Casson invariant is a surjective map
λ from oriented integral homology 3-spheres to Z satisfying the following properties:

formula_1
is independent of "n". Here formula_2 denotes formula_3 Dehn surgery on Σ by "K". formula_4
The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant. formula_5. formula_6
where formula_7 is the coefficient of formula_8 in the Alexander–Conway polynomial formula_9, and is congruent (mod 2) to the Arf invariant of "K". formula_11
where
formula_12 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 formula_13 where formula_14 denotes the space of irreducible SU(2) representations of formula_15. For a Heegaard splitting formula_16 of formula_17, the Casson invariant equals formula_18 times the algebraic intersection of formula_19 with formula_20. 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":
formula_21 
where:


where "x", "y" are generators of "H"1(∂"N"("K"), Z) such that formula_25, "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: formula_26. 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:
formula_27.
formula_28 
where Δ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.
formula_29 
where γ is the oriented curve given by the intersection of two generators formula_30 of formula_31 and formula_32 is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by formula_33.
formula_35.
The Casson–Walker–Lescop invariant has the following properties:
formula_40
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. formula_41 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 formula_42, where formula_43 is the space of SU(2) connections on "M" and formula_44 is the group of gauge transformations. He regarded the Chern–Simons invariant as a formula_45-valued Morse function on formula_42 and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. ()
H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.
Generalizations
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.