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| 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-sphere]]s, introduced by [[Andrew Casson]].
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| Kevin Walker (1992) found an extension to [[rational homology 3-sphere]]s, called the '''Casson-Walker invariant''', and Christine Lescop (1995) extended the invariant to all [[closed manifold|closed]] oriented [[3-manifold]]s.
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| ==Definition==
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| A Casson invariant is a surjective map
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| λ from oriented integral homology 3-spheres to '''Z''' satisfying the following properties:
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| *λ('''S'''<sup>3</sup>) = 0.
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| *Let Σ be an integral homology 3-sphere. Then for any knot ''K'' and for any integer ''n'', the difference
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| ::<math>\lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)</math>
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| :is independent of ''n''. Here <math>\Sigma+\frac{1}{m}\cdot K</math> denotes <math>\frac{1}{m}</math> [[Dehn surgery]] on Σ by ''K''.
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| *For any boundary link ''K'' ∪ ''L'' in Σ the following expression is zero:
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| ::<math>\lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n+1}\cdot L\right) -\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n+1}\cdot L\right)-\lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n}\cdot L\right) +\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n}\cdot L\right)</math>
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| The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.
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| ==Properties==
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| *If K is the trefoil then
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| ::<math>\lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)=\pm 1</math>.
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| *The Casson invariant is 1 (or −1) for the [[Poincaré homology sphere]].
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| *The Casson invariant changes sign if the orientation of ''M'' is reversed.
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| *The [[Rokhlin invariant]] of ''M'' is equal to the Casson invariant mod 2.
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| *The Casson invariant is additive with respect to connected summing of homology 3-spheres.
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| *The Casson invariant is a sort of [[Euler characteristic]] for [[Floer homology]].
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| *For any integer ''n''
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| ::<math>\lambda \left ( M + \frac{1}{n+1}\cdot K\right ) - \lambda \left ( M + \frac{1}{n}\cdot K\right ) = \alpha (K), </math>
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| :where α(''K'') is the [[Arf invariant]] of ''K''.
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| *The Casson invariant is the degree 1 part of the [[LMO invariant]].
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| *The Casson invariant for the [[Seifert manifold]] <math>\Sigma(p,q,r)</math> is given by the formula:
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| ::<math> \lambda(\Sigma(p,q,r))=-\frac{1}{8}\left[1-\frac{1}{3pqr}\left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\right)
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| -d(p,qr)-d(q,pr)-d(r,pq)\right]</math>
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| :where
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| ::<math>d(a,b)=-\frac{1}{a}\sum_{k=1}^{a-1}\cot\left(\frac{\pi k}{a}\right)\cot\left(\frac{\pi bk}{a}\right)</math>
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| ==The Casson invariant as a count of representations==
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| 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.
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| The representation space of a [[compact space|compact]] oriented 3-manifold ''M'' is defined as <math>\mathcal{R}(M)=R^{\mathrm{irr}}(M)/SO(3)</math> where <math>R^{\mathrm{irr}}(M)</math> denotes the space of irreducible SU(2) representations of <math>\pi_1 (M)</math>. For a [[Heegaard splitting]] <math>\Sigma=M_1 \cup_F M_2</math> of <math>M</math>, the Casson invariant equals <math>\frac{(-1)^g}{2}</math> times the algebraic intersection of <math>\mathcal{R}(M_1)</math> with <math>\mathcal{R}(M_2)</math>.
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| ==Generalizations==
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| ===Rational homology 3-spheres===
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| Kevin Walker found an extension of the Casson invariant to [[rational homology 3-sphere]]s. A Casson-Walker invariant is a surjective map λ<sub>''CW''</sub> from oriented rational homology 3-spheres to '''Q''' satisfying the following properties:
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| '''1.''' λ('''S'''<sup>3</sup>) = 0.
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| '''2.''' For every 1-component [[Dehn surgery]] presentation (''K'', μ) of an oriented rational homology sphere ''M''′ in an oriented rational homology sphere ''M'':
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| :<math>\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)</math> | |
| where:
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| *''m'' is an oriented meridian of a knot ''K'' and μ is the characteristic curve of the surgery.
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| *ν is a generator the kernel of the natural map ''H''<sub>1</sub>(∂''N''(''K''), '''Z''') → ''H''<sub>1</sub>(''M''−''K'', '''Z''').
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| *<math>\langle\cdot,\cdot\rangle</math> is the intersection form on the tubular neighbourhood of the knot, ''N''(''K'').
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| *Δ is the Alexander polynomial normalized so that the action of ''t'' corresponds to an action of the generator of <math>H_1(M-K)/\text{Torsion}</math> in the infinite [[cyclic cover]] of ''M''−''K'', and is symmetric and evaluates to 1 at 1.
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| *<math>\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}</math>
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| :where ''x'', ''y'' are generators of ''H''<sub>1</sub>(∂''N''(''K''), '''Z''') such that <math>\langle x,y\rangle=1</math>, ''v'' = δ''y'' for an integer δ and ''s''(''p'', ''q'') is the [[Dedekind sum]]. | |
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| Note that for integer homology spheres, the Walker's normalization is twice that of Casson's: <math> \lambda_{CW}(M) = 2 \lambda(M) </math>.
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| ===Compact oriented 3-manifolds===
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| Christine Lescop defined an extension λ<sub>''CWL''</sub> of the Casson-Walker invariant to oriented compact [[3-manifolds]]. It is uniquely characterized by the following properties:
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| *If the first [[Betti number]] of ''M'' is zero,
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| ::<math>\lambda_{CWL}(M)=\tfrac{1}{2}\left\vert H_1(M)\right\vert\lambda_{CW}(M)</math>.
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| *If the first Betti number of ''M'' is one,
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| ::<math>\lambda_{CWL}(M)=\frac{\Delta^{\prime\prime}_M(1)}{2}-\frac{\mathrm{torsion}(H_1(M,\mathbb{Z}))}{12}</math>
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| :where Δ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.
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| *If the first Betti number of ''M'' is two,
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| ::<math>\lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M))\right\vert\mathrm{Link}_M (\gamma,\gamma^\prime)</math>
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| :where γ is the oriented curve given by the intersection of two generators <math>S_1,S_2</math> of <math>H_2(M;\mathbb{Z})</math> and <math>\gamma^\prime</math> is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by <math>S_1, S_2</math>. | |
| *If the first Betti number of ''M'' is three, then for ''a'',''b'',''c'' a basis for <math>H_1(M;\mathbb{Z})</math>, then
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| ::<math>\lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M;\mathbb{Z}))\right\vert\left((a\cup b\cup c)([M])\right)^2</math>.
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| *If the first Betti number of ''M'' is greater than three, <math>\lambda_{CWL}(M)=0</math>.
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| The Casson-Walker-Lescop invariant has the following properties:
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| *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.
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| *For connect-sums of manifolds
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| ::<math>\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)</math>
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| ===[[SU(N)]]===
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| Boden and Herald (1998) defined an [[SU(3)]] Casson invariant.
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| ==References==
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| *S. Akbulut and J. McCarthy, ''Casson's invariant for oriented homology 3-spheres— an exposition.'' Mathematical Notes, 36. Princeton University Press, Princeton, NJ, 1990. ISBN 0-691-08563-3
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| *M. Atiyah, ''New invariants of 3- and 4-dimensional manifolds.'' The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 285-299, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.
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| *H. Boden and C. Herald, ''The SU(3) Casson invariant for integral homology 3-spheres.'' J. Differential Geom. 50 (1998), 147–206.
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| *C. Lescop, ''Global Surgery Formula for the Casson-Walker Invariant.'' 1995, ISBN 0-691-02132-5
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| *N. Saveliev, ''Lectures on the topology of 3-manifolds: An introduction to the Casson Invariant.'' de Gruyter, Berlin, 1999. ISBN 3-11-016271-7 ISBN 3-11-016272-5
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| *K. Walker, ''An extension of Casson's invariant.'' Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992. ISBN 0-691-08766-0 ISBN 0-691-02532-0
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| [[Category:Geometric topology]]
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