Casson invariant
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In 3-dimensional topology, a part of the mathematical field of
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originate ...
, 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
oriented 3-manifolds.


Definition

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 ::\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right) :is independent of ''n''. Here \Sigma+\frac\cdot K denotes \frac
Dehn surgery In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...
on Σ by ''K''. *For any boundary link ''K'' ∪ ''L'' in Σ the following expression is zero: ::\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) -\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right)-\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) +\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.


Properties

*If K is the trefoil then ::\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right)=\pm 1. *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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
for Floer homology. *For any integer ''n'' ::\lambda \left ( M + \frac\cdot K\right ) - \lambda \left ( M + \frac\cdot K\right ) = \phi_1 (K), :where \phi_1 (K) is the coefficient of z^2 in the
Alexander–Conway polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed ...
\nabla_K(z), and is congruent (mod 2) to the
Arf invariant In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf ...
of ''K''. *The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant. *The Casson invariant for the Seifert manifold \Sigma(p,q,r) is given by the formula: :: \lambda(\Sigma(p,q,r))=-\frac\left -\frac\left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\right) -d(p,qr)-d(q,pr)-d(r,pq)\right/math> :where ::d(a,b)=-\frac\sum_^\cot\left(\frac\right)\cot\left(\frac\right)


The Casson invariant as a count of representations

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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
oriented 3-manifold ''M'' is defined as \mathcal(M)=R^(M)/SU(2) where R^(M) denotes the space of irreducible SU(2) representations of \pi_1 (M). For a
Heegaard splitting In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', an ...
\Sigma=M_1 \cup_F M_2 of M, the Casson invariant equals \frac times the algebraic intersection of \mathcal(M_1) with \mathcal(M_2).


Generalizations


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 In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...
presentation (''K'', μ) of an oriented rational homology sphere ''M''′ in an oriented rational homology sphere ''M'': :\lambda_(M^\prime)=\lambda_(M)+\frac\Delta_^(M-K)(1)+\tau_(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 ''H''1(∂''N''(''K''), Z) → ''H''1(''M''−''K'', Z). *\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 H_1(M-K)/\text in the infinite cyclic cover of ''M''−''K'', and is symmetric and evaluates to 1 at 1. *\tau_(m,\mu;\nu)= -\mathrm\langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle)+\mathrm\langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle)+\frac :where ''x'', ''y'' are generators of ''H''1(∂''N''(''K''), Z) such that \langle x,y\rangle=1, ''v'' = δ''y'' for an integer δ and ''s''(''p'', ''q'') is the
Dedekind sum In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They ha ...
. Note that for integer homology spheres, the Walker's normalization is twice that of Casson's: \lambda_(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 In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...
. It is uniquely characterized by the following properties: *If the first Betti number of ''M'' is zero, ::\lambda_(M)=\tfrac\left\vert H_1(M)\right\vert\lambda_(M). *If the first Betti number of ''M'' is one, ::\lambda_(M)=\frac-\frac :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, ::\lambda_(M)=\left\vert\mathrm(H_1(M))\right\vert\mathrm_M (\gamma,\gamma^\prime) :where γ is the oriented curve given by the intersection of two generators S_1,S_2 of H_2(M;\mathbb) and \gamma^\prime 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;\mathbb), then ::\lambda_(M)=\left\vert\mathrm(H_1(M;\mathbb))\right\vert\left((a\cup b\cup c)( \right)^2. *If the first Betti number of ''M'' is greater than three, \lambda_(M)=0. The Casson–Walker–Lescop invariant has the following properties: *When the orientation of ''M'' changes the behavior of \lambda_(M) depends on the first Betti number b_1(M) = \operatorname H_1(M;\mathbb)of ''M'': if \overline is ''M'' with the opposite orientation, then ::\lambda_(\overline) = (-1)^\lambda_(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 ::\lambda_(M_1\#M_2)=\left\vert H_1(M_2)\right\vert\lambda_(M_1)+\left\vert H_1(M_1)\right\vert\lambda_(M_2)


SU(N) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the spec ...

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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of \mathcal/\mathcal, where \mathcal is the space of SU(2) connections on ''M'' and \mathcal is the group of gauge transformations. He regarded the Chern–Simons invariant as a S^1-valued Morse function on \mathcal/\mathcal 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.


References

* Selman Akbulut and John McCarthy, ''Casson's invariant for oriented homology 3-spheres— an exposition.'' Mathematical Notes, 36. Princeton University Press, Princeton, NJ, 1990. * Michael 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. *Hans Boden and Christopher Herald, ''The SU(3) Casson invariant for integral homology 3-spheres.''
Journal of Differential Geometry The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in b ...
50 (1998), 147–206. *Christine Lescop, ''Global Surgery Formula for the Casson-Walker Invariant.'' 1995, *Nikolai Saveliev, ''Lectures on the topology of 3-manifolds: An introduction to the Casson Invariant.'' de Gruyter, Berlin, 1999. * *Kevin Walker, ''An extension of Casson's invariant.'' Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992. {{ISBN, 0-691-02532-0 Geometric topology