In 3-dimensional topology, a part of the mathematical field of geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson

Andrew Casson

Andrew John Casson FRS is a mathematician, an expert on geometric topology.Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University in the United States where he served as department chair between 2004 and 2007. His Ph.D. advisor at the University of Liverpool was C. T. C...

.

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 manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

 oriented 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

s.

Definition

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

• .
• Let be an integral homology 3-sphere. Then for any knot K and for any , the difference

is independent of n. Here denotes Dehn surgery

Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...

 on by K.




is equal to zero for any boundary link in .

The Casson invariant is unique up to sign.

Properties

• If K is the trefoil then .
• The Casson invariant is 2 (or − 2) 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
  Euler characteristic
  In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
  
   for Floer homology
  Floer homology
  Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
  
  .
• For any let be the result of Dehn surgery
  Dehn surgery
  In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
  
   on M along K. Then the Casson invariant of minus the Casson invariant of

is the Arf invariant

Arf invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...

 of .

• The Casson invariant is the degree 1 part of the LMO invariant.
• The Casson invariant for the Seifert manifold  is given by the formula:

where

The Casson invariant as a count of representations

Informally speaking, the Casson invariant counts the number of conjugacy classes of representations of the fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 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 space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 oriented 3-manifold M is defined as
where denotes the space
of irreducible SU(2) representations of .
For a Heegaard splitting

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:...

  of , the Casson invariant equals
times the algebraic intersection of with .

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
from oriented rational homology 3-spheres to satisfying the following properties:

• .
• For every 1-component Dehn surgery
  Dehn surgery
  In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
  
   presentation of an oriented rational homology sphere in an oriented rational homology sphere M:

where:

• • m is an oriented meridian of a knot K and is the characteristic curve of the surgery.
  • is a generator the kernel of the natural map from to .
  • is the intersection form on the tubular neighbourhood of the knot, N(K).
  • is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of in the infinite cyclic cover of M-K, and is symmetric and evaluates to 1 at 1.
  • 

where x, y are generators of such that , and for an integer . is the Dedekind sum

Dedekind sum

In mathematics, Dedekind sums, named after Richard Dedekind, 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 have subsequently been much...

.

Compact oriented 3-manifolds

Christine Lescop defined an extension of the Casson-Walker invariant to oriented compact 3-manifolds. It is uniquely characterized by the following properties:

• If the first Betti number
  Betti number
  In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
  
  of M is zero, .
• If the first Betti number of M is one, where 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 is the oriented curve given by the intersection of two generators of and is the parallel curve to induced by the trivialization of the tubular neighbourhood of determined by .
• If the first Betti number of M is three, then for a,b,c a basis for , then .
• If the first Betti number of M is greater than three, .

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