# Casson's Invariant for Oriented Homology Three-Spheres: An Exposition. (MN-36)

Selman Akbulut
John D. McCarthy
Pages: 200
https://www.jstor.org/stable/j.ctt7ztjbw

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-x)
4. Introduction
(pp. xi-xviii)

Let M³ be an oriented homology 3-sphere and K be a knot in M³. Let N(K) be a closed regular neighborhood of K, so that N(K) (symbol) S¹ x D². Let M³(K) be the complement of the interior of N(K) in M³. M³(K) is a 3-manifold with boundary. The boundary is a torus T². Since M³ is a homology 3-sphere, we can choose the identification of N(K) with S¹ x D² so that S¹ x {1} is homologically trivial in M³(K). Under this assumption, let μ be the meridian {1} x ∂D² and λ be the preferred longitude S¹ x...

5. CHAPTER I: BEPRBSENTATION SPACES
(pp. 1-32)

Let

\mathrm{M(2,\mathbb{C}) = \left \{[_{c\; d}^{a\: b}\]\; |\; a,b,c,d\:\; \epsilon \; \mathbb{C}\left. \right \}}

\mathrm{SU(2,\mathbb{C}) = \left \{ A\;\; \epsilon\:\; M(2,\mathbb{C})\; |\; A\bar{A^{t}} = 1,\; det(A) = 1\left. \right \}}.

\mathrm{S^{3}=\left \{ \right.(a,b)\; \epsilon \; \mathbb{C}^{2}\; |\; a\bar{a}+b\bar{b}=\; 1\left. \right \}.}

There is a natural identification of S³ with\mathrm{SU(2,\mathbb{C})}:

\mathrm{S^{3}\; \overset{\cong }{\rightarrow}\; SU(2,\mathbb{C})}

\mathrm{(a,b)\; \rightarrow \; \begin{bmatrix} \; \: \: a\; \; \; b \\-\bar{b}\; \; \; \bar{a} \end{bmatrix}.}

(1) We shall identify S³ and\mathrm{SU(2,\mathbb{C})}by this fixed diffeomorphism.

(2) We shall consider S³ to have a fixed orientation throughtout our discussion.

By differentiating the defining equations for S³, we obtain the tangent bundle of S³:

\mathrm{TS^{3}=\left [\left [\begin{bmatrix} \; \; \mathrm{a}\; \; \; \mathrm{b}\\ -\bar{\mathrm{b}}\; \; \; \bar{\mathrm{a}} \end{bmatrix},\; \begin{bmatrix} \; \; \mathrm{u}\; \; \; \mathrm{v} \\ -\bar{\mathrm{v}}\; \; \; \bar{\mathrm{u}} \end{bmatrix}\right ]\begin{bmatrix}\mathrm{a\bar{u}+\bar{a}u+b\bar{v}+\bar{b}v=0\\ \mathrm{a\bar{a}+b\bar{b}=1 \end{bmatrix}.}}}

In particular, the Lie algebra of S³ is given as:

\mathrm{S=SU(2,\mathbb{C})=T_{I}(S^{3})}

\mathrm{s}=\left [\begin{bmatrix} \mathrm{is}\; \; \; \; \mathrm{v} \\ \mathrm{-\bar{v}-is } \end{bmatrix} \left [ \mathrm{s}\; \epsilon \; \mathrm{R},\; \mathrm{v}\; \epsilon \; \mathbb{C} \right ].

The action of S³ on itself by left translations provides a trivialization of TS³:

\mathrm{s^{3}\; x\; S\; \overset{\cong }{\rightarrow}\; TS^{3}}

\mathrm{(A,X)\rightarrow (A,AX)}.

We identify TS³ with S³ x S by the natural trivialization provided above....

6. CHAPTER II: HEBGARD DECOMPOSITIONS AND STABLE BQUIVALBNCE
(pp. 33-46)

Let:

W = standard (model) handlebody of genus g (g ≥ 1)

F = ∂W = boundary of W

D = embedded 2–disk in F

O = basepoint of F on ∂D

F*= F \ interior (D), S¹ = ∂D.

We may choose a family of loops on (F*,0), as in Figure 4, such that:

π1(F*,0) = 1

,...,ag,b1,...,bg>

π1(F, 0) = 1

,...,ag, b1,,...,bg| [a1, b1] ···[ag, bg] = 1 >

where[\mathrm{a}, \mathrm{b}] = \mathrm{aba}^{-1}\mathrm{b}^{-1}

π1(W, 0) = 1

,...,ag, b1,...,bg| b1= ··· = bg= 1 >

=1

,...,ag>.

Let ciand didenote the homology classes...

7. CHAPTER III: REPRESENTATION SPACES ASSOCIATED TO HEEGARD DECOMPOSITIONS
(pp. 47-62)

Given an Heegard decomposition (W1,W2) of a closed oriented 3-manifold M³, we have an associated commutative diagram of inclusions:

where:

F = ∂W1= ∂W2= W1∩ W2

F*is as in section II.1(a).

Note that each inclusion except for i is obtained by attaching 2 and 3 cells. Hence, by applying π1, we obtain a commutative diagram of groups where all homomorphisms except i*are surjective. “All the fundamental group is in the punctured surface”.:

Clearly, i*is an inclusion.

Finally, by applying the representation functor R, we obtain a commutative diagram of spaces:

where:

R= R(π1(∂F*,0))...

8. CHAPTER IV: CASSON’S INVARIANT FOR ORIENTED HOMOLOGY 3-SPHERES
(pp. 63-79)

Let M³ be an homology 3-sphere:

\mathrm{H_{0}}(\mathrm{M}^{3},\mathbb{Z}) \cong \mathrm{H}_{3} (\mathrm{M}^{3}, \mathbb{Z}) \cong \mathbb{Z}

\mathrm{H_{1}}(\mathrm{M}^{3},\mathbb{Z}) \cong \mathrm{H}_{2} (\mathrm{M}^{3}, \mathbb{Z}) = \{0\}.

Let (W1, W2) be a Heegard decomposition of M³. As an immediate consequence of Proposition III.1.1, we obtain:

(a) 1

, Q2> R* = ± 1

(b) Q1↑ Q2at 1

By Proposition 1.2.1, every reducible representation is conjugate to a diagonal representation. Hence, every reducible representation of π1(M³, 0) factors through \mathrm{H}_{1}(\mathrm{M}^{3}, \mathbb{Z}). We deduce the following corollary:

(a) S(π1(M³,0)) = R(π1(M³,0)) ∩ S = Q1∩ Q2∩ S = {1}.

(b) (Q1\S) ∩ (Q2\S) is compact.

(c)\hat{\mathrm{Q}}_{1}and\hat{\mathrm{Q}}_{2}are properly embedded open submanifolds...

9. CHAPTER V: CASSON’S INVARIANT FOR KNOTS IN HOMOLOGY 3-SPHERES
(pp. 80-149)

Let M³ be an homology 3-sphere.

Let K be a knot in M³.

Let N(K) be a closed regular neighborhood of K.

Let T² be the boundary of N(K), ∂N(K), so that T² is a torus.

Let M³(K) be M³ \ interior(N(K)).

Now:

\mathrm{H_{1}\; (M^{3}(K))\; \cong\; H_{1}(N(K))\; \cong\; \mathbb{Z}}.

Consider the inclusions:

\mathrm{i:T^{2}\rightarrow M^{3}(K)}

\mathrm{j:T^{2}\rightarrow N(K)}

and the associated surjections:

Letμbe a generator of the kernel of j*. Letλbe a generator of the kernel of i*. Sinceμandλare primitive elements, we may represent them by nontrivial simple closed curves on T² which we also denote...

10. CHAPTER VI: THE TOPOLOGY OF THE SPACE OF REPRESENTATIONS
(pp. 150-180)

As in previous sections, we identify R* with\mathrm{(S^{3})^{2g}}. Since we shall be inducting on g at various points, we denote the boundary map as follows:

\mathrm{\partial _{g}:R^{*}\rightarrow S^{3}}

\mathrm{(A_{1},\ldots,A_{g},B_{1},\ldots,B_{g})\; \rightarrow\; \prod_{i=1}^{g}\; [A_{i}B_{i}]}.

Likewise, we use the following notation:

\mathrm{R_{+}^{g}=R^{g}=\partial _{g}^{-1}(I)}

\mathrm{N_{+}^{g}=\partial _{g}^{-1}(D_{+}^{3})}

\mathrm{R_{-}^{g}=\partial _{g}^{-1}(-I)}

\mathrm{N_{-}^{g}=\partial _{g}^{-1}(D_{-}^{3})}.

Clearly:

\mathrm{R^{*}=N_{+}^{g}\; \cup \; N_{-}^{g}}.

The singular set of\mathrm{\partial _{g}}S is contained in R+. Hence,\mathrm{\partial _{g}}is regular on R*\S. In particular, we have a fibre bundle with fibre R-:

\mathrm{\partial _{g}|:N_{-}^{g}\rightarrow D_{-}^{3}}

Hence, there is a trivialization:

\mathrm{w_{g}:(R_{-}^{g}\; x\; D_{-}^{3},R_{-}^{g}\; x\; S^{2})\overset{\cong }{\rightarrow}(N_{-}^{g},\partial N_{-}^{g}}

with the following properties:

(1)\mathrm{\partial _{g}\cdot w_{g}=P_{2}}(projection onto second factor)

(2)\mathrm{w_{g}|_{R_{-}^{g}x(-I)}}is the usual inclusion\mathrm{R_{-}^{g}\subset N_{-}^{g}}.

\mathrm{R_{-}^{g}}is connected for...

11. References
(pp. 181-182)