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

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

Selman Akbulut
John D. McCarthy
Copyright Date: 1990
Pages: 200
https://www.jstor.org/stable/j.ctt7ztjbw
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    Casson's Invariant for Oriented Homology Three-Spheres: An Exposition. (MN-36)
    Book Description:

    In the spring of 1985, A. Casson announced an interesting invariant of homology 3-spheres via constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives surprising corollaries in low-dimensional topology. In the fall of that same year, Selman Akbulut and John McCarthy held a seminar on this invariant. These notes grew out of that seminar. The authors have tried to remain close to Casson's original outline and proceed by giving needed details, including an exposition of Newstead's results. They have often chosen classical concrete approaches over general methods. For example, they did not attempt to give gauge theory explanations for the results of Newstead; instead they followed his original techniques.

    Originally published in 1990.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-6062-3
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (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:

    \mathrm{i_{*}:H_{1}(T^{2})\twoheadrightarrow H_{1} (M^{3}(K))}

    \mathrm{j_{*}:H_{1}(T^{2})\twoheadrightarrow H_{1} (N(K))}

    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)