A Casson-Lin invariant for knots in homology 3-spheres [Elektronische Ressource] / vorgelegt von Jochen Kroll

Kroll, Jochen Hoock

Sujets

• Mathematics

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• Publié par: universitat_siegen
• Publié le: 01 janvier 2003
• Nombre de lectures: 17
• Langue: Deutsch

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A Casson-Lin invariant for knots
in homology 3-spheres
DISSERTATION
zur Erlangung des Grades eines Doktors
der Naturwissenschaften
vorgelegt von
Dipl.-Math. Jochen Kroll
aus Siegen
eingereicht beim Fachbereich 6
der Universit¨at Siegen
Siegen 2003Gutachter der Dissertation: Prof. Dr. U. Koschorke
PD. Dr. U. Kaiser
Datum der Disputation: 25. April 2003
Internetpublikation der Universit¨at Siegen: urn:nbn:de:hbz:467-413Zusammenfassung: Im Jahr 1985 definierte A. Casson eine topologische Invariante λ(Σ)
fur¨ Homologie-3-Sph¨aren Σ, die, vereinfacht formuliert, nicht-abelsche SU(2)-Darstellungen der
Fundamentalgruppe von Σ mit Vorzeichen z¨ahlt. X.-S. Lin griff das konstruktive Prinzip Casson’s
3auf und definierte 1992 eine Invariante h(k) fur¨ Knoten k in der 3-Sph¨are S . Die Berechnung von
h(k)fuhrt¨ aufeinenvonLinals“mysteri¨os”eingesch¨atztenZusammenhangmitderKnotensignatur
σ , einer klassischen Seifert-Invariante.k
In der vorliegenden Arbeit werden beide Ans¨atze zusammengefuhrt,¨ um unter Verwendung von
αSU(2)-DarstellungenderKnotengruppeπ (Σ−k)eineSchnittzahls (k⊂Σ)zudefinieren(wobeiα1
anzeigt, dass fur¨ die Darstellung der Knotenmeridiane SU(2)-Matrizen mit Spur 2cosα, α∈(0,π),
αverwendet werden). Das zentrale Resultat ist die Berechnung von s (k⊂Σ) mit Hilfe eines Skein-
αAlgorithmus. Aus dieser folgt, dass s (k⊂Σ) eine Invariante fur¨ Knoten in Homologie-3-Sph¨aren
ist. Es ergibt sich
1α 2iαs (k⊂Σ)=2λ(Σ)+ σ (e ),k⊂Σ
2
2iαwobei die ¨aquivariante Signatur σ (e ) eine Verallgemeinerung der Knotensignatur darstellt.k⊂Σ
αDamit bildet s (k⊂Σ) das topologische Gegenstuc¨ k zu der von C. Herald auf analytischem Wege
definierten Invariante h (Σ,k). Die Invarianten von Casson und Lin ergeben sich als Spezialf¨alleα
α π/2 3lim s (k⊂Σ) bzw. s (k⊂S ).α→0,π
2iαDasResultatderBerechnungwirdzeigt, dassdieBedingung σ (e )=0hinreichendfur¨ diek⊂Σ
ExistenzeinesabelschenLimesnicht-abelscherDarstellungenvon π (Σ−k)ist. Insbesonderefolgt,1
dassπ (Σ−k)ineinemsolchenFallnicht-abelscheSU(2)-Darstellungenerm¨oglicht. Darub¨ erhinaus1
αwerden die Zusammenh¨ange, die man zwischen s (k ⊂ Σ) und der klassischen Seifert-Invariante
2iασ (e ) beobachten kann, begrundet.¨k⊂Σ
Abstract: In 1985 A. Casson defined a topological invariant λ(Σ) for homology 3-spheres.
Roughly speaking, λ(Σ) counts the irreducible SU(2)-representations of the fundamental group of
Σ with signs. In 1992, motivated by Casson’s construction, X.-S. Lin defined an invariant h(k)
3for knots k in the 3-sphere S . The computation yields a correlation to the knot signature σ , ak
classical Seifert invariant, which seemed “mysterious” to Lin.
αCombining both constructions, we define an intersection number s (k ⊂ Σ) using the repre-
sentations of the knot group π (Σ−k) (where α indicates that SU(2)-matrices with trace 2cosα,1
α ∈ (0,π), are used to represent the knot meridians). Our main result is the computation of
α αs (k ⊂ Σ) by using a skein algorithm. The computation implies that s (k ⊂ Σ) is actually an
invariant for knots in homology 3-spheres. It yields
1α 2iαs (k⊂Σ)=2λ(Σ)+ σ (e ),k⊂Σ
2
2iαwhere the equivariant signature σ (e ) is a generalization of the knot signature. It turns outk⊂Σ
αthat s (k⊂ Σ) is the topological counterpart of the knot invariant h (Σ,k) defined by C. Heraldα
along the lines of the analytical interpretation of Casson’s invariant. The invariants of Casson and
α π/2 3Lin appear as the special cases lim s (k⊂Σ) and s (k⊂S ) respectively.α→0,π
2iαUsing the results of the computation we show that the condition σ (e ) = 0 ensures thek⊂Σ
existence of an abelian limit of non-abelian representations of π (Σ−k). In particular this im-1
plies that π (Σ−k) admits non-abelian SU(2)-representations. Furthermore the computation of1
α αs (k ⊂ Σ) provides an explanation of the correlations between the invariant s (k ⊂ Σ) based on
2iαrepresentation spaces and the classical Seifert invariant σ (e ).k⊂Σ
66Acknowledgments: First I would like to express my gratitude to Prof. Dr. U. Koschorke.
In addition to his helpful advice and motivating interest, he offered me a position as a scientific
assistantinthetopologyworkgroup. AlsoIwouldliketothankProf.Dr.M.Heusenerforinitiating
the topic of this thesis and the intensive discussions during my two visits in Clermont-Ferrand.
Many thanks to PD Dr. A. Fel’shtyn and Dipl.-Math. A. Pilz for their inspiring questions and for
qualified answers, not only in regard to this work. My sincere thanks go to PD Dr. U. Kaiser for
carefully reading the manuscript and suggesting many improvements. Last but not least I want to
thank Dipl.-Math. U. Bauer whose hints concerning the English language were so very helpful.
The work was partially supported by the Graduiertenf¨orderung NRW.Contents
1 Introduction 2
2 Facts and Notations 8
2.1 Homology 3-spheres and surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Seifert-invariants of knots in homology 3-spheres . . . . . . . . . . . . . . . . . . . . 9
2.3 Representation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
g2.3.1 The representation space of π (H ). . . . . . . . . . . . . . . . . . . . . . . . 151
g2.3.2 Thetation space of π (F ) . . . . . . . . . . . . . . . . . . . . . . . . 151
2.3.3 The pillow case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4 Local properties of representation spaces . . . . . . . . . . . . . . . . . . . . . 17
2.4 The Casson invariant (Sketch of definition) . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
α3 The definition of s (k⊂Σ) 22
g03.1 The definition of the standard position for k ⊂H . . . . . . . . . . . . . . . . . . . 221
g 03.2 The representation space of π (H −k ) . . . . . . . . . . . . . . . . . . . . . . . . . 241 1
3.3 Orientations of the manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
α3.4 The intersection number s (k⊂Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
α4 The computation of s (k⊂Σ) 34
g+104.1 The definition of the computational position k ⊂H . . . . . . . . . . . . . . . . 34c 1
α 34.2 An important example: the computation of s (k ⊂S ) . . . . . . . . . . . . . . . . 36n
4.2.1 The case k =k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36n 1
α 34.2.2 Computation of s (k ⊂S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 45n
α 04.3 Comparison of s (k⊂Σ) and the Casson invariant λ(k) . . . . . . . . . . . . . . . . 48
α4.4 Computation of s for arbitrary k⊂Σ . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1Chapter 1
Introduction
In1985AndrewCassondefinedaninvariantλ(Σ)ofhomology3-spheresΣviaelegantconstructions
on SU(2)-representation spaces. The idea of Casson’s invariant comes from the observation that
g g g gfor a given Heegaard splitting Σ = H ∪H (where H and H are solid handlebodies of genus g1 2 1 2
i j∗k g ∗kg gmeeting along their common boundary F ) the surjections π (F ) → π (H ) → π (Σ), k = 1,2,1 1 1k
of the fundamental groups give rise to inclusions of spaces of conjugacy classes of non-abelian
representations of these groups into SU(2):
gbR(H )1
PPn P bb ni P j1 n 1Pn Pnn Pn Pn Pn Pn
bgb R(Σ)R(F )
P pP b pbP j pi 2P 2 pP pP pP pP pPP pp
gbR(H )2
gb bIt turns out that R :=R(H ) are complementary (and equal) dimensional smooth open subman-k k
gbifolds of R(F ). Then Casson’s invariant is roughly “the algebraic intersection number” of the
g gb b b e b bmanifolds R in R(F ). The isotopy R ˆR tR ⊂R(F ) which is used to obtain a transversal1 1 2k
gbintersection can be compactly supported away from the abelian singularities of R(F ). This is
possible because the condition H (Σ,Z)=0 guarantees that the trivial representation of Σ (which1
gbis the only abelian one) is isolated in R(F ).
1 1Let k be a knot in Σ and denote the manifold obtained by -surgery on k by Σ+ k. Then itn n
turns out that the difference
1 1 10 00λ(k):=λ(Σ+ k)−λ(Σ+ k)= Δ (1) (1.1)k⊂Σn+1 n 2
0is independent of n and therefore an invariant of the knot k. In fact 2λ(k) is determined by the
secondderivativeofthesymmetrizedAlexanderpolynomialΔ (t)evaluatedat1. Togetherwithk⊂Σ
3the initial value λ(S )=0 equation (1.1) allows us to compute λ(Σ) for any homology 3-sphere.
There are many remarkable consequences due to the properties of Casson’s invariant. One of the
most important corollaries states that any homotopy sphere has zero Rohlin invariant. Another
3 00beautiful corollary is that a knot k ⊂ S has property P if Δ (1) = 0. Here a non-trivial knot isk
said to have property P if no non trivial Dehn surgery on k yields a homotopy sphere.
2
vhhhhx6vxIn 1992 K. Walker generalized Casson’s construction to the case of rational homology 3-spheres
(cf. [Wak92]). Five years later C. Lescop published a global surgery formula which determines
3the Casson-Walker invariant as a function of the surgery data given for a link in S (cf. [Les97]).
Because all closed orientable 3-manifolds can be presented by a surgery on a link her definition can
be extended to this general case.
In 1990 C. H. Taube