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

by Selman Akbulut, John D. McCarthy

Overview

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.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover 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.

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ISBN-13:	9781400860623
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Mathematical Notes , #36
Sold by:	Barnes & Noble
Format:	eBook
Pages:	202
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Casson's Invariant for Oriented Homology Three-Spheres

An Exposition

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By Selman Akbulut, John D. McCarthy

PRINCETON UNIVERSITY PRESS

Copyright © 1990 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08563-0

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CHAPTER 1

REPRESENTATION SPACES

1. The special unitary group - SU(2, C).

(a) Identification with S3

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There is a natural identification of S3 with SU(2, C):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conventions:

(1) We shall identify S3 and SU(2, C) by this fixed diffeomorphism.

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

(b) The tangent bundle of S3 TS3

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Convention: We identify TS3 with S3 × S by the natural trivialization provided above.

A natural Riemannian metric arises via translation from the inner product on S:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Associated to this metric, we have a natural exponential map:

exp S -> S3

The natural identification of S3 with SU(2, C) is an isometry of the standard metric on S3 and the above metric on SU(2, C).

(c) The conjugation actions on S3 and TS3

S3 acts on S3 by conjugation:

*: S3 × S3 -> S3

(A, B) [??] ABA-1.

The derivative of this action is simply:

* : S3 × TS3 -> TS3

(A, (B, X)) [??] (ABA-1, AXA-1).

From the definition of the inner product on S:

= .

Hence, these actions are actions by isometries and:

exp(AXA-1) = A exp(X)A-1.

Let:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Every element of S3 is conjugate to a diagonal element:

Proposition 1.1: (l) S0 [subset] S1 [subset] S3

(2) S3 = [union] {CS1C-1 | C [member of] S3}

Proof: (1) This is, of course, obvious. It is stated in anticipation of an analogous statement.

(2) Suppose A [member of] S3. Since C is algebraically closed, A has an eigenspace of eigenvalue λ. Choose a unit eigenvector in this eigenspace:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since C [member of] SU(2, C),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is to say, A [member of] CS1C-1.

From the previous facts and the fact that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we deduce that the exponential map is a diffeomorphism on the ball of radius π about 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Clearly, the injectivity radius of S3 is π.)

Notation: We shall use the following abuses of notation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, the natural contraction of Bπ(0):

G : Bπ(0) × [0, 1] -> Bπ(0)

(X, t) [??] tX.

exponentiates to a natural contraction of S3 \(-I):

G: (S3\(-I)) × [0, 1] -> S3\(-I)

(A, t) [??] At.

Note: By the previous observations, these contractions are equivariant with respect to the conjugation actions.

We note that this contraction extends to continuous maps:

G: Bπ(0) × R -> S

(x, t) [??] tX

G: (S3\(-I)) × R -> S3

(A, t) [??] At.

Note: G does not extend continuously over S3 × [0, 1].

Note: The fact that exp(iθ) = eiθ follows from the previous observation concerning the metrics on S3 and SU(2, C). We shall frequently take advantage of these two perspectives (as well as others to be introduced below).

From Proposition 1.1 and the previous observations, we conclude that the extension of the contraction defined above is given by the following rule:

G (S3\{-I}) × R -> S3

(CeiθC-1, t) [??] CeitθC-1

where 0 ≤ θ < π.

Note: The restriction on θ is possible due to the fact that eiθ is conjugate to e-iθ:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The next proposition describes the stabilizers of various elements in the "stratification" of S3 above. For this reason, we use the standard notation:

B*A = BAB-1

C*S1 = {CAC-1 | A [member of] S1}

S3A = {B [member of] S3 | B*A = A}.

Proposition 1.2:

(1) If A [member of] S0, then S3A = S3.

(2) If A [member of] (C*S1)\S0, then S3A = C*S1.

Proof: (1) This is obvious.

(2) Suppose A [member of] (C*S1)\S0. Since S1 is an abelian group, it is clear that C*S1 [subset] S3A. We need to establish the reverse inclusion.

By part (1), C-1*A [member of] S1\S0. That is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose that B [member of] S3A, so that:

BAB-1 = A or (C-1BC) (C-1 AC) = (C-1AC) (C-1BC).

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with a[bar.a] + b[bar.b] = 1, then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, b = 0, and B [member of] CS1C-1 as before.

(d) The trace and argument maps

We have the usual trace function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, this function is SO3 invariant:

tr(CAC-1) = tr(A).

Hence, by the results of the previous section, we have a surjection:

tr: S3 ->> [-2, 2].

Hence, we may define the argument function via the inverse cosine:

arg : S3 ->> [0, π]

A [??] cos-1(tr(A)/2).

Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The argument respects the natural contraction of the previous section:

arg(At) = t · arg(A) A ≠ -I 0 ≤ t ≤ 1.

Let:

D3+ + {A | 0 ≤ arg(a) ≤ π/2} = {A | 0 ≤ trace(A) ≤ 2}

S2 = {A | arg(A) = π/2| = {A | trace(A) = 0}

D3- = {A | π/2 ≤ arg(A) ≤ π = {A | -2 ≤ trace(A) ≤ 0}.

There is a natural involution of this decomposition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We observe that this involution is SO3 equivariant:

i(CAC-1) = Ci(A)C-1.

In addition, it fixes S2 pointwise:

i(A) = A iff A [member of] S2.

Moreover, it is respected by the argument:

arg(-A-1) = π - arg(A).

(e) The unit quaternions Sp(l) and Sp3,

We may identify C2 with the quaternions H:

C2 -> H

(a,b) [??] a + bj.

This identifies S3 with the unit quaternions:

S3 = Sp(l) = unit quaternions.

Under this identification, the diffeomorphism of part (a) identifying S3 with SU(2, C) is an isomorphism of group structures and an isometry of metrics.

The action of S3 on H by conjugation is a linear action by isometries of S3. Hence, we obtain a representation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note: The representation is in SO4 because S3 is connected.

Now R [subset] H and R is fixed by the action. Hence, we can reduce r to a representation in SO3:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Proposition 1.2, we conclude that the kernel of r1 is S0. Hence, r1 is onto:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Proposition 1.2, it follows that the conjugation action descends to a well-defined faithful action of SO3 on S3:

* : SO3 × S3 -> S3

([A], B) [??] ABA-1.

Convention:

(1) We consider all of the above isomorphisms as identifications.

We observe that the natural involution of S3 defined in the previous section, under the above identification with the unit quaternions, is given by the rule:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note: We shall primarily work with SU(2, C). It is, however, convenient to have these alternative models. Frequently an argument is more transparent in one model than another.

(f) The pure unit guaternions S2

Under the identification of S3 with Sp(l) given above, we see that S2, (as defined in part (d)), is the set of pure unit quaternions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly, S3 acts on S2 by conjugation:

S3 × S2 -> S2

(A, B) [??] ABA-1.

This action yields an identification of S2 with the right coset space of S3 by S1:

Proposition 1.3: S3/S1 [congruent to] S2

Proof: Let:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, from Proposition 1.1, we conclude that S2 = S3 · i. On the other hand, from Proposition 1.2, S3i = S1. Hence, S2 = S3 · [congruent to] S3/S3i = S3/S1.

2. The space of representations of G, R(G).

(a) the functor R

Let G be a group. The space of representations of G in S3 is defined as follows:

R(G) = Hom(G, S3)

= {ρ : -> S3 | ρ is a representation}.

We give R(G) the compact open topology, where G has the discrete topology and S3 the usual topology,

To each homomorphism λ : G1 -> G2, there is an associated continuous map:

R(λ) : R(G2) -> R(G1) ρ [??] ρ•λ.

Clearly, we have the following identities:

F(λ•μ) = R(μ)•R(λ), R(Id) = Id.

In other words, R is a contravariant functor from the category of groups and homomorphisms to the category of topological spaces and continuous maps.

(b) The action of SO3 on R(G)

By our previous remarks, the conjugation action of S3 on S3 descends to a faithful action of SO3 on S3:

* : SO3 × S3 -> S3 (faithful).

Given a group G, we may define an action of SO3 on R(G) (or an action of SU2(C)):

· : SO3 × R(G) -> R(G)

([A] · ρ)(g) = A*ρ(g) = Aρ(g)A-1.

This is a natural action in the following sense. For any homomorphism λ : G -> H, the following diagram commutes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let:

S0 (G) = {ρ | ρ : G -> S0}

S1 (G) = {ρ | ρ : G -> S1} = diag. reps

S(G) = {ρ | ρ : G -> S3 is reducible).

Proposition 2.1: (1) S0 (G) [subset] S1(G) [subset] S(G) [subset] R(G)

(2) S(G) = [union] {C·S1(G) | C [member of] SU2 (C)}

Proof: (1) This is immediate from the definitions.

(2) Since the conjugate of a reducible representation is clearly reducible, part (l) implies that:

S(G) [contains] [union] {C·S1(G) | C [member of] SU2 (C)}.

It remains to show that a reducible representation is conjugate to a diagonal representation. Suppose, therefore, that we have a reducible representation:

ρ : G -> SU(2, C) reducible.

By definition, ρ(G) leaves a proper subspace V of C2 invariant. The proof of Proposition 1.1 (2) yields an element C in SU2(C) such that:

C-1 ρ(g)C [member of] S1 for all g [member of] G.

Equivalently, ρ [member of] C·S1.

The next corollary follows immediately from the definitions, Proposition 2.1 and the naturality of the SO3 action.

Corollary 2.2: If λ : G -> H is a homomorphism, then we have a commutative diagram:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note: In fact, the decomposition of Proposition 1.1(2) is also natural.

The next proposition describes the stabilizers of representations in this "stratification" of R(G). For that purpose, we use the standard notation:

S3ρ = {A [member of] S3 | A·ρ = ρ}.

Clearly:

A [member of] S3ρ iff A [member of] S3ρ(g) for all g [member of] G

A [member of] S3ρ iff ρ(g) [member of] S3A for all g [member of] G.

Proposition 2.3:

(1) If ρ [member of] S0 (G), then S3ρ = S3.

(2) If ρ [member of] C·S1(G)\S0(G), then S3ρ = CS1C-1.

(3) If ρ [member of] R(G)\S(G), then S3ρ = S0.

Let:

[bar.R](G) = SO3\R(G)

= space of conjugacy classes of reps. of G in S3.

By the previous observations, we conclude that if λ : G -> H is a homomorphism, there is a continuous map:

[bar.R](λ) : [bar.R](H) -> [bar.R](G)

such that the following diagram commutes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let:

[bar.S]0(G) = π(S0(G))

[bar.S]1(G) = π(S1(G))

[bar.S](G) = π(S(G)) = SO3\S(G).

The diagram below commutes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) The algebraic set associated to a set of generators, S1 of G

Suppose that G is a finitely generated group. Consider a finite set of generators for G:

S = {g1, ..., gn}

We have a natural map:

f = fS : R(G) -> (S3)n

ρ [??] (ρ(g1), ..., ρ(gn)).

Let:

V(S) = fS(R(G)).

Now, if R is a set of relations for G, so that:

G =

is a presentation of G, then:

V(S) = {(A1, ..., An) [member of] (S3)n | r(A1, ..., An) = 1 r [member of] R}.

Hence, we have a natural homeomorphism from R(G) to a real algebraic set:

fS : R(G) -> V(S)

ρ [??] (ρ(g1), ..., ρ(gn)).

(Continues...)

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Excerpted from Casson's Invariant for Oriented Homology Three-Spheres by Selman Akbulut, John D. McCarthy. Copyright © 1990 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

• FrontMatter, pg. i
• Contents, pg. v
• Preface, pg. ix
• Introduction, pg. xi
• Chapter I: Representation Spaces, pg. 1
• Chapter II: Heegard Decompositions and Stable Bquivalence, pg. 33
• Chapter III: Representation Spaces Associated to Heegard Decompositions, pg. 47
• Chapter IV; Casson’s Invariant for Oriented Homology 3-spheres, pg. 63
• Chapter V: Casson’s invariant for knots in homology 3-spheres, pg. 80
• Chapter VI: The Topology of the Space of Representations, pg. 150
• References, pg. 181

Editorial Reviews

In the spring of 1985, A. Casson announced an interesting invariant of homology 3-spheres via beautiful constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives surprising corollaries in low dimensional topology. These notes grew out of a 1985 seminar conducted by the authors on this invariant. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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