Andrew Ranicki
High-dimensional knot
theory
Algebraic surgery in codimension 2
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
V
In memory of J. F. Adams
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VI
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Preface
On my first day as a graduate student at Cambridge in October, 1970 my
official Ph. D. supervisor Frank Adams suggested I work on surgery theory.
In September he had attended the International Congress at Nice, where
Novikov had been awarded the Fields Medal for his work in surgery. Novikov
was prevented by the Soviet authorities from going to the Congress himself,
and his lecture on hermitian K-theory [219] was delivered by Mishchenko.
As usual, Frank had taken meticulous notes, and presented me with a copy.
He also suggested I look at ‘Novikov’s recent paper in Izvestia’. This recommendation was quite mysterious to me, since at the time I knew of only one
publication called Izvestia, and I couldn’t imagine that journal publishing
an article on topology. I was too shy to ask, but a visit to the library of
the Cambridge Philosophical Society soon enlightened me. I started work on
Novikov’s paper [218] under the actual supervision of Andrew Casson, using
a translation kindly provided by Dusa McDuff. (Andrew could not be my
official supervisor since he did not have a Ph. D. himself). Ultimately, my
reading of [218] became my Ph. D. thesis, which was published as Ranicki
[230], [231]. Frank had remained my official supervisor, being ever helpful in
answering my many queries on algebraic topology, and generally keeping me
under his protective wing. I dedicate this book to his memory, as a token of
my gratitude to him.*
Edinburgh, June 1998
This is a reprint of the published version of the book, which includes the
corrections and additional comments posted on
aar/books/knoterr.pdf
August 2004
* After the book was published I came across a statement of Frank Adams
which makes the dedication of the book even more appropriate: ‘Of course,
from the point of view of the rest of mathematics, knots in higher-dimensional
space deserve just as much attention as knots in 3-space’ (article on topology,
in ’Use of Mathematical Literature’ (Butterworths (1977)).
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Preface
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV
Knot theory high and low – Knotty but nice – Why 2? – The
fundamental group – Local flatness – Polynomials – Seifert
surfaces – Fibred knots and open books – Knot cobordism
– Simple knots – Surgery theory – Algebraic transversality
– The topological invariance of rational Pontrjagin classes –
Localization – Algebraic K- and L-theory invariants of knots –
Codimension q surgery – Fredholm localization – Structure
Part One Algebraic K-theory
1. Finite structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1A. The Wall finiteness obstruction . . . . . . . . . . . . . . . . . . . . . . . . .
1B. Whitehead torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1C. The mapping torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
5
8
2. Geometric bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3. Algebraic bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4
Localization and completion in K-theory . . . . . . . . . . . . . . . .
4A. Commutative localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4B. The algebraic K-theory localization exact sequence . . . . . . .
19
20
21
5. K-theory of polynomial extensions . . . . . . . . . . . . . . . . . . . . . .
27
6. K-theory of formal power series
........................
41
7. Algebraic transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
8. Finite domination and Novikov homology . . . . . . . . . . . . . . .
57
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9. Noncommutative localization
...........................
71
10. Endomorphism K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10A. The endomorphism category . . . . . . . . . . . . . . . . . . . . . . . . . . .
−1
−1
10B. The Fredholm localizations Ω+
A[z], Ω+
A[z] . . . . . . . . . . . .
−1
10C. The A-contractible localization Π A[z, z −1 ] . . . . . . . . . . . . .
79
80
82
88
11. The characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
12. Primary K-theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
13. Automorphism K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
13A. The Fredholm localization Ω −1 A[z, z −1 ] . . . . . . . . . . . . . . . . . 120
13B. The automorphism category . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
14. Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
15. The fibering obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16. Reidemeister torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
17. Alexander polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
18. K-theory of Dedekind rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
19. K-theory of function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Part Two Algebraic L-theory
20. Algebraic Poincar´
e complexes . . . . . . . . . . . . . . . . . . . . . . . . . . .
20A. L-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20B. Γ -groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20C. Thickenings, unions and triads . . . . . . . . . . . . . . . . . . . . . . . . .
207
207
217
219
21. Codimension q surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
21A. Surgery on submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
21B. The splitting obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
22. Codimension 2 surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22A. Characteristic submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22B. The antiquadratic construction . . . . . . . . . . . . . . . . . . . . . . . . .
22C. Spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22D. The homology splitting obstruction . . . . . . . . . . . . . . . . . . . . .
237
237
239
246
255
23. Manifold and geometric Poincar´
e bordism of X × S 1 . . . . 261
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24. L-theory of Laurent extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 265
25. Localization and completion in L-theory . . . . . . . . . . . . . . . . 271
26. Asymmetric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
27. Framed codimension 2 surgery . . . . . . . . . . . . . . . . . . . . . . . . . .
27A. Codimension 1 Seifert surfaces . . . . . . . . . . . . . . . . . . . . . . . . .
27B. Codimension 2 Seifert surfaces . . . . . . . . . . . . . . . . . . . . . . . . .
27C. Branched cyclic covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27D. Framed spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
307
307
312
318
322
28. Automorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28A. Algebraic Poincar´e bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28B. Duality in automorphism K-theory . . . . . . . . . . . . . . . . . . . . .
28C. Bordism of automorphisms of manifolds . . . . . . . . . . . . . . . . .
28D. The automorphism signature . . . . . . . . . . . . . . . . . . . . . . . . . . .
28E. The trace map χz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28F. Automorphism and asymmetric L-theory . . . . . . . . . . . . . . . .
325
326
328
332
334
343
347
29. Open books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
29A. Geometric open books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
29B. The asymmetric signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
30. Twisted doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30A. Geometric twisted doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30B. Algebraic twisted doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30C. Algebraic open books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30D. Twisted double L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
377
379
386
393
396
31. Isometric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
31A. Isometric structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
31B. The trace map χs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
32. Seifert and Blanchfield complexes . . . . . . . . . . . . . . . . . . . . . . .
32A. Seifert complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32B. Blanchfield complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32C. Fibred Seifert and Blanchfield complexes . . . . . . . . . . . . . . . .
32D. Based Seifert and Blanchfield complexes . . . . . . . . . . . . . . . . .
32E. Minimal Seifert complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
425
427
429
436
440
446
33. Knot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33A. The Seifert complex of an n-knot . . . . . . . . . . . . . . . . . . . . . . .
33B. Fibred knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33C. The Blanchfield complex of an n-knot . . . . . . . . . . . . . . . . . . .
453
453
455
456
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33D. Simple knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
33E. The Alexander polynomials of an n-knot . . . . . . . . . . . . . . . . . 460
34. Endomorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
34A. Endometric structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
34B. The trace map χx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
35. Primary L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35A. Endomorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35B. Isometric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35C. Automorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35D. Asymmetric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
477
478
482
484
487
36. Almost symmetric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
37. L-theory of fields and rational localization . . . . . . . . . . . . . . 501
37A. Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
37B. Rational localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
38. L-theory of Dedekind rings
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
39. L-theory of function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39A. L-theory of F (x) (x = x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39B. L-theory of F (s) (s = 1 − s) . . . . . . . . . . . . . . . . . . . . . . . . . . .
39C. L-theory of F (z) (z = z −1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39D. The asymmetric L-theory of F . . . . . . . . . . . . . . . . . . . . . . . . .
39E. The automorphism L-theory of F . . . . . . . . . . . . . . . . . . . . . . .
521
522
534
542
553
560
40. The multisignature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40A. Isometric multisignature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40B. Asymmetric multisignature . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40C. The ω-signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40D. Automorphism multisignature . . . . . . . . . . . . . . . . . . . . . . . . . .
571
573
575
577
583
41. Coupling invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41A. Endomorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41B. Isometric L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41C. Automorphism L-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
593
594
604
609
42. The knot cobordism groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
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Appendix
The history and applications of open books
(by H.E.Winkelnkemper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
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Table of Contents
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Introduction
Knot theory high and low
An n-dimensional knot (M, N, k) is an embedding of an n-dimensional manifold N n in an (n + 2)-dimensional manifold M n+2
k : N n ⊂ M n+2 .
An n-knot is a knot of the type (S n+2 , S n , k). In general, n can be any positive
integer. A classical knot is a 1-knot k : S 1 ⊂ S 3 .
The homological algebra methods of surgery theory apply to n-dimensional
knots for all n ≥ 1. This book is mainly concerned with knots in the high
dimensions n ≥ 4, for which there is a much closer correspondence between
this type of algebra and the topology than in the low dimensions n = 1, 2, 3.
However, the story begins with the classical case n = 1.
The mathematical study of knots started in the 19th century, with the
work of Gauss.1 Towards the end of the century, P. G. Tait (working in Edinburgh) tabulated all the classical knots with ≤ 10 crossings. It was only
in the 20th century that a systematic theory of knots was developed. Many
algebraic and geometric techniques have been invented and used to deal with
classical knots : a large number of invariants is available, including the fundamental group of the knot complement, the Alexander polynomial, the Seifert
matrix, assorted signatures, the Jones polynomial, the Vassiliev invariants,
. . . , although we are still short of a complete classification. The last 20 years
have seen a particular flourishing of classical knot theory, involving deep
connections with 3-manifold topology, physics and biology. There is a large
literature at various levels, including the books by C. Adams [2], Atiyah [12],
Burde and Zieschang [34], Crowell and Fox [59], Kauffman [121], Lickorish
[166], Livingston [168], Murasugi [207], Reidemeister [250], and Rolfsen [253].
Although high-dimensional knot theory does not have such glamorous
applications as classical knot theory, it has many fascinating results of its own,
which make use of a wide variety of sophisticated algebraic and geometric
methods. This is the first book devoted entirely to high-dimensional knot
1
See Epple [68], [69], [70] for the history of knot theory.
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XVI
Introduction
theory, which previously has been largely confined to research and survey
papers. The book actually has two aims :
(i) to serve as an introduction to high-dimensional knot theory, using
surgery theory to provide a systematic exposition,
(ii) to serve as an introduction to algebraic surgery theory, using highdimensional knots as the geometric motivation.
The topological properties of high-dimensional knots are closely related
to the algebraic properties of modules and quadratic forms over polynomial
extensions. The main theme of the book is the way in which this relationship
is essential to both (i) and (ii). High-dimensional knot theory has a somewhat deserved reputation as being an arcane geometric machine – I hope
that aim (i) is sufficiently achieved to demystify the geometry, and make it
more accessible to algebraic topologists. Likewise, surgery theory has a somewhat deserved reputation as being an arcane algebraic machine – I hope that
aim (ii) is sufficiently achieved to demystify the algebra, and make it more
accessible to geometric topologists.
Knot theory is a good introduction to surgery since it is easier to visualize
knots than manifolds. Many surgery invariants can be viewed as generalizations of (high-dimensional) knot invariants. For example, the self intersection
quadratic form µ used by Wall [304] to define the surgery obstruction of a
normal map is a generalization of the matrix associated by Seifert [263] to
a spanning surface for a classical knot. Moreover, the plumbing construction in [304, Chap. 5] of normal maps with prescribed quadratic form is a
generalization of the construction in [263] of classical knots with prescribed
matrix.
Artin [8] produced the first non-trivial examples of 2-knots S 2 ⊂ S 4 , by
spinning classical knots S 1 ⊂ S 3 .
The theory of n-knots S n ⊂ S n+2 and more general n-dimensional knots
N ⊂ M n+2 evolved with the work of Whitney on removing singularities in
the 1940s, Thom’s work on transversality and cobordism in the 1950s, the
h-cobordism theorem for manifolds of dimension n ≥ 5 of Smale in 1960
and the consequent surgery theory of high-dimensional manifolds and their
submanifolds. The last 35 years have seen the growth of a large body of research literature on codimension 2 embeddings of high-dimensional manifolds
in the differentiable, piecewise linear and topological categories. However, it
is certainly not the aim of this book to provide a comprehensive account of
all the methods and results of high-dimensional knot theory!2 The book has
the more limited objective of providing an exposition of the algebraic surgery
n
2
In particular, there is very little about links ∪S n ⊂ S n+2 , and not much about
the connections between knots and singularities, or about the homotopy-theoretic
aspects of high-dimensional knot theory.
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Knotty but nice
XVII
method for the construction and classification of high-dimensional knots in
the topological category.
The pre-1981 applications of high-dimensional surgery theory to codimension 2 embeddings were considered in Chap. 7 of Ranicki [237] – however, at
the time algebraic surgery was not so highly developed, and the treatment
still relied on geometric transversality. The current treatment makes full use
of the algebraic analogues of transversality obtained by the author since 1981.
Knotty but nice
Two embeddings k0 , k1 : N n ⊂ M m are concordant (or cobordant) if there
exists an embedding
: N × [0, 1] −−→ M × [0, 1] ; (x, t) −−→ (x, t)
such that
(x, 0) = (k0 (x), 0) ,
(x, 1) = (k1 (x), 1) (x ∈ N ) .
Two embeddings k0 , k1 : N ⊂ M are isotopic if there exists a cobordism
which is level-preserving, with
(N × {t}) ⊂ M × {t} (t ∈ [0, 1]) .
Isotopy is a considerably stronger equivalence relation than concordance –
much in the way that the isomorphism of quadratic forms is stronger than
stable isomorphism.
Two embeddings k0 , k1 : N ⊂ M are equivalent if there exists a homeomorphism h : M −−→M such that
hk0 (N ) = k1 (N ) ⊂ M .
If h is isotopic to the identity then k0 , k1 are isotopic. Every orientationpreserving homeomorphism h : S m −−→S m is isotopic to the identity, so for
embeddings k : N n ⊂ S m
equivalent = isotopic =⇒ cobordant .
An embedding k : S n ⊂ S m is unknotted if it is equivalent to the trivial
knot k0 : S n ⊂ S m defined by the standard embedding
k0 : S n −−→ S m ; (x0 , x1 , . . . , xn ) −−→ (x0 , x1 , . . . , xn , 0, . . . , 0) .
These definitions are particularly significant for knots, that is embeddings
with codimension m − n = 2.
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XVIII Introduction
Why 2?
A knot is a codimension 2 embedding. What about embeddings k : N n ⊂ M m
with codimension m − n = 2 ? Zeeman [316] and Stallings [274] proved that
embeddings k : S n ⊂ S m with codimension m − n ≥ 3 are unknotted in the
piecewise linear and topological categories.3 Thus topological knotting only
starts in codimension 2. Many algebraic invariants have been developed to
classify the knotting properties of codimension 2 embeddings. Codimension
1 embeddings are almost as interesting as codimension 2 embeddings, although they do not have the intuitive appeal of classical knot theory. In fact,
codimension 1 embeddings deserve a book of their own! In any case, many
of the techniques used in codimension 2 make crucial use of codimension 1
embeddings, such as spanning surfaces.
The fundamental group
The study of knots necessarily involves the fundamental group, as well as
the higher homotopy groups and homology. The isomorphism class of the
fundamental group π1 (X) of the complement of a knot (M, N, k)
X = M \k(N )
is an invariant of the equivalence class. The algebraic theory developed in
this book works with modules over an arbitrary ring, reflecting the major
role of the fundamental group π1 (X) and the group ring Z[π1 (X)] in the
classification of knots (M, N, k).
The fundamental group π1 (X) of the complement X = S 3 \k(S 1 ) of a
classical knot k : S 1 ⊂ S 3 was the first knot invariant to be studied by the
methods of algebraic topology, serving to distinguish many classical knots
k : S 1 ⊂ S 3 . Dehn’s Lemma (formulated in 1910, but only finally proved
by Papakyriakopoulos in 1956) states that a classical knot k : S 1 ⊂ S 3 is
unknotted if and only if π1 (X) = Z. The complement X = S n+2 \k(S n ) of
any n-knot k : S n ⊂ S n+2 has the homology of a circle by Alexander duality,
H∗ (X) = H∗ (S 1 ). Levine [153] proved that for n ≥ 4 an n-knot k : S n ⊂ S n+2
is unknotted if and only if the complement X is homotopy equivalent to a
circle, i.e. if and only if π∗ (X) = π∗ (S 1 ). This unknotting criterion also holds
for n = 3 by Levine [158] and Trotter [292], and for n = 2 by Freedman [85]
(in the topological category). Thus homology does not see knotting, while
homotopy detects unknotting.
3
There is codimension ≥ 3 knotting in the differentiable category. Differentiable
embeddings k : S n ⊂ S m with m − n ≥ 3 were classified by Haefliger [100] and
Levine [154].
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Local flatness
XIX
Local flatness
An n-dimensional knot (M n+2 , N n , k) is locally flat at x ∈ N if x has a
neighbourhood in (M, k(N )) which is homeomorphic to (Rn+2 , Rn ). The knot
is locally flat if it is locally flat at every x ∈ N .
Given an n-dimensional P L knot (M n+2 , N n , k) there is defined at every
point x ∈ N a P L embedding
starM (x) ∩ N = starN (x) = Dn ⊂ starM (x) = Dn+2
which could be knotted, i.e. not P L equivalent to the standard embedding
Dn ⊂ Dn+2 . The restriction to the links defines a P L (n − 1)-knot
kx : linkM (x) ∩ N = linkN (x) = S n−1 ⊂ linkM (x) = S n+1 .
The knot type of kx is a measure of the local singularity of the topology of
k at x : if k is locally flat at x then kx : S n−1 ⊂ S n+1 is unknotted (Fox and
Milnor [82], with n = 2).4
Unless specified otherwise, from now on knots (M n+2 , N n , k) will be taken
to be locally flat, i.e. only knots which are locally unknotted will be considered.
Surgery theory can also deal with non-locally flat knots, such as arise from
singular spaces (Cappell and Shaneson [42], [46], [47]). However, non-locally
flat knots require special techniques, such as intersection homology.
For any (locally flat) knot (M n+2 , N n , k) the codimension 2 submanifold
k(N ) ⊂ M has a normal bundle, that is a closed regular neighbourhood
(P, ∂P ) which is the total space of a bundle
(D2 , S 1 ) −−→ (P, ∂P ) −−→ k(N ) .
Unless specified otherwise, from now on knots (M n+2 , N n , k) will be taken
to be oriented, with M, N compact and the normal bundle of k : N ⊂ M
compatibly oriented.
The exterior of a knot (M n+2 , N n , k) is the codimension 0 submanifold
of M
(X, ∂X) = (closure(M \P ), ∂P )
with a homotopy equivalence to the knot complement
X
A locally flat knot (M
n+2
M \k(N ) .
n
, N , k) is homology framed if
k[N ] = 0 ∈ Hn (M )
and the normal bundle of k(N ) ⊂ M is framed
(P, ∂P ) = k(N ) × (D2 , S 1 ) ,
4
High-dimensional knots arise in a similar way at isolated singular (i.e. nonmanifold) points of a complex hypersurface V 2i ⊂ S 2i+1 ⊂ C Pi+1 – see the
section Fibred knots and open books further below in the Introduction.
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XX
Introduction
with a given extension of the projection ∂X = ∂P −−→S 1 to a map p :
X−−→S 1 . Every n-knot (S n+2 , S n , k) is homology framed, in an essentially
unique manner. Homology framed knots are particularly tractable – for example, the knot complement has a canonical infinite cyclic cover, and the
knot admits a codimension 1 spanning (= Seifert) surface.
Polynomials
Many knot invariants involve the Laurent polynomial extension ring A[z, z −1 ]
of a ring A, in the first instance for A = Z.
Given a homology framed knot (M, N, k) with exterior X = cl.(M \N ×
D2 ) let X = p∗ R be the infinite cyclic cover of X classified by the map
p : X−−→S 1
/R
X
X
p /
S1
and let ζ : X−−→X be a generating covering translation. The Laurent polynomial ring Z[z, z −1 ] acts on the homology groups H∗ (X), with z = ζ∗ :
H∗ (X)−−→H∗ (X), and also on the fundamental group π1 (X). The fundamental group of X is the ζ∗ -twisted extension of π1 (X) by Z
π1 (X) = π1 (X) ×ζ∗ Z = {gz j | g ∈ π1 (X), j ∈ Z} ,
with ζ∗ : π1 (X)−−→π1 (X) the induced automorphism and gz = zζ∗ (g). The
group ring of π1 (X) is the ζ∗ -twisted Laurent polynomial extension ring of
the group ring of π1 (X)
Z[π1 (X)] = Z[π1 (X)]ζ∗ [z, z −1 ] .
In particular, if ζ∗ = 1 : π1 (X)−−→π1 (X) then
π1 (X) = π1 (X) × Z , Z[π1 (X)] = Z[π1 (X)][z, z −1 ] .
The Alexander polynomial ([4]) of a classical knot k : S 1 ⊂ S 3
∆(z) ∈ Z[z, z −1 ]
is the basic polynomial invariant of the knot complement X = S 3 \k(S 1 ),
with
∆(1) = 1 , ∆(z)H1 (X) = 0 .
(See Chaps. 17, 33 for the Alexander polynomials of n-knots.) The Alexander
polynomial was the first application of polynomial extension rings to knot theory. Many of the abstract algebraic results in this book concern modules and
quadratic forms over Laurent polynomial extensions, which are then applied
to high-dimensional knots (M, N, k) by considering the algebraic topology of
the complement X = M \k(N ).
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Seifert surfaces
XXI
Seifert surfaces
Transversality is a key ingredient of knot theory. For example, every classical
knot k : S 1 ⊂ R3 (regarding R3 as a subset of S 3 ) has plane projections,
that is functions f : R3 −−→R2 such that f k : S 1 −−→R2 is an immersion
with a finite number of transverse self-intersections. Plane projections are
not unique. The minimum number of self-intersections in a plane projection
is the crossing number of k. This was the first knot invariant, measuring the
‘knottedness’ of k.
A Seifert (or spanning) surface for a homology framed knot (M n+2 , N n , k)
is a codimension 1 submanifold F n+1 ⊂ M n+2 with boundary ∂F = k(N )
and trivial normal bundle F × [0, 1] ⊂ M . Seifert surfaces can be regarded
as higher-dimensional analogues of knot projections. Seifert surfaces are in
fact the main geometric tool of high-dimensional knot theory, with surgery
theory providing the means for transforming one Seifert surface into another.
Codimension 1 transversality guarantees the existence of Seifert surfaces
for a homology framed knot (M n+2 , N n , k) – make the map on the knot
exterior
p : (cl.(M \(k(N ) × D2 )), k(N ) × S 1 ) −−→ S 1
transverse regular at 1 ∈ S 1 and set
(F n+1 , ∂F ) = (p−1 (1), k(N )) ⊂ M n+2 .
Isolated instances of singular (i.e. immersed) spanning surfaces for classical
knots already feature in the work of Tait [283], as in the following example
of an ‘autotomic’ surface spanning the trefoil knot :
Nonsingular spanning surfaces for classical knots were first obtained by
Frankl and Pontrjagin [83]. Seifert [263] obtained a spanning surface of a knot
k : S 1 ⊂ S 3 from any knot projection, and defined the genus of k to be
genus(k) = min {genus(F )} ,
the minimum genus of a spanning surface F 2 ⊂ S 3 with
genus(F ) =
1
rank H1 (F ) ≥ 0 .
2
The genus of an alternating knot was shown in [263] to be the genus of the
Seifert surface determined by an alternating knot projection.
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XXII
Introduction
The intersection properties of H1 (F ) were used in [263] to construct the
Seifert matrix of a Seifert surface F of a classical knot k : S 1 ⊂ S 3 , a 2g × 2g
matrix V over Z with g = genus(F ). The difference between V and the
transpose V t is an invertible 2g×2g matrix V −V t , so that det(V −V t ) = ±1,
with
V − V t : H1 (F ) −−→ H1 (F )∗ = HomZ (H1 (F ), Z) = H 1 (F )
the Poincar´e duality isomorphism of F . The Seifert matrix determines the
Alexander polynomial by
∆(z) = ±det(V − zV t ) ∈ Z[z, z −1 ] .
The signature of k
σ(k) = signature (H1 (F ), V + V t ) ∈ Z
is another classical knot invariant which can be defined using the Seifert form.
The Seifert matrix is the most versatile algebraic artefact of a classical knot,
although the high degree of non-uniqueness of Seifert surfaces has to be taken
into account in the applications.
According to Kervaire and Weber [134] the existence of Seifert surfaces for
high-dimensional n-knots ‘seems to have become public knowledge during the
Morse Symposium at Princeton in 1963. . . . It appears in print in Kervaire
[131] and Zeeman [316]’. (See also Kervaire [130, Appendix]). Seifert surfaces
for arbitrary high-dimensional homology framed knots were obtained by Erle
[71].
Every n-knot k : S n ⊂ S n+2 is of the form k = x (constructed as in the
section Local flatness above) for some non-locally flat knot : Gn+1 ⊂ S n+3
with unique singular point x ∈ G, as follows. Let i : F n+1 ⊂ Dn+3 be the
locally flat embedding of a codimension 2 Seifert surface obtained by pushing
a Seifert surface F n+1 ⊂ S n+2 for k into Dn+3 relative to the boundary
∂F = k(S n ). For x = 0 ∈ Dn+3 identify the cone x ∗ k(S n ) with Dn+1 , and
let
j : x ∗ k(S n ) = Dn+1 −−→ Dn+3
be the inclusion. The union
= i ∪ j : Gn+1 = F n+1 ∪∂ Dn+1 ⊂ Dn+3 ∪∂ Dn+3 = S n+3
defines a non-locally flat embedding with
x
= k.
The results of Levine [153], [156] show that for n ≥ 3 an n-knot k : S n ⊂
S
is unknotted (resp. null-cobordant) if and only if k admits a contractible
Seifert surface in S n+2 (resp. Dn+3 ). One way of applying surgery to knot
theory is to start with an arbitrary Seifert surface F for k, and then try to
modify the codimension 1 (resp. 2) submanifold F ⊂ S n+2 (resp. Dn+3 ) by
surgery5 , making F as contractible as possible.
n+2
5
See the section Codimension q surgery further below in the Introduction for the
basic definitions.
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Fibred knots and open books XXIII
There are two types of invariants for n-knots k : S n ⊂ S n+2 : the intrinsic
invariants of the infinite cyclic cover of the knot complement, and the extrinsic
ones associated with the Seifert surfaces. There is a similar distinction for the
invariants of arbitrary homology framed knots (M, N, k). The two types of
invariants determine each other, although in practice it is not always easy to
work out the details of the correspondence. One of the aims of this book is
to show how algebraic K- and L-theory can be used to define both types of
invariants, and also to relate them to each other.
Fibred knots and open books
A knot (M n+2 , N n , k) is fibred if it is homology framed and the canonical
projection p : X−−→S 1 on the exterior X = cl.(M \(k(N )×D2 )) is (homotopic
to) a fibre bundle, or equivalently if there exists a Seifert surface F n+1 ⊂ M
with a monodromy self homeomorphism h : F −−→F such that :
(i) h| = 1 : ∂F = k(N )−−→∂F ,
(ii) M = k(N ) × D2 ∪k(N )×S 1 T (h), with
T (h) = (F × [0, 1])/{(x, 0) ∼ (h(x), 1) | x ∈ F }
the mapping torus of h, so that X = T (h).
Following Winkelnkemper [311], M is called an open book with page F and
binding N .6
For the sake of simplicity only fibre bundles over S 1 and open books with
h∗ = 1 : π1 (F ) −−→ π1 (F )
will be considered, so that
π1 (M \k(N )) = π1 (X) = π1 (T (h)) = π1 (F ) × Z .
The discovery of exotic spheres by Milnor [190], the h-cobordism theorem
of Smale, and the surgery classification of exotic spheres in dimensions ≥ 5
by Kervaire and Milnor [133] were powerful incentives to the extension of
classical knot theory to knots (S n+2 , N n , k) (n ≥ 1), initially for differentiable
n-knots by Kervaire [130], [131]. It was proved in [130, Appendix] that an
exotic n-sphere Σ n admits a (differentiable) embedding k : Σ n ⊂ S n+2 if
and only if Σ n is the boundary of a parallelizable (n + 1)-manifold (e.g. a
Seifert surface F n+1 ⊂ S n+2 ), in which case Σ n represents an element of the
group bPn+1 of [133].
6
See the Appendix by Winkelnkemper for the history and applications of open
books.
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XXIV Introduction
Singular points of complex hypersurfaces (the title of Milnor [196]) provided a large supply of differentiable knots (S n+2 , Σ n , k) with Σ n an exotic
n-sphere : given complex variables z0 , z1 , . . . , zi and integers a0 , a1 , . . . , ai ≥ 2
there is defined a fibred knot (S 2i+1 , N, k) with
k : N 2i−1 = Σ(a0 , a1 , . . . , ai ) = V (f ) ∩ S 2i+1 ⊂ S 2i+1
where
f : C i+1 −−→ C ; (z0 , z1 , . . . , zi ) −−→ z0a0 + z1a1 + . . . + ziai ,
V (f ) = {(z0 , z1 , . . . , zi ) ∈ C i+1 | f (z0 , z1 , . . . , zi ) = 0} ,
S 2i+1 = {(z0 , z1 , . . . , zi ) ∈ C i+1 | |z0 |2 + |z1 |2 + . . . + |zi |2 = 1} .
The (2i − 1)-dimensional manifold N is (i − 2)-connected, and the fibre
of X = cl.(S 2i+1 \(N × D2 ))−−→S 1 is an (i − 1)-connected 2i-dimensional
Seifert surface F 2i ⊂ S 2i+1 . The precise topological and differentiable nature
of N 2i−1 ⊂ S 2i+1 has been the subject of many investigations, ever since
Brieskorn [26] identified certain N = Σ(a0 , a1 , . . . , ai ) as exotic spheres.
Fibred knots (M, N, k) play an especially important role in the development of high-dimensional knot theory. On the one hand, the complex hypersurface knots (S 2i+1 , N 2i−1 , k) are fibred, and the invariants of fibred
knots have many special features, e.g. the fibre is a minimal Seifert surface, with unimodular Seifert matrix, and the extreme coefficients of the
Alexander polynomials are ±1 ∈ Z. On the other hand, surgery theory
provided ways of recognizing algebraically if a high-dimensional homology
framed knot (M n+2 , N n , k) is fibred. In general, the fundamental group
π1 (X) = ker(p∗ : π1 (X)−−→Z) of the infinite cyclic cover X of the exterior
X = cl.(M \(k(N ) × D2 )) is infinitely generated, but for a fibred knot X is
homotopy equivalent to the fibre F and π1 (X) = π1 (F ) is finitely presented.
For an n-knot (S n+2 , S n , k)
H1 (X) = H1 (S 1 ) = Z , π1 (X) = [π1 (X), π1 (X)] .
Stallings [273] proved that a classical knot k : S 1 ⊂ S 3 is fibred if and only
if π1 (X) is finitely generated. Browder and Levine [31] proved that for n ≥ 4
a homology framed knot (M n+2 , N n , k) with π1 (X) = Z is fibred if and
only if H∗ (X) is finitely generated over Z. Farrell [78] and Siebenmann [267]
generalized this result to the non-simply-connected case, using the finiteness
obstruction of Wall [302] and Whitehead torsion : the fibering obstruction
Φ(Y ) ∈ W h(π1 (Y )) is defined for a finite CW complex Y with a finitely
dominated infinite cyclic cover Y (a band), and for n ≥ 4 a homology framed
knot (M n+2 , N n , k) is fibred if and only if the infinite cyclic cover X of the
exterior X is finitely dominated and Φ(X) = 0 ∈ W h(π1 (X)).
Winkelnkemper [311] and Quinn [227] applied surgery theory to investigate the existence and uniqueness of open book decompositions for manifolds
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Knot cobordism
XXV
of dimension ≥ 6, and the closely related bordism groups of diffeomorphisms
∆∗ . These results are reproved in Chap. 29 using the algebraic Poincar´e complexes of Ranicki [235], [236].
Knot cobordism
The connected sum of n-knots k1 , k2 : S n ⊂ S n+2 is the n-knot
k1 #k2 : S n #S n = S n ⊂ S n+2 #S n+2 = S n+2 .
An n-knot k : S n ⊂ S n+2 is slice if there exists an (n + 1)-dimensional
knot : Dn+1 ⊂ Dn+3 with | = k, or equivalently if k is null-cobordant. The
hypothesis of local flatness is crucial here, since the cone on k(S n ) ⊂ S n+2 is
a non-locally-flat null-cobordism Dn+1 ⊂ Dn+3 .
Fox and Milnor [82] defined the abelian group C1 of cobordism classes
of 1-knots k : S 1 ⊂ S 3 , with addition by connected sum, and the trivial
knot k0 : S 1 ⊂ S 3 as the zero element. The group C1 is countably infinitely
generated. The motivation for the definition of C1 came from the construction
of the 1-knots (S 3 , S 1 , kx ) (already recalled in the section Locally flat above)
from non-locally flat P L knots (M 4 , N 2 , κ) (x ∈ N ). It was proved in [82]
that the connected sum of 1-knots k1 , k2 , . . . , kr : S 1 ⊂ S 3 is slice if and only
if there exists a non-locally flat 2-knot κ : S 2 ⊂ S 4 with ki = kxi (1 ≤ i ≤ r)
the 1-knots defined at the points x1 , x2 , . . . , xr ∈ S 2 where κ is not locally
flat.
Kervaire [131] defined cobordism for n-knots k : S n ⊂ S n+2 for all n ≥ 1.
The group of cobordism classes of n-knots with addition by connected sum is
denoted by Cn . An n-knot k is such that [k] = 0 ∈ Cn if and only if k is slice.
If : N n+1 ⊂ M n+3 is a non-locally flat embedding with a single non-locally
flat point x ∈ N there is defined a (locally flat) n-knot k = kx : S n ⊂ S n+2
as before. The n-knot k is slice if and only if the singularity of at x can be
‘resolved’, with replaced near x by a locally flat embedding.
The algebraic determination of the knot cobordism groups C∗ was a major
preoccupation of high-dimensional knot theorists in the 1960s and 1970s. The
algebraic structure of Cn for n ≥ 3 was worked out in [131], Levine [156], [157]
and Stoltzfus [278], with
Cn = Cn+4 , C2i = 0
and both C4∗+1 and C4∗+3 are countably infinitely generated of the type
Z⊕
∞
Z2 ⊕
∞
Z4 .
∞
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XXVI
Introduction
(See Chap. 42 for the structure of C2i+1 for i ≥ 1). The classical knot cobordism group C1 is still fairly mysterious, with the kernel of the natural surjection C1 −−→C4j+1 known to be non-trivial, by virtue of the invariants of
Casson and Gordon [48].
Simple knots
An n-knot k : S n ⊂ S n+2 is simple if it satisfies one of the following equivalent
conditions :
(i) the knot complement X = S n+2 \k(S n ) is such that
πr (X) = πr (S 1 ) for 1 ≤ r ≤ (n − 1)/2 ,
(ii) k admits a Seifert surface F n+1 ⊂ S n+2 such that π1 (F ) = {1} and
Hr (F ) = 0 for 1 ≤ r ≤ (n − 1)/2 .
Every classical knot k : S 1 ⊂ S 3 is simple. For n ≥ 2 every n-knot k : S n ⊂
S n+2 is cobordant to a simple n-knot. The (2i − 1)-knots k : S 2i−1 ⊂ S 2i+1
constructed by Milnor [196] from singular points of complex hypersurfaces
are simple and fibred, with (i − 1)-connected fibre F 2i .
Seifert surfaces and simply-connected surgery theory were used by Kervaire [131] to characterize the homotopy groups π∗ (X) of the complements
X = S n+2 \k(S n ) of high-dimensional n-knots k : S n ⊂ S n+2 (see also Wall
[304, p. 18]), and to prove that Cn is isomorphic to the cobordism group of
simple n-knots, with C2i = 0 (i ≥ 2). Levine [155] obtained polynomial invariants of an n-knot k from the homology H∗ (X) of the infinite cyclic cover
X of X, generalizing the Alexander polynomial. The high-dimensional knot
polynomials were used in [155] to characterize the rational homology groups
H∗ (X; Q) as Q[z, z −1 ]-modules, and (working in the differentiable category)
were related to the exotic differentiable structures on spheres. Much work was
done in the mid-1960s and early 1970s on the isotopy classification of simple
n-knots S n ⊂ S n+2 , using the Alexander polynomials, the Seifert matrix and
the Blanchfield linking form – see Chaps. 33, 42 for references.
Surgery theory
The Browder–Novikov–Sullivan–Wall surgery theory developed in the 1960s
brought a new methodology to high-dimensional knot theory, initially for
n-knots S n ⊂ S n+2 and then for arbitrary codimension 2 embeddings
N n ⊂ M n+2 . Surgery theory was then extended to 4-dimensional manifolds with certain fundamental groups by Freedman and Quinn [86], but
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Surgery theory XXVII
low-dimensional manifolds have so many distinctive features that the highdimensional theory is too limited in dimensions 3 and 4 – accordingly, ndimensional knots N n ⊂ M n+2 are much harder to classify for n = 1, 2 than
n ≥ 3.
It will be assumed that the reader is already familiar with the basics of
surgery, at least in the simply-connected case considered by Browder [30].
The basic surgery operation on manifolds starts with an n-dimensional
manifold N n and an embedding
S r × Dn−r ⊂ N .
The effect of the surgery is the n-dimensional manifold
N
n
= (N \S r × Dn−r ) ∪ Dr+1 × S n−r−1
which is related to N by an elementary cobordism (W ; N, N ), with
W = N × [0, 1] ∪ Dr+1 × Dn−r .
Conversely, every cobordism of manifolds is a union of elementary cobordisms
(Milnor [192]) – for a closed manifold this is just a handle decomposition.
There is a similar decomposition for knot cobordisms.
An n-dimensional geometric Poincar´e complex X is a finite CW complex
with n-dimensional Poincar´e duality H ∗ (X) ∼
= Hn−∗ (X). The fundamental
problem of surgery is to decide if such an X is homotopy equivalent to a
compact n-dimensional manifold. The traditional method is to break down
the problem into two stages. In the first stage there is a topological K-theory
obstruction to the existence of a normal map7 (f, b) : M −−→X, that is a degree 1 map f : M −−→X from a compact n-dimensional manifold M together
with a map b : νM −−→η a map from the stable normal bundle of M to some
bundle over X. In the second stage there is an algebraic L-theory obstruction,
the surgery obstruction of Wall [304]
σ∗ (f, b) ∈ Ln (Z[π1 (X)])
such that σ∗ (f, b) = 0 if (and for n ≥ 5 only if) (f, b) : M −−→X can be
modified by surgeries on M to a normal bordant homotopy equivalence.8 The
algebraic L-groups L∗ (A) of [304] are defined for any ring with involution A,
and are 4-periodic, Ln (A) = Ln+4 (A). By construction, L2i (A) is the Witt
group of nonsingular (−)i -quadratic forms over A, and L2i+1 (A) is a group
of automorphisms of nonsingular (−)i -quadratic forms over A.
7
8
See the section Algebraic K- and L-theory invariants of knots further below in
the Introduction for the basic constructions of normal maps from n-knots.
See Ranicki [245] for a more streamlined approach, in which the two stages are
united in the total surgery obstruction.
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XXVIII Introduction
An n-dimensional quadratic Poincar´e complex over A is an A-module
chain complex C with a quadratic Poincar´e duality, H n−∗ (C) ∼
= H∗ (C). The
algebraic L-groups Ln (A) are the cobordism groups of quadratic Poincar´e
complexes over A, by the algebraic theory of surgery of Ranicki [235], [236],
[237]. The quadratic kernel of an n-dimensional normal map (f, b) : M −−→X
is an n-dimensional quadratic Poincar´e complex (C, ψ) with C = C(f ! ) the
algebraic mapping cone of the Umkehr Z[π1 (X)]-module chain map
f ! : C(X)
f∗
C(X)n−∗ −−→ C(M )n−∗
C(M )
with X the universal cover of X and M = f ∗ X the pullback cover of M , and
with a Z[π1 (X)]-module chain equivalence
C(M )
C(f ! ) ⊕ C(X) .
The surgery obstruction of (f, b) is the quadratic Poincar´e cobordism class
σ∗ (f, b) = (C(f ! ), ψ) ∈ Ln (Z[π1 (X)]) .
There is also the notion of geometric Poincar´e pair (X, ∂X), with a rel
∂ surgery obstruction σ∗ (f, b) ∈ Ln (Z[π1 (X)]) for a normal map (f, b) :
(M, ∂M )−−→(X, ∂X) from a manifold with boundary and with ∂f = f | :
∂M −−→∂X a homotopy equivalence.
Algebraic transversality
The main technique used in the book is algebraic transversality, an analogue
of the geometric transversality construction
F n+1 = p−1 (1) ⊂ M n+2
of a Seifert surface of a homology framed knot (M n+2 , N n , k) from a map
p : X−−→S 1 on the knot exterior X = cl.(M \(k(N )×D2 )) which is transverse
regular at 1 ∈ S 1 – cutting X along F results in a fundamental domain
(XF ; F, zF ) for the infinite cyclic cover X = p∗ R of X. The technique extracts
finitely generated A-module data from finitely generated A[z, z −1 ]-module
data, with A[z, z −1 ] the Laurent polynomial extension of a ring A.
Algebraic transversality is a direct descendant of the linearization trick of
Higman [110] for converting a matrix in A[z, z −1 ] by stabilization and elementary transformations to a matrix with linear entries a0 + a1 z (a0 , a1 ∈ A). As
explained by Waldhausen [301] and Ranicki [244] linearization corresponds
to the geometric transversality construction of a fundamental domain for
an infinite cyclic cover of a compact manifold. The algebraic transversality
methods used to prove the theorem of Bass, Heller and Swan [14] on the
Whitehead group of a polynomial extension
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