ALGEBRAIC AND GEOMETRIC SURGERY by Andrew Ranicki

374 Pages·2014·1.85 MB·English

ALGEBRAIC AND GEOMETRIC SURGERY
by Andrew Ranicki
Oxford Mathematical Monograph (OUP), 2002
This electronic version (March 2014) incorporates the errata which were
included in the second printing (2003) as well as the errata found subsequently,
and some additional comments. Note that the pagination of the electronic
version is somewhat different from the printed version. The list of errata is
maintained on http://www.maths.ed.ac.uk/˜aar/books/surgerr.pdf
For Frank Auerbach
CONTENTS
Preface vi
1 The surgery classification of manifolds 1
2 Manifolds 14
2.1 Differentiable manifolds 14
2.2 Surgery 16
2.3 Morse theory 18
2.4 Handles 22
3 Homotopy and homology 29
3.1 Homotopy 29
3.2 Homology 32
4 Poincar´e duality 48
4.1 Poincar´e duality 48
4.2 The homotopy and homology effects of surgery 53
4.3 Surfaces 61
4.4 Rings with involution 66
4.5 Universal Poincar´e duality 71
5 Bundles 85
5.1 Fibre bundles and fibrations 85
5.2 Vector bundles 89
CONTENTS iii
5.3 The tangent and normal bundles 105
5.4 Surgery and bundles 112
5.5 The Hopf invariant and the J-homomorphism 118
6 Cobordism theory 124
6.1 Cobordism and transversality 124
6.2 Framed cobordism 129
6.3 Unoriented and oriented cobordism 133
6.4 Signature 135
7 Embeddings, immersions and singularities 143
7.1 The Whitney Immersion and Embedding Theorems 143
7.2 Algebraic and geometric intersections 149
7.3 The Whitney trick 156
7.4 The Smale-Hirsch classification of immersions 161
7.5 Singularities 167
8 Whitehead torsion 170
8.1 The Whitehead group 170
8.2 The h- and s-Cobordism Theorems 175
8.3 Lens spaces 185
9 Poincar´e complexes and spherical fibrations 193
9.1 Geometric Poincar´e complexes 194
9.2 Spherical fibrations 198
9.3 The Spivak normal fibration 205
iv CONTENTS
9.4 Browder-Novikov theory 209
10 Surgery on maps 218
10.1 Surgery on normal maps 220
10.2 The regular homotopy groups 226
10.3 Kernels 230
10.4 Surgery below the middle dimension 238
10.5 Finite generation 240
11 The even-dimensional surgery obstruction 246
11.1 Quadratic forms 246
11.2 The kernel form 255
11.3 Surgery on forms 280
11.4 The even-dimensional L-groups 287
11.5 The even-dimensional surgery obstruction 294
12 The odd-dimensional surgery obstruction 301
12.1 Quadratic formations 301
12.2 The kernel formation 305
12.3 The odd-dimensional L-groups 316
12.4 The odd-dimensional surgery obstruction 319
12.5 Surgery on formations 322
12.6 Linking forms 332
13 The structure set 338
13.1 The structure set 338
CONTENTS v
13.2 The simple structure set 342
13.3 Exotic spheres 344
13.4 Surgery obstruction theory 356
References 361
PREFACE
Surgery theory is the standard method for the classification of high-dimen-
sional manifolds, where high means (cid:62)5. The theory is not intrinsically difficult,
butthewidevarietyofalgebraicandgeometrictechniquesrequiredmakesheavy
demands on beginners. Where to start?
Thisbookaimstobeanentrypointtosurgerytheoryforareaderwhoalready
has some background in topology. Familiarity with a book such as Bredon [10]
or Hatcher [31] is helpful but not essential. The prerequisites from algebraic and
geometric topology are presented, along with the purely algebraic ingredients.
Enough machinery is developed to prove the main result of surgery theory: the
surgery exact sequence computing the structure set of a differentiable manifold
M of dimension (cid:62)5 in terms of the topological K-theory of vector bundles over
M and the algebraic L-theory of quadratic forms over the fundamental group
ring Z[π (M)]. The surgery exact sequence is stated in Chapter 1, and finally
1
provedinChapter13.Alongtheway,therearebasictreatmentsofMorsetheory,
embeddings and immersions, handlebodies, homotopy, homology, cohomology,
Steenrod squares, Poincar´e duality, vector bundles, cobordism, transversality,
Whitehead torsion, the h- and s-Cobordism Theorems, algebraic and geometric
intersections of submanifolds, the Whitney trick, Poincar´e complexes, spherical
fibrations,quadraticformsandformations,exoticspheres,aswellasthesurgery
obstruction groups L (Z[π]).
∗
Thistextintroducessurgery,concentratingonthebasicmechanicsandwork-
ing out some fundamental concrete examples. It is definitely not an encyclope-
diaofsurgerytheoryanditsapplications.Manyresultsandapplicationsarenot
covered,includingsuchimportantitemsasNovikov’stheoremonthetopological
invarianceoftherationalPontrjaginclasses,surgeryonpiecewiselinearandtopo-
logicalmanifolds,thealgebraiccalculationsoftheL-groupsforfinitegroups,the
geometriccalculationsoftheL-groupsforinfinitegroups,theNovikovandBorel
conjectures, surgery on submanifolds, splitting theorems, controlled topology,
knots and links, group actions, stratified sets, the connections between surgery
and index theory, ... . In other words, there is a vast research literature on
surgery theory, to which this book is only an introduction.
The books of Browder [14], Novikov [65] and Wall [92] are by pioneers of
surgery theory, and are recommended to any serious student of the subject.
However, note that [14] only deals with the simply-connected case, that only a
relatively small part of [65] deals with surgery, and that the monumental [92] is
CONTENTS vii
notoriouslydifficultforbeginners,probablyevenwiththecommentaryIhadthe
privilegetoaddtothesecondedition.ThepaperscollectedinFerry,Ranickiand
Rosenberg [24], Cappell, Ranicki and Rosenberg [17] and Farrell and Lu¨ck [23]
give a flavour of current research and include many surveys of topics in surgery
theory, including the history. In addition, the books of Kosinski [42], Madsen
andMilgram[45],Ranicki[70],[71],[74]andWeinberger[94]provideaccountsof
various aspects of surgery theory.
On the afternoon of my first day as a graduate student in Cambridge, in
October, 1970 my official supervisor Frank Adams suggested that I work on
surgerytheory.Thisisstillsurprisingtome,sincehewasaheavydutyhomotopy
theorist.Inthemorninghehadindeedproposedthreetopicsinhomotopytheory,
but I was distinctly unenthusiastic. Then at tea-time he said that I might look
at the recent work of Novikov [64] on surgery theory and hamiltonian physics,
draining thephysics outto see what mathematics was left over. Novikov himself
had not been permitted by the Soviet authorities to attend the Nice ICM in
September, but Frank had attended the lecture delivered on Novikov’s behalf
byMishchenko.Themathematicsandthecircumstancesofthelecturedefinitely
sparked my interest. However, as he was not himself a surgeon, Frank suggested
that I actually work with Andrew Casson. Andrew explained that he did not
have a Ph.D. himself and was therefore not formally qualified to be a supervisor
of a Ph.D. student, though he would be willing to answer questions. He went on
tosaythatinanycasethiswasthewrongtimetostartworkonhigh-dimensional
surgery theory! There had just been major breakthroughs in the field, and what
was left to do was going to be hard. This brought out a stubborn streak in me,
and I have been working on high-dimensional surgery theory ever since.
Itisworthremarkingherethatsurgerytheorystartedin1963withtheclassi-
ficationbyKervaireandMilnor[38]oftheexoticspheres,whicharethedifferen-
tiablemanifoldswhicharehomeomorphicbutnotdiffeomorphictothestandard
sphere. Students are still advised to read this classic paper, exactly as I was
advised to do by Andrew Casson in 1970.
This book grew out of a joint lecture course with Jim Milgram at G¨ottingen
in 1987. I am grateful to the Leverhulme Trust for the more recent (2001/2002)
Fellowship during which I completed the book. I am grateful to Markus Banagl,
Jeremy Brookman (who deserves special thanks for designing many of the dia-
grams), Diarmuid Crowley, Jonathan Kelner, Dirk Schuetz, Des Sheiham, Joerg
Sixt,ChrisStark,IdaThompsonandShmuelWeinbergerforvarioussuggestions.
Any comments on the book subsequent to publication will be posted on the
website
http://www.maths.ed.ac.uk/ aar/books
(cid:101)
2nd June, 2002
1
THE SURGERY CLASSIFICATION OF MANIFOLDS
Chapter 1 is an introduction to the surgery method of classifying manifolds.
Manifolds are understood to be differentiable, compact and closed, unless oth-
erwise specified.
A classification of manifolds up to diffeomorphism requires the construction
of a complete set of algebraic invariants such that:
(i) the invariants of a manifold are computable,
(ii) twomanifoldsarediffeomorphicifandonlyiftheyhavethesameinvariants,
(iii) there is given a list of non-diffeomorphic manifolds realizing every possible
set of invariants.
Onecouldalsoseekahomotopyclassificationofmanifolds,askingforacomplete
set of invariants for distinguishing the homotopy types of manifolds. Diffeomor-
phic manifolds are homotopy equivalent.
ThemostimportantinvariantofamanifoldMm isitsdimension,thenumber
m (cid:62) 0 such that M is locally diffeomorphic to the Euclidean space Rm. If
m=n then Rm is not diffeomorphic to Rn, so that an m-dimensional manifold
(cid:54)
Mm cannot be diffeomorphic to an n-dimensional manifold Nn. The homology
and cohomology of an orientable m-dimensional manifold M are related by the
Poincar´e duality isomorphisms
H∗(M) ∼= Hm (M) .
−∗
Any m-dimensional manifold M has Z -coefficient Poincar´e duality
2
H∗(M;Z2) ∼= Hm (M;Z2) ,
−∗
with
H (M;Z ) = Z , H (M;Z ) = 0 for n>m .
m 2 2 n 2
ThedimensionofamanifoldM isthuscharacterisedhomologicallyasthelargest
integer m (cid:62) 0 with H (M;Z ) = 0. Homology is homotopy invariant, so that
m 2
(cid:54)
thedimensionisalsoahomotopyinvariant:ifm=nanm-dimensionalmanifold
(cid:54)
Mm cannot be homotopy equivalent to an n-dimensional manifold Nn.
2 THE SURGERY CLASSIFICATION OF MANIFOLDS
There is a complete diffeomorphism classification of m-dimensional mani-
folds only in the dimensions m = 0,1,2, where it coincides with the homotopy
classification. For m (cid:62) 3 there exist m-dimensional manifolds which are homo-
topyequivalentbutnotdiffeomorphic,sothatthediffeomorphismandhomotopy
classificationsmustnecessarilydiffer.Form=3completeclassificationsarethe-
oretically possible, but have not been achieved in practice – the Poincar´e con-
jecturethatevery3-dimensionalmanifoldhomotopyequivalenttoS3 isactually
diffeomorphic to S3 remains unsolved!
For m (cid:62) 4 group-theoretic decision problems prevent a complete classifica-
tion of m-dimensional manifolds, by the following argument. Every manifold M
canbetriangulatedbyafinitesimplicialcomplex,sothatthefundamentalgroup
π (M)isfinitelypresented.Homotopyequivalentmanifoldshaveisomorphicfun-
1
damentalgroups.Everyfinitelypresentedgrouparisesasthefundamentalgroup
π (M) of an m-dimensional manifold M. It is not possible to have a complete
1
setofinvariantsfordistinguishingtheisomorphismclassofagroupfromafinite
presentation. Group-theoretic considerations thus make the following questions
unanswerable in general:
(a) Is M homotopy equivalent to M ?
(cid:48)
(b) Is M diffeomorphic to M ?
(cid:48)
since already the question
(c) Is π (M) isomorphic to π (M )?
1 1 (cid:48)
is unanswerable in general.
Thesurgerymethodofclassifyingmanifoldsseekstoansweradifferentques-
tion:
Given a homotopy equivalence of m-dimensional manifolds f : M M is f
(cid:48)
→
homotopic to a diffeomorphism?
Every homotopy equivalence of 2-dimensional manifolds (= surfaces) is ho-
motopictoadiffeomorphism,bythe19thcenturyclassificationofsurfaceswhich
is recalled in Chapter 3.
A homotopy equivalence of 3-dimensional manifolds is not in general homo-
topic to a diffeomorphism. The first examples of such homotopy equivalences
appeared in the classification of the 3-dimensional lens spaces in the 1930’s: the
Reidemeister torsion of a lens space is a diffeomorphism invariant which is not
homotopy invariant. Algebraic K-theory invariants such as Reidemeister and