Algebraic and geometric surgery

229 Pages·2000·0.943 MB·English

by  Milgram R.J.

Algebraic and Geometric Surgery
R. James Milgram
Department of Mathematics, Stanford University, Stanford,
California, 94305
E-mail address: [email protected]
The author was supported in part by Grant from the National Science
Foundation.
Copyright 2000, R. James Milgram. Permission granted to reproduce for educational
purposes and distribute for cost of reproduction and distribution.

Contents
Chapter 1. Differentiable Manifolds 1
1. Differentiable maps on Euclidean spaces 1
2. Singular points and Morse functions 2
3. Differentiable manifolds and differentiable maps 7
4. Differentiable manifolds with boundary 9
Chapter 2. Bundles 13
1. Fibre bundles 13
2. Vector bundles 18
3. Associated bundles 22
4. Reduction of the group of a fibre bundle 23
5. Classifying spaces for fibre bundles 24
6. Thom spaces and transversality 29
Chapter 3. Immersions and embeddings 33
1. Embeddings and Immersions in Rm 33
2. Constructing immersions f: Nn−→Mm with m ≥ 2n 35
3. The Singular Set of an Immersion 36
4. Immersions of Sn in R2n 37
5. The Whitney trick 42
Chapter 4. Signature Invariants and characteristic classes 47
1. Some Two Dimensional Examples – the Fermat Surfaces 49
2. The Multisignature for Finite Group Actions 54
3. The Wall Embedding Theorem 56
Chapter 5. The Homology of Fibre Bundles 59
1. Four homology theories 59
2. The homology and cohomology of a covering 61
3. Poincar´e duality 68
4. The homology of a fibration 72
Chapter 6. Cobordism and Handle Decompositions 77
1. Foundations of cobordism 77
2. Handles 79
3. Lagrangians and Even Forms 84
iii
iv CONTENTS
Chapter 7. The Homotopy and PL Classification of Generalized Lens
Spaces 87
1. The homotopy classification of Generalized Lens Spaces 88
2. The Torsion of Chain Complexes with Trivial Homology 93
3. The Invariance of Torsion Under Cellular Subdivision 99
4. The Reidemeister-Whitehead Classification of Lens Spaces 101
Chapter 8. Surgery in Low Dimensions 103
1. Types of Modifications of Manifolds 103
2. Further Examples of the effect of surgery 106
3. The effect of surgery on homology 109
4. Normal maps and surgery below the middle dimension 112
5. Degree one normal maps. 113
Chapter 9. Surgery for simply connected manifolds 121
1. The case n ≡ 0 mod (4) 121
2. The case n = 4k+2 and π (M) = 0 123
1
3. Surgery on degree one maps for n odd and π (X) = 0 126
1
4. The geometric moves 128
5. Plumbing and the Browder-Novikov Theorem 130
6. The Browder-Novikov Theorem 136
7. The Arf-invariant for surgery in dimension 4k+2 is well defined138
8. The classification of homotopy spheres 139
Chapter 10. The Algebraic Analysis of Surgery Groups when
π (X) = 0 143
1
1. The Vanishing of the Odd Surgery Groups when π (X) = 0 143
1
2. Torsion forms and signature 150
3. The construction of an exact sequence for the surgery groups 159
Chapter 11. The Global Structure of Surgery when π (X) = 0 161
1
1. The Spivak normal fibration in the simply connected case 162
2. Browder’s extension of the Spivak normal bundle to π (X) 6= 0 165
1
3. The basic properties of homotopy sphere bundles 168
4. The Spivak bundle and degree one normal maps 171
Chapter 12. Surgery When π (X) 6= {1} 183
1
1. Preliminaries on Modules over Rings with Involutions 184
2. The Even Surgery Obstruction Groups 186
3. The Even Dimensional Surgery Obstruction 195
4. The Odd Dimensional Surgery Obstruction Groups 198
5. The Odd Dimensional Surgery Obstruction 204
6. Realizing the Surgery Obstructions 207
7. Surgery Exact Sequence and Bordism when π (X) 6= 0 211
1
Chapter 13. The Instant Surgery Obstruction and Product Formulae 215
1. Review of Iterated Loop Space Theory 215
CONTENTS v
2. The quadratic reduction of the surgery kernel 222
3. 222

CHAPTER 1
Differentiable Manifolds
1. Differentiable maps on Euclidean spaces
Consideradifferentiablemapf : Rn−→Rm. Thederivativeoff atx ∈ Rn
is the linear map defined by the Jacobian m × n matrix of first partial
derivatives
(cid:181) (cid:182)
df(x) = ∂fi : Rn −→ Rm ; h = (h ,h ,...,h ) −→
∂xj (cid:181) 1 2 n (cid:182)
(cid:80)n (cid:80)n (cid:80)n
df(x)(h) = ∂f1h , ∂f2h ,..., ∂fmh .
∂xj j ∂xj j ∂xj j
j=1 j=1 j=1
If g: Rm → Rs is a second differentiable map and gf: Rn → Rs is the
composition then
d(gf) = dgdf.
Definition 1. Let f : Rn → Rm be a differentiable map.
(1) A regular point of f is a point x ∈ Rn where the linear map df(x)
is of maximal rank, i.e.
rank(df(x)) = min(m,n) .
(2) A critical point of f is a point x ∈ Rn which is not regular.
(3) A regular value of f is a point in the image y ∈ Rm such that
every x ∈ f−1(y) ⊆ N is regular or f−1(y) is empty.
(4) A critical value of f is a point y ∈ Rn which is not regular.
Theorem 1. (Implicit Function) For any differentiable map f : Rn →
Rm and any regular point x ∈ Rn of f there exist a neighborhood N ⊂ Rn
x
and local coordinates (z ,...,z ) in N with x at (0,...,0), and a neighbor-
1 n x
hood N ⊂ Rm of f(x) with local coordinates (w ,...,w ) with f(x) ∼
f(x) 1 m
(0,...,0), such that :
• if n ≤ m, then
f(z ,...,z ) = (z ,...,z , 0,...,0 )
1 n 1 n (cid:124) (cid:123)(cid:122) (cid:125)
m−n times
• if n ≥ m then
f(z ,...,z ) = (z ,...,z ) .
1 n 1 m
1
2 1. DIFFERENTIABLE MANIFOLDS
Thus, in a neighborhood of a regular point the map either looks like an
embedding or a projection. In particular, the set of regular points is open
in Rn.
As regards the regular and singular values of f we have the fundamental
theorem of Sard :
Theorem 2. (Sard) The set of singular values of f has measure 0 in
Rm for any C∞ map f: Rn → Rm.
In the case where n < m this says that the image of f cannot be some-
thing like a space filling curve. But in the case where n ≥ m it is even more
restrictive. For example the implicit function theorem immediately implies
:
Corollary 1. Let f: Rn → Rn be any C∞ map for which the measure
of im(f) > 0. Then there is a regular value, y ∈ Rn of f, and f−1(y) is a
discrete set in Rn.
In particular, if f: Rn → Rn is such that im(f) contains an open set in
Rn then im(f) has measure > 0.
More generally, the implicit function theorem 1 gives :
Corollary 2. Let f: Rn → Rm be any C∞ map for which the measure
of im(f) > 0, with n ≥ m. Then there is a regular value y ∈ Rm for
f and f−1(y) has the property that for each x ∈ f−1(y) there is an open
neighborhood N ⊂ Rn of x and local coordinates (z ,...,z ) there so that
x (cid:80) 1 n
f−1(y)∩N = {(z ,...,z ,0,...,0)} with mz2 < †.
x 1 m 1 i
2. Singular points and Morse functions
In a neighborhood of a singular point things are much more complex.
Indeed, in general the situation is far from being understood. However, in
the extreme case where f is a function f: Rn → R we have a fairly good
understandingofwhathappens–atleastwhenthesingularpointisisolated!
The Taylor expansion through degree k of a differentiable function f :
Rn−→R at x ∈ Rn is given by
(cid:80)k (cid:80)
f(x+h) = f(x)+ 1 ∂jf h h ...h
j=1 j! 1≤i1,i2,...,ij≤n ∂xi1∂xi2...∂xij i1 i2 ij
+O(|h|k+1) ∈ R
(x = (x ,x ,...,x ),h = (h ,h ,...,h ) ∈ Rn) ,
1 2 n 1 2 n
so that
f(x+h) = f(x)+df(x)(h)+... .
The linear term is determined by the gradient vector
(cid:181) (cid:182)
∂f ∂f ∂f
df(x) = ... ∈ Rn ,
∂x ∂x ∂x
1 2 n
2. SINGULAR POINTS AND MORSE FUNCTIONS 3
corresponding to the linear map
(cid:88)n ∂f
df(x) : Rn → R ; h = (h ,h ,...,h ) → h ,
1 2 n i
∂x
i
i=1
which is either 0 or has the maximal rank 1. Thus x ∈ Rn is a regular point
of f if and only if df(x) 6= 0, and x ∈ Rn is a critical point if df(x) = 0.
The quadratic term in the Taylor expansion is the quadratic function of the
symmetric bilinear form (Rn,λ) defined by the Hessian matrix of second
partial derivatives
(cid:181) (cid:182)
λ = ∂2f : Rn×Rn → R ;
∂x1∂x2
(cid:80)
(u,v) = ((u ,u ,...,u ),(v ,v ,...,v )) → ∂2f u v ,
1 2 n 1 2 n ∂xi∂xj i j
1≤i,j≤n
namely
µ : Rn → R ;
(cid:80)
h = (h ,h ,...,h ) → λ(h,h)/2 = 1 ∂2f h h .
1 2 n 2 ∂xi∂xj i j
1≤i,j≤n
We say that the critical point x of the function f: Rn → R is non-
degenerate if the determinant of the Hessian matrix
(cid:181) (cid:182)
∂2f
H (x) =
f
∂x ∂x
i j x
is non-zero, Det(H (x)) 6= 0. Note that if we change coordinates near x so
f
(x ,...,x ) = K(y ,...,y )
1 n 1 n
= (K (y ,...,y ),...,K (y ,...,y ))
1 1 n n 1 n
with dK invertible at x, then
∂2(fh) (cid:88) ∂2f ∂K ∂K
r s
=
∂y ∂y ∂x ∂x ∂y ∂y
i j r s i j
r,s
∂f
since = 0 at x (1 ≤ i ≤ n). Consequently,
∂x
i
H = (dK)tH dK
fk f
and H actually transforms like a symmetric bilinear form. In particular :
f
(1) non-degeneracy for the critical point x is invariant under local co-
ordinate changes,
(2) since we can diagonalize a symmetric matrix by an orthogonal ma-
trix, T, there is an orthogonal transformation centered at x which