Manifolds — Amsterdam 1970: Proceedings of the Nuffic Summer School on Manifolds Amsterdam, August 17–29, 1970

235 Pages·1971·4.15 MB·English

by  R. C. Kirby

Lecture Notes ni
Mathematics
A collection of informal reports dna seminars
detidE yb .A Heidelberg Dold, dna .B ,nnamkcE Z0rich
197
Manifolds - Amsterdam 1970
Proceedings of the Nuffic Summer School on Manifolds
Amsterdam, August 17-29, 1970
Edited yb Nicotaas .H Kuiper
Universiteit nav Amsterdam/Nederland Amsterdam,
galreV-regnirpS
Berlin.Heidelberg. New York 1791
AMS Subject Classifications (1970): 57Axx, 57Bxx, 57Cxx, 57Dxx, 58C25, 58D05, 58D 51
ISBN 7-76450-045-3 Springer Verlag Berlin • Heidelberg • New York
ISBN 7-76450-783-0 Springer Verlag New York • Heidelberg • Berlin
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© by Springer-Vertag Berlin • Heidelberg 1971. Library of Congress Catalog Card Number 74-160173. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach
PREFACE
These are the proceedings of the International Summer School on
Manifolds organized by the Netherlands Universities Foundation for
International Cooperation (NUFFIC), The Hague, and held at the
Mathematical Institute of the University of Amsterdam, August 17-29,
1970.
Also included is a set of problems presented by the lecturers and
members, and discussed in a special problem session.
The conference was attended by one hundred participants, specialists
as well as students, from various countries of the world.
The goal of the summer school was a survey and introduction to the
theory of manifolds, a field which is in a tremendous development
at present. The speakers, in particular those who gave series of
lectures, are most heartily thanked for the great effort they have
made to meet this goal in thelr lectures, as well as in the manu-
scripts they have sent in.
Finally I thank all persons who have cooperated so intensely to
make the summer school and these proceedings a success.
Nicolaas H. Kuiper
Amsterdam, December 1970
CONTENTS
Geometric Topology
Kirby, R.C. and Siebenmann, L.C.: Some Theorems on Topological
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .
Glaser, L.C.: On Suspensions of Homology Spheres ........
Algebraiq Topology
Browder, W.: Manifolds and Homotopy Theory . . . . . . . . . . . 17
Hsiang, W.C.: Manifolds with ~I =~k . . . . . . . . . . . . . . 36
Sullivan, D.: Geometric Periodicity and the Invariants of
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Differentiable Maps
Cerf, J.: The Pseudo-Isotopy Theorem for Simply Connected
Differentiable Manifolds . . . . . . . . . . . . . . . . . . 76
Kervaire, M.A.: Knot Cobordism in Codimension Two ........ 83
Po@naru, V.: Homotopy Theory and Differentiable Singularities . .106
Haefliger, A.: Homotopy and Integrability . . . . . . . . . . . . 133
Hausmann, J.Cl.: Extension d'une Homotopie A un Feuilletage
Topologique (Appendice aux Expos@s de A.Haefliger et
V. Po@naru) . . . . . . . . . . . . . . . . . . . . . . . . . 164
Singularities
Mather, J.N.: Stable Map-Germs and Algebraic Geometry . . . . . .176
Smale, S.: Problems on the Nature of Relative Equilibria in
Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . 194
Shub, M.: Appendix to Smale's Paper: Diagonals and Relative
Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 199
Thom, R.: The Bifurcation Subset of a Space of Maps ....... 202
Takens, F.: On Zeeman's Tolerance Stability Conjecture ..... 209
Problems Concerning Manifolds . . . . . . . . . . . . . . . . . . 220
A Solution (by F. Takens) . . . . . . . . . . . . . . . . . . . . 231
SOME THEOREMS ON TOPOLOGICAL MANIFOLDS
R.C. Kirby and L.C. Siebenmann
We list here some theorems about (metrlzable) topological manifolds, most of which
were announced in [7], [8], with proofs to appear in [9], [10], [11]. In addition,
see [12], [13], D5], [6].
First is the theorem on existence and uniqueness of PL (manifold) structures, which
settles the triangulation problem and the Hauptvermutung for manifolds. (Cohomo!ogy
will be Cech cohomology throughout,)
Theorem :I Let Qq be a q-dimensional topological manifold and let C be a closed
subset of Q. Suppose that a neighborhood of C has a PL structure 0. Z Let q ~ 6,
or q ~ 5 if ~Q~ C.
If Hh(Q, C; Z )2 = O, then Q has a PL structure Z which agrees with 0 Z near C.
Given the PL structure Z, then the PL structures (up to isotopy rel C) on Q
which agree with Z near C are in one to one correspondence with the elements of
H3(Q, C; Z2).
Definition: Let Z and ~ be two PL structures on Q. Then Z and ~ are said
to be equivalent (up to homotopy, or up to isotopy) if there exists a PL homeomor-
phism :f QZ ÷ Q~ (which is homotopic, or isotopic, to the identity). In the
relative case, Z =8 near C and the homotopy, or isotopy, fixes a neighbourhood
of C, and respects both BQ and Q-C.
S livan proved the following theorem on =iqueness up to homotopy ['], [17]
Theorem 2: For the data of Theorem I suppose H4(Q, C; Z) has no 2-torsion.
Suppose s~so that Q is compact and C is a eodimension I (locally flat) submani-
fold and suppose that for each connected component 'Q of Q-C either
(i) ~Q' is connected and w1~Q' ~ I~ 'Q by inclusion, or
(ii) 8Q' = @ and ln Q' = O.
Then any two PL structures on Q which agree near C are homotopic (rel C).
The two uniqueness theorems are related as follows: let :8 H3(Q, 2) ÷ C; Z Hh(Q, C;Z)
be the Bockstein homomorphism coming from the exact sequence of coefficients
0 ÷ Z ~ 2 ~ Z 0 ÷ Z (which is the sequence O ~ w4(G/PL) + ~4(G/TOP) ÷ ~3(TOP/PL) + 0).
Assume the conditions @f Theorem .2 Then if Q has PL structures Z and
corresponding to [Z], ]9~[ ~ H3(Q, C; Z2) , then Z and ~ are equivalent (rel C)
up to homotopy iff 8([Z]) = (6 le[ .)
This fails however for × 1 3 × S S I S which has two PL structures up to isomorphism
homotopy or isotopy.
The uniqueness half of Theorem I fails for 3-manifolds, where PL structures unique
up to isotopy (Moise). But it is a reasonable conjecture that Theorem I holds whenever
dim Q # 3 # dim ~Q.
Theorem I is elucidated by two more detailed theorems 3 and h below, which also deal
with DIFF (smooth C ~) manifold structures.
Let CAT(q) be the semi-simplieial group of CAT automorphisms of q R fixing the
origin. Here CAT = DIFF, PL or TOP. Recall that DIFF(q) ~ Ol(q) ~ O(q), [lh~.
Theorem 3. [7], . [13], ] [15],111 Let r ~ 5 throughout. Then the stabilization map
:s ~k(TOP(r), PL(r)) ÷ Wk(TOP(r+1), PL(r+I)) is an isomorphism. Further
~k(TOP(r), PL(r)) = #k(TOP, PL) is zero if k # 3 and is 2 Z if k = 3. For DIFF
in place of PL, the stabilization s is bijective if k ~ r+1, and surjective if
k = r+2 (using Cerf's pseudo-isotopy result).
Thus, for example, the classifying map :f Q + BTOP(q) for the tangent q-micro-
bundle rQ of Q lifts to hBpL(q) if and only if an obstruction in
Hh(Q; w3(TOP(q), PL(q))) = H~(Q; 2) Z is zero. By the classification theorem below,
this is the obstruction of Theorem I to imposing a PL manifold structure on .Q
Let y: BCAT(q) + BTOP(q) be the map of classifying spaces for CAT and TOP
q-mierobundles that corresponds to forgetting CAT structure. Form a triangle
)q(TAeB ~ )q(POTB
\ i
BCAT(q)
where i is a homotopy equivalence and 'y is a Hurewicz fibration. For example
B~AT(q) can be the space of maps (BTOP(q) ' BCAT(q))([0,1], O)
Consider a TOP q-manifold Q, a closed subset C CQ and a CAT structure 0 Z
on a neighbourhood of C. Define Cat(Q rel C; ~0 ) to be the (semi-simplicial, Kan)
complex of which a typical d-simplex is a CAT structure F on d x Q A such that
projection onto d A is a CAT submersion ~) and F = A dx ~0 near Ad x C. If
C = ¢, we write Cat (Q).
Similarly, let Lift(f rel C, fo ) be the (semi-simplicial, Kan) complex of lifts
of :f Q ÷ BTO (q) P to maps f': Q ~ B~A T (q), where 'f agrees near C with a fixed
lifting fo of f near C induced by 2 .0
~) A map f: X ÷ Y of CAT manifolds is a CAT submersion if f-1(y) is a CAT sub-
manifold of X for each y in Y and for each point x ~ X there exists an open
neighbourhood V of f(x) in Y, an open neighbourhood U of x in the submanifold
f-lf(x) and a CAT isomorphism $ of U x V onto an open neighbourhood of x in X
such that f$(u, v) = v for each (u, v) in U x V.
Theorem h. (Classification Theorem) [10]. There is a natural map (defined up to
homotopy) f: Cat (Q rel C, Z O) ÷ Lift (f rel C, fo ) that is a homotopy equivalence
if q # h and ~Q c C.
If ~Q ~ C but q => 6, Lift (f rel C, fo ) must be replaced by a new more complica-
ted complex that takes account of liftings of the map ~Q + BTOP(q_ ) I classifying
TaQ; also q must be > 6. But happily, for q > 6, the new complex has the same WO'
and in case CAT = PL it is actually equivalent to Lift (f rel C, fo ) by Theorem 3.
Lashof [12], Morlet [15], and C. Rourke have also proved versions of the classifica-
tion theorem. We discuss ingredients of the proof below.
Let F be a CAT structure on I x Q and let 0 x Z be its restriction to 0 x Q.
Suppose F = I x Z near I × C.
Theorem 5. (Concordance implies isotopy) [9]. Let q _> 6, or q > 5 if ~Q ~ C.
There exists an isotopy ht: I × Q + I x Q, t ~ ,0[ I], fixing O × Q and a
neighbourhood of I x C, such that 0 = h identity and hi: I x QZ ÷ (I x Q)F is a
CAT isomorphism.
Furthermore t h can be chosen arbitrarily close to the identity.
Theorem .6 (Sliced concordance implies isotopy) [9]. Suppose q # h, and q # 5 if
~Q ~9 C. Also suppose the projection (I x Q)F ~ I is a CAT submersion. Then t h
exists as in Theorem .5 Furthermore ht(s x Q) = s x Q for all s ~ I. This assertion
d
holds if, more generally, a pair (A , A) replaces the pair (I, O), where d A is
the standard d-simplex and A is a contractible subcomplex such that
F I
A x Q = A x -7 •
Theorem 6 readily implies the sliced concordance extension theorem:
Theorem 7. Let Q' be an open subset of Q. Suppose q # h, and q # 5 if ~Q # @.
Then the restriction map Cat(Q) ÷ Cat(Q' ) is a Kan fibration.
Theorems 5 and 6 are established first for ~Q = ~, by decomposing ~ into small
handles and then applying inductively a version where Q is an open k-handle
k x R R n, q = k+n, and k - C = (R int B k) n. x R This version differs in that I h
needs to be a CAT imbedding only near k x I B x (R n) and the smallness condition is
replaced by the condition that t h fix all points outside a compactum (which is
independent of t .)
The handle version of Theorem 5 is most efficiently established by use of a variant
of the Main Diagram of ]7[ (see also [9], [6]) which uses only the s-cobordism
theorem. In case CAT = PL (but not DIFF), the Alexander isotopy device (invalid
for DIFF) and the s-cobordism theorem can be used quite directly in the Main Diagram
to strengthen Theorem 5 to read: Given the data of Theorem 5, the semi-simpllcial
space of concordances of the given structure Z on Q, rel I x C, is contractible
(a d-simplex is a PL structure F on d x I x A Q such that projection on d A is
a submersion, F AI d x O d x x Q 0 = x A E, and F equals d x I A x E near AdxIxC).
For the handle version of Theorem 6, first note that the (many parameter) TOP iso-
topy extension theorem will deduce it from the statement that k d x R x B (A n)F is
CAT isomorphic to d x 6 (B k x En)z by a map respecting projection to &d. This
statement follows from a similar isomorphism of 6( n k d _ x x B (R O))F with
d x 8 (B n k _ x (R 0)) Z which is established via a useful technical lemma.
Lemma. (This lemma will hold for CAT equal TOP as well as PL or DIFF.)
Consider a CAT manifold E with two ends equipped with
a) a CAT submersion p: d E ~ A (the leaves F ~ p-1(u), u ~ A d, are CAT mani-
u
folds, possibly with boundary),
b) a propereontinuous #: E ~ R such that for each pair of integers a, b with
a < b the prelmage Fu(a, b) ~ (# I Fu I-) (a, b) ~ F u~ -I (a, b) of the open inter-
val (a, b) of real numbers is a CAT product of a compact manifold with R, or at
least has the following engulfing property:
(a) Fu(a , b) has two ends E , C+ and if U_, U+ are given open neighborhoods
of E , e+, then there exists a CAT self-isomorphism h of Fu(a , b) fixing points
outside some compactum such that h(U ~ ) U+ = F (a, b).
- u
Then p: d E + A is a CAT product bundle.
To prove this one can apply the d-parameter CAT isotopy extension theorem and an
elementary engulfing argument to prove a "global, sliced" version of (*) namely the
version with E in place of Fu(a , b), where h is supposed to respect each u- F
Then glue together the ends of E as in [181 to deduce the result from the known
fact that every proper CAT submersion is a CAT bundle map.
We remark that Theorems h, 6, 7 shun dimension h precisely because for CAT = DIFF
or PL we are unable to verify (*) given that Fu(a , b) is topologically 3 x R. S
The classification Theorem now follows from the immersion theory machine [2].
Theorem 7 is the key tool~ with it, the machine works easily, establishing the homo-
topy equivalence handle by handle, once we observe that the classification theorem
holds for zero-handles. But this amounts to observing that the complex Cat q, O) (R
of CAT manifold structures on q R respecting the origin is identical to the complex
of CAT mierobundle structures on the trivial q R bundle over a point, which in
turn is TOP(q)/CAT(q). This proof of Theorem I can be regarded as a semi-simplicial
version (with improvements) of the handle-by-handle argument sketched in [7].
A very different method of proving theorem I involves a stable version of the
classification theorem.
By Milnor's argument in [lh] a lifting Q + BCA T of the stable classifying map
Q + BT0 P of TQ gives a CAT structure on z Q x R for some z; and concordance
classes of such liftings correspond to concordance classes of CAT structures on
Q × R ,z Application of the concordance-implies-lsotopy Theorem (Theorem 5) and the
Product Structure Theorem below finish this proof of Theorem .I
Theorem 8. (Product Structure.Theorem .) Let q __> 6 or q = 5 if ~Q c C, and
let Z0 be a CAT structure near C. Let 69 be a CAT structure on Q s x R which
agrees with ZO s x R near C S. × R Then Q has a CAT structure ~, extending ~0
near C, so that ~ s x R is concordant to ~9 modulo C s. x R
Moreover, there is an s-isotopy ht: QZ x RI~ (Q x RS~ with 0 = h identity, h! CAT,
and t = h identity near C s × . R Here c: Q s + x R R+ is a given continuous function.
Note that the theorem fails for closed B-manifolds; e.g. B 2 x S R has two PL
structures but B S has only one.
The Product Structure Theorem is equivalent to the Concordanee-implies-isotopy
Theorem plus the Annulus Theorem [5]; the equivalence is not too hard to prove [9].
The classical PL-DiFF Product Structure Theorem (Cairns-Hirsch Theorem) ]3[ follows
easily from the TOP-CAT versions of the Product Structure Theorem and Concordance-
implies-isotopy Theorem. By using the strengthened Concordance-implies-isotopy
Theorem in addition, we can also recover the PL-DIFF version of Concordance-implies-
isotopy [h], [16].
To be sure,we land up with the same restrictions to high dimensions that we have for
the TOP-CAT versions, whereas in fact no dimension restrictions are necessary [4].
The Product Structure Theorem is particularly significant because of Theorems 9, 10,
and 11 below which follow easily, (see [11], [6]).
Theorem 9. Let m M be a TOP manifold. Then M has a well defined simple homo-
topy type (infinite if M is non-compact [191 ) which agrees with the usual definition
if M is PL or is a handlebody. This implies that the Whitehead torsion of a
homeomorphism is zero [7].
Theorem 10. (Transversality). Let ~n : E(~n) ~ X be TOP n R bundle over a
topological space X and let m f: ~ M E(~ n) be a continuous function. Then if m # 4
m-n#4 and aM = ~, f is homotopic to a map fl which is transverse to the zero-
I fl ~ossibly em1~ty}
section of ~ . This means - (0-section) is an (m-n)-manifold ~ t h a normal
TOP n R bundle v such that the restriction of fl to a neighborhood of P gives
a (micro-)bundle map v ~ ~. If f is transverse near a closed set C c_ M, then the
homotopy can equal f near C. (A version with BM # ~ results).
Theorem 1 .I Every closed TOP manifold M MT of dimension m ~ 6 admits a Morse
2 2 2 + .+x 2 .
function f: M ~- R; that is, f is locally of the form -Xl-...-x~+x~+ I .. m
Equivalently M is a TOP handlebody. If M ,MT m ~ 6, is compact with non-empty
boundary BM and V, V' are given compact (m-1)-submanifolds of BM such that
~M - int(V ~ V') % 3V x [0, I], then there exists a Morse function f: M ~ R such
that f-1(O) = V, f-I(I) = V' on ~M - int(V ~V') f is projection
aV x [0, ]I ÷ [0, ]1 C R, and all critical points lie in M - ~M .
On the other hand, in dimension h or 5 (or both) there exists a manifold which
is not a handlebody, and thus has no Morse function.
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