Combinatorial Symmetries of the M-Dimensional Ball

133 Pages·1986·9.699 MB·English

by  Lowell Jones

Memoirs of the American Mathematical Society
Number 352
Lowell Jones
Combinatorial symmetries
of the m-dimensional ball
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
July 1986 • Volume 62 • Number 352 (end of volume)
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Abstract
The problem of determining which subsets K c Bm of the m-dimensional
ball can be the fixed point set of a semi-free PL group action 7L xBm -> B™
is completely solved when m-dim(K) >• 6. P. A. Smith proved that K must
be a Z -homology ball, and the pair (K,KnBm) must be a Z -homology
manifold pair [27]. If n has an odd number divisor it is also known
that m-dim(K) = 0 (mod 2). The author has shown that these conditions
are also sufficient, when n = even, to realize any PL subset K c Bm which
satisfies them as the fixed point subset of a PL semi-free action
TL xBm -* Bm [13]. The author has also shown, in the case that n = odd,
that for any PL Z -homology manifold pair (K,8K) and any prime factor p
of n there is a characteristic class I h?(K) € E H .. ,((K,9K),Z)
v
± 1 i K+41-1
which must vanish if K is to be the fixed point set of a PL semi-free
action ZxBm -• Bm with 8K = K n3Bm (cf. [10]). In this paper it is
shown that all the above necessary conditions which a fixed point
set K c B1 of a PL semi-free odd order group action must satisfy are
also sufficient to realize any K c Bm which satisfies them as the fixed
point set of such an action. In addition, all such actions are classified
up to the concordance equivalence.
1980 Mathematics Subject Classification. 57S17, 57R65, 57R67.
Library of Congress Cataloging-in-Publication Data
Jones, Lowell, 1945—
Combinatorial symmetries of the m-dimensional ball.
(Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 352)
Bibliography: p.
1. Surgery (Topology) 2. Manifolds (Topology) 3. Unit ball. I. Title.
II. Title: m-Dimensional ball. III. Series.
QA3.A57 no. 352 510 s [514'.3] 86-17500
[QA613.658]
ISBN 0-8218-2414-7
This page intentionally left blank
§0. Introduction to the Problem
Notation: Z ; additive cyclic group of order n.
TL ; multiplicative cyclic group of order n.
B ; unit ball in m-dimensional Euclidean space,
cp: Z xBm -• Bm; P.L. group action on Bm by TL .
n
A semi-free group action cp: ZSxB -+ Bm is one such that for any
x G B the orbit set {cp(t,x) |t GZZ } contains exactly one point or n
points. The set of fixed points {xGBm |cp(t ,x)=x vtG TL } will be denoted K.
Set aK = Kn8Bm.
P. A. Smith has studied semi-free group actions cp: TL *Bm -> Bm in
n
terms of their fixed point set (see [2 7]). He has proven that K satisfies
these two properties.
0.1. (K,9K) is a Z -homology manifold pair, of dimension k <^ m.
0 if i>0, and
0 ±. Hi(K,Zn) =
z if i-o
n
0 if i^k
H ((K,9K),Z )
i n
Z if i-k
n
It is also well known that K must satisfy the following
0.3. If n has an odd divisor, then m-k = 0 mod 2.
The author has shown in [13] that if K <= B satisfies 0.1, 0.2, 0.3,
and in addition satisfies H*(K,Z ) = H (K,Z ) = Z , then K <= Bm is the
2 Q 2 2
fixed point set of some P.L. semi-free group action cp: TL xB -> Bm
(provided m-k _> 6). For example, if K c Bm satisfies 0.1, 0.2, 0.3, and
n = even, then H^(K,Z ) = H (K,Z ) = Z can be deduced from 0.2.
2 Q 2 2
In the rest of this paper it is assumed that n is an odd integer.
Received by the editors May 3, 1982 and, in revised form May 2, 1986.
The author was supported in part by the NSF.
2 LOWELL JONES
If n = odd integer there is a further restriction on the fixed point
set K of cp: TL^Bm -* Bm, which is not implied by 0.1-0.3, which shall be
recalled now. Let p denote an odd prime number, and (M,3M) a finite
simplical pair which is an orientable (with respect to Z -homology)
Z -homology manifold pair. The author introduces a characteristic class
EhP(M) € E H ((M,3M) Z)
m+4i-1 >
in [15], [10], and proves that h£(M) vanishes if M is the fixed point set
of PL TL -action on an oriented PL manifold. In particular
P
Z h?(K) € I H _ ((K,3K),Z)
k+41 1
is well defined for any odd prime divisor p of n, and for any P.L.
subset (K,3K) c (Bm,3Bm) which satisfies 0.1, 0.2; and the following is
true.
0.4. If K c Bm is the fixed point set for a semi-free PL action
cp: TL xBm -> Bm, then h£(K) = 0 for all odd prime divisors p of n.
In this paper the following characterization of fixed point sets of
odd order actions cp: TL *Bm •-» Bm is proven.
Theorem 0.5. Let K c Bm denote a PL subset of the m-dimensional ball,
and n an odd positive integer. Suppose K satisfies 0,1-0,4, and
dim(Bm) - dim 00 >_ 6. Then K c Bm is the fixed point set of a semi-free
PL action cp: Z *Bm c Bm.
n
The above theorem is a special case of the following more general
theorem. Let (N,9N) denote a compact PL manifold pair, and K c N denote
a compact PL subset of N, with 3K = K n3N, satisfying:
0.6 (a) ir(N} = 0, TrC3N) = 0 for i-1,2,
i i
(b) H*(N,Z ) = H (N,Z ) - Z ,
n Q n n
(c) (K,3K) satisfies 0.1, 0.2,
(d) dim(N)-dim(K) is even and greater than 5.
COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 3
Theorem 0.7. Let K c N be as in 0.6, and n an odd positive integer.
Then there is a semi-free PL action tp: 7L xN -» N having K c N for fixed
point set if and only if h§(K) = 0 for all odd prime divisors p of n.
A complete classification of the group actions of 0.7 is given in §6.
Organization of Paper
There are six sections to the paper, which shall be outlined in a
moment.
The following reading procedure is recommended. First read the
outlines of the sections provided immediately below. Next read section 1;
read step 2 and lemmas 2.5, 2.6 (but not proofs) of section 2; read
lemma 3.2 (but not proof) of section 3; read lemma 4.0 and the first two
steps in its proof in section 4; read lemmas 5.1, 5.2 (but not proofs)
and the completion of the proof of 0.7 given in section 5. Finally read
all the steps in the proofs for lemmas 2.5, 2.6, 3.2, 4.0, 5.1. 5.2.
Outline of Section 1. The author has reduced the proof of 0.5
to completing surgery on a "blocked" normal map t (see [13]). This
reduction is reviewed in detail in this section, and also adapted to
the proof of 0.7.
Both the image blocks and the domain blocks of t are Poincare
duality pairs with fundamental group 7L . The domain blocks are not
manifolds, and the framing information is given in the category of
spherical fibrations. Thus the surgery procedures on the various blocks
of t need to be carried out in the Poincare duality category as discussed
in [12].
In the special case that K is a PL manifold, both the domain and
range of t are block spaces over a cell structure for K, but having
Poincare duality pairs for blocks instead of PL manifold pairs for blocks
as in [26]. Thus t can be identified with a mapping f: K ->» l.i,.i ? )
n
into F. Quinn's surgery classifying spaces (see [5] and [14] for a
description of these spaces).
If K is not a PL manifold, then the block structure of t is somewhat
more exotic. Choose a triangulation T for N which also triangulates K.
4 LOWELL JONES
Let C denote the dual cell structure of T, let R denote the union of all
cells in C which intersect T, and let R denote the topological boundary
of R in N. A blocked space structure £ is given to R by taking the
intersections of the dual cells of C with R to be the blocks of R. The
notion of blocked space (which generalizes the notion of block bundle)
is given in [[9], section 1], The blocks of t and those of | are in a
one-one correspondence, in a way which is consistent with the boundary
operation. In fact, the 2Z -covering of the range of t is homotopy
equivalent to R via a mapping that maps each block of this TL -covering
homotopy equivalently to the corresponding block of £. So t may be
identified with an element [t] € L (£,Z ), where L (£,Z ) is a surgery
group defined in [[13], 3.3]. Roughly speaking L (i ,TL ) is the group
Q
of blocked normal maps which have fundamental group TL in each block,
and are equipped with a one-one correspondence from their blocks to the
blocks of £, which is consistent with the boundary operator and shifts
dimensions down by %. The superscript "h" denotes that surgery is to
be completed only up to homotopy equivalence (not up to simple-homotopy
equivalence). The groups L„(£,Z ) are discussed in more detail in
n
[[13], 3.3], and similar surgery groups are described in [4].
Outline of Section 2. The blocked surgery problem t of section 1
can be studied by using the author's generalization of D. Sullivan's
Characteristic Variety Theorem (see [14]). The problem of completing
surgery on t (block by block) is thus replaced by the problem of
completing surgery on a finite set of more elementary surgery problems
t-y , tn > • • • > to • The surgery problems t- are more elementary than t,
because t- has a Poincare duality space (or Z -Poincare duality space,
r=positive integer) for range and domain, and thus has at most two blocks,
where as t can have a very large number of blocks. The notion of
Z -manifold is defined in [21] ; the notion of Z -Poincare duality space
is defined similarly.
In the special case that K is a PL manifold, the t- are constructed
as follows. Choose a characteristic variety for K, {g^: M. -+• K
i=l,2,3,...,£}, consisting of mappings from oriented smooth manifolds or
smooth Z -manifold (see [[14], 1.3]). Then pull the universal
COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 5
surgery problem back along the composition
(where f is the classifying map for t noted in the outline of Step 1),
and amalgamate this pull-back first over 6M- (the codimension 1 singular
set of M.) and then over M. to get a Z -surgery problem (£.,<$£.). It
follows directly from the Characteristic Variety Theorem (see [[14] , 1.4])
that surgery can be completed on t (block by block) if surgery can be
completed on each (t. ,<5t-).
In the general case, when K may not be a PL manifold, there is a more
indirect procedure for constructing (£.,<$£.) which will give (up to
normal Z -cobordism equivalence) the same (t.,6£.) as constructed in
the last paragraph for K a PL manifold. Begin by choosing a character
istic variety for the quotient space R/R, denoted {g,: M.+R/R|i=l,2,...,£},
consisting of mappings from oriented smooth manifolds or smooth Z -
manifolds. Let K*- ' denote the first barycentric subdivision of the
triangulation of K by T (recall T is a triangulation of N which also
triangulates K c N). First putting g.: 6M. -> R/R into transverse position
to every simplex of K^ , and then extending this to a transversality of
g-: M- •+ R/R to every simplex of K** \ we obtain "correspondences"
6c : 6n + K^, c : n -* K ^ as described in [[9], 1.2]. Here 6 n »n
i ± i ± ± ±
are the block space structures for (g., „ ) (K) , g. (K) having for
blocks (g.I.w ) (A), g- (A) where A is any simplex of K^ ' ; and
6ciCCgil6M )"1^A^ = A> c (gT1(A)) = A. Note that K(1) is the "base
1 i i
space", for the blocked space structure £ (see [[9], pg. 490]). Since
the blocks of £ and t are in a one-one correspondence in a way that is
consistent with boundary operators, it follows that £; and t have the
same base space, K^ ^. Thus t can be pulled back along 5c- and c. (see
[[9], pg. 491]) to get blocked surgery problems 5c #(t), c #(t), having
i i
ordinary surgery problems and Z -surgery problems as blocks respectively.
The Z -surgery problem (t.,St.) is obtained by amalgamating the blocks
of 6c.#(£) to get 6t. and by amalgamating the blocks of c^#(t) to get t^.
Surgery can be completed on t if it can on all the (t.,61^) (see below).
This fact is not an immediate consequence of the Characteristic Variety