LOCAL HOMOTOPY PROPERTIES OF TOPOLOGICAL
EMBEDDINGS IN CODIMENSION TWO
Gerard A. Venema
Suppose N is an (n − 2)-manifold topologically embedded in an n-manifold M .
Chapman and Quinn have shown that N is locally flat in M provided that the local homotopy
groups of M − N at points of N are infinite cyclic and that all the higher local homotopy
groups of M − N at points of N vanish. The main theorem in this paper asserts that it is only
necessary to assume the homotopy conditions in dimensions below the middle dimension; i.e.,
if the local fundamental groups are good and if M − N is locally k-connected at points of N
for 2 ≤ k < n/2, then N is locally flat.
A b str a ct .
1. Introduction
Let N n−2 be a topological (n − 2)-submanifold of the topological n-manifold M n . (All
manifolds are without boundary.) In this paper we wish to study the problem of characterizing local flatness for N in terms of local homotopy properties of the complement
M − N . We begin with the relevant definitions.
Definitions. Let x ∈ N ⊂ M. We say that N is locally flat at x if there exists a neighborhood U of x in M such that the pair (U, U ∩ N) is homeomorphic with the standard pair
(Rn , Rn−2 × {0}). We say that N is locally 1-alg at x if for every neighborhood U of x in
M there exists a neighborhood V of x in U such that the image of π1 (V − N ) in π1 (U − N)
is abelian. We say that N is locally k-co-connected (abbreviated k-LCC) at x if for every
neighborhood U of x in M there exists a neighborhood V of x in U such that the image
of πk (V − N ) in πk (U − N) is trivial. We say that N is locally homotopically unknotted at
x if N is locally 1-alg at x and N is k-LCC at x for every k ≥ 2. ¤
Remark. The “alg” in 1-alg stands for “abelian local groups.” Other authors (e.g. [FQ])
use the terminology “good local π1 ” to mean the same thing. It is not difficult to see that
if N is locally 1-alg at x, then the image of π1 (V − N ) in π1 (U − N ) is infinite cyclic for
sufficiently small U and V . (See Lemma 2.1, below.) ¤
The basic theorem relating these concepts is due to Chapman and Quinn. Chapman
first proved that locally homotopically unknotted codimension two embeddings are locally
flat as long as n ≥ 5 [Ch] but Quinn later gave a different proof of this result [Q1] and also
proved the theorem in case n = 4 [Q2]. The theorem is true in dimension 3 as well [Ca],
but that case is not of interest to us here.
1991 Mathematics Subject Classification. 57M30, 55Q05, 55M05.
Key words and phrases. Local homotopy groups, local π1 , locally flat, topological embedding, codimension two, duality.
Typeset by AMS-TEX
1
2
GERARD A. VENEMA
Theorem [Chapman-Quinn]. Let N n−2 be a topological (n − 2)-submanifold of the
n-manifold M n . If N is locally homotopically unknotted at x for every x ∈ N, then N is
locally flat in M .
Note that the definition of locally homotopically unknotted involves assumptions about
the local homotopy groups in all dimensions. In this paper we wish to investigate the
question of whether or not it might be possible to weaken the hypotheses in the ChapmanQuinn Theorem and still conclude that the embedding is locally flat. In order to understand
the question it is helpful to recall the analogous global problem for codimension two knots:
If Σn−2 ⊂ S n is a smooth knot and πi (S n − Σn−2 ) ∼
= πi (S 1 ) for i < n/2, then Σn−2 is
unknotted [Le]. On the other hand, for each k < n/2 there exists a smooth slice knot
Σn−2 ⊂ S n such that πi (S n − Σn−2 ) ∼
= πi (S 1 ) for i < k but πk (S n − Σn−2 ) À πk (S 1 ) [Su].
The analogy above leads us to conjecture that it should suffice to assume that N is
k-LCC for k in the range 2 ≤ k < n/2. This is the best we can hope for because taking
the suspension of one of the knots mentioned at the end of the previous paragraph gives
the following example.
Example 1.1. For each n and k with 1 ≤ k < (n − 1)/2 there exists a piecewise linear
(n − 2)-sphere Σ ⊂ S n such that Σ is locally 1-alg and j-LCC at x for every x ∈ Σ and
for every j < k but there exists a point x0 ∈ Σ such that Σ is not k-LCC at x0 . The
embedding is locally flat except at two points.
But we must be careful with this analogy because we are dealing with topological embeddings, not piecewise linear or smooth ones. In the topological category we can have
the following more surprising example.
Example 1.2 [Ve]. For each n and k with 1 ≤ k ≤ n − 2 there exists a topological (n − 2)sphere Σ ⊂ S n and a point x ∈ Σ such that Σ is locally 1-alg at x and Σ is j-LCC at x for
every j < k but Σ is not k-LCC at x. The embedding is locally flat except at a sequence
of points; the sequence of nonlocally flat points has x as its only limit point.
These examples show that we must be very careful about how we ask our question. In
particular, Example 1.2 shows that if we focus our attention on a single point x ∈ N , all
the k-LCC conditions are independent of each other in the range 2 ≤ k ≤ n − 2. So it
is not the case that locally 1-alg at x together with k-LCC at x for 2 ≤ k < n/2 imply
that N is locally homotopically unknotted at x. However, the examples constructed in
[Ve] do not have just one point at which they fail to be locally flat. If k ≥ n/2, then the
examples are forced to have a sequence of points converging to x at which the codimension
two sphere is not j-LCC for some j < n/2. Thus there is still hope for a positive answer
to the question. In fact, if we assume the condition at all points in a neighborhood of x,
then the answer to the question follows the expected pattern. That fact is the main result
of this paper. Since the conclusion we seek is local we may as well assume that the open
set on which the hypothesis holds is all of N .
Theorem 1.3. Let N n−2 be a topological (n − 2)-submanifold of the n-manifold M n . If
N is locally 1-alg at x for every x ∈ N and N is k-LCC at x for every x ∈ N and for
2 ≤ k < n/2, then N is locally flat in M .
LOCAL HOMOTOPY PROPERTIES
3
Notice that the inequality k < n/2 in the theorem is a strict inequality. This is the
real novelty of the theorem. The fact that it suffices to assume k-LCC for k in the range
2 ≤ k ≤ n/2 was apparently known to Hollingsworth and Rushing [HR]. Although they did
not explicitly state a theorem about this, they incorporated that fact into their definition
of locally homotopically unknotted [HR, page 393].
A particularly interesting special case of the theorem is the case n = 4. That is the
case which inspired the work on the present paper. In the 4-dimensional case it is only
necessary to assume a π1 condition.
Corollary. Let Σ be a surface (without boundary) topologically embedded in the 4manifold M 4 . If Σ is locally 1-alg at x for every x ∈ Σ, then Σ is locally flat in M.
This result has already appeared as part of Theorem 9.3A in [FQ] but the proof given
in [FQ] contains a gap. In particular, [FQ, Lemma 9.3B] purports to prove that, for a fixed
x ∈ Σ, if Σ is locally 1-alg at x, then Σ is 2-LCC at x. Example 1.2 and the example in
[LV] show that this is not the case.
Acknowledgment. The author wishes to express his thanks to Vo Thanh Liem for many
helpful conversations related to the results in this paper.
2. Homotopy properties of codimension two embeddings
Let N n−2 ⊂ M n be a pair of topological manifolds-without-boundary as in the Introduction. In this section we develop the basic homotopy properties enjoyed by such pairs if
they have good local homotopy properties at every point. We begin by quoting a lemma
of Chapman which allows us to specify exactly which neighborhoods of points of N are
good for π1 of M − N . The proof is an application of Alexander Duality and can be found
in [Ch, Lemma 3.1]. All homomorphisms of homotopy and homology groups are assumed
to be inclusion-induced unless otherwise indicated.
Lemma 2.1 [Ch]. Suppose N is locally 1-alg at x ∈ N . Let V1 ⊃ V2 ⊃ V3 be three
neighborhoods of x such that
(1) Vi ∩ N ∼
= Rn−2 for each i,
(2) Vi is homotopic to a point in Vi−1 for i = 2, 3, and
(3) Vi satisfies the 1-alg hypothesis with respect to Vi−1 for i = 2, 3.
Then both im[π1 (V2 − N ) → π1 (V1 − N)] and im[π1 (V3 − N ) → π1 (V2 − N)] are infinite
cyclic and the inclusion induces an isomorphism from im[π1 (V3 − N) → π1 (V2 − N)] to
im[π1 (V2 − N) → π1 (V1 − N )].
We next prove three technical lemmas which show the key way in which our hypothesis
of k-LCC at every point is used. The techniques of proof are very similar to those in [Ch]
and [HR]. The first of these lemmas (Lemma 2.2) says that homotopies of j-dimensional
polyhedra can be pushed off N . The second (Lemma 2.3) says roughly that if N is locally
j-co-connected, then it is globally j-co-connected. The final lemma (Lemma 2.4) says
that j-dimensional polyhedra in the complement of N can be pushed close to N with
homotopies supported in the complement of N .
4
GERARD A. VENEMA
Lemma 2.2. Fix j ≥ 1 and an open set V0 such that V0 ∩ N ∼
= Rn−2 . Suppose N is
locally 1-alg at x for every x ∈ N and N is k-LCC at x for every x ∈ N and for all k in the
range 2 ≤ k ≤ j. For every positive function ² defined on V0 there exists a neighborhood
V of V0 ∩ N such that if K j is a compact j-dimensional polyhedron and f : K × [0, 1] → V
is a map with f(K × {0}) ⊂ V − N, then there is a map g : K × [0, 1] → V0 − N
such that g(x, 0) = f (x, 0) for every x ∈ K and dist(f(x, t), g(x, t)) < ²(f(x, t)) for every
(x, t) ∈ K × [0, 1].
Proof. First consider the case j = 1. For each x ∈ N ∩ V0 there exists a neighborhood
Vx of x such that any loop in Vx − N which is null-homologous in Vx − N bounds a disk
of diameter less than ² in V0 − N . Let V = ∪x∈V0 ∩N Vx . Suppose f : K 1 × [0, 1] → V
with f(K × {0}) ⊂ V − N. Take a triangulation of K with small mesh and a partition
0 = t0 < t1 < · · · < tm = 1 such that for each 1-simplex σ in K and each i, the diameter of
f (σ × [ti , ti+1 ]) is small relative to a Lebesgue number for the cover {Vx } of f(K × [0, 1]).
Let K (0) denote the 0-skeleton of K. Use the fact that N does not separate any open set
to shift f (K (0) × [0, 1]) off N with a very small move. This defines g|K (0) × [0, 1]. Let σ 1
be a 1-simplex in K. We next push each of the arcs f(σ 1 × {ti }) off N in such a way that
each g(∂(σ 1 × [ti , ti+1 ])) is null-homologous in some Vx − N . This is done by pushing them
off one at a time, starting with f(σ 1 × {t1 }). Push f (σ 1 × {t1 }) off with a small move
to define g|σ 1 × {t1 }. If g(∂(σ 1 × [t0 , t1 ])) links N, add a small loop to g(σ 1 × {t1 }) to
make g(∂(σ 1 × [t0 , t1 ])) null-homologous missing N . Then proceed to f (σ 1 × {t2 }). Push
it off and add a loop to it, if necessary, to make g(∂(σ 1 × [t1 , t2 ])) null-homologous. This
process is continued inductively. Once g(∂(σ 1 × [ti , ti+1 ])) ∩ N = ∅ for every i, we can use
the choice of Vx to extend g to each of the 2-cells f(σ 1 × [ti , ti+1 ]).
Next consider the case j = 2. For each x ∈ N ∩ V0 there exists a neighborhood Vx of x
such that any singular 2-sphere in Vx − N bounds a singular 3-cell of diameter less than
²/2 in V0 − N . Let δ be a Lesbegue function for the cover {Vx }. For each x ∈ V0 ∩ N
we can use Lemma 2.1 to choose a neighborhood Vx0 such that any loop in Vx0 − N that is
null-homologous in V0 − N bounds a singular disk of diameter less than δ(x)/2 in V0 − N .
Let δ 0 be a Lesbegue function for the cover {Vx0 }. Finally use the case j = 1 of the Lemma
to choose a neighborhood V such that any homotopy of a 1-dimensional polyhedron in
V is (δ 0 /2)-homotopic to a homotopy in the complement of N. Now consider a map
f : K 2 × [0, 1] → V such that f (K 2 × {0}) ⊂ V − N . Take a triangulation of K with
small mesh and a partition 0 = t0 < t1 < · · · < tm = 1 such that for each simplex σ in
K and each i, the diameter of f(σ × [ti , ti+1 ]) is small relative to δ 0 /2. Use the choice of
V to define g|K (1) × [0, 1]. Consider a 2-simplex σ 2 in K. For each i, g((∂σ 2 ) × {ti }) is
null-homotopic in the complement of N (it bounds g(σ 2 × {0} ∪ (∂σ 2 ) × [0, ti ]) ), so the
choice of Vx0 allows us to extend g to σ 2 × {ti }. Then the choice of Vx allows us to extend
g to each of the 3-cells σ 2 × [ti , ti+1 ].
The cases j ≥ 3 are handled inductively. They are a little easier than either of the cases
j = 1 or j = 2 because there is no problem of linking in these higher dimensions. ¤
Lemma 2.3. Fix j ≥ 1. Suppose N is locally 1-alg at x for every x ∈ N and N is k-LCC
at x for every x ∈ N and for every k in the range 2 ≤ k ≤ j. If V1 is an open set such that
V1 ∩ N ∼
= Rn−2 , then there exists an open set V2 ⊂ V1 with V1 ∩ N = V2 ∩ N such that
LOCAL HOMOTOPY PROPERTIES
5
any map f : S j → V2 − N is null-homotopic in V1 − N . (If j = 1, we must also assume
that f is null homologous in V1 − N.)
Proof. Consider the case j = 1. For each x ∈ V1 ∩ N we can use Lemma 2.1 to find a
positive number ²(x) such that any loop in the ² neighborhood of x that misses N and is
null-homologous in V1 − N is null-homotopic in V1 − N. Choose V 0 using this ² and Lemma
2.2. Then choose V2 ⊂ V 0 so that V2 strong deformation retracts to N in V 0 . Suppose
f : S 1 × {0} → V2 − N and f is null-homologous in V1 − N . The choice of V2 allows us to
extend f to f : S 1 × [0, 1] → V 0 so that f (S 1 × {1}) is a single point. Using Lemma 2.2
we can find an ² approximation g : S 1 × [0, 1] → V1 − N . Now g(S 1 × {1}) has diameter
less than ² and g(S 1 × {1}) is null-homologous in V1 − N (since g(S 1 × {0}) was), so the
choice of ² allows us to find a singular 2-cell which g(S 1 × {1}) bounds in V1 − N. Thus
f (S 1 × {0}) bounds a singular 2-cell in V1 − N .
The proof of the higher dimensional cases is similar. ¤
Lemma 2.4. Fix j ≥ 1 and an open set V0 such that V0 ∩ N ∼
= Rn−2 . Suppose N is
locally 1-alg at x for every x ∈ N and N is k-LCC at x for every x ∈ N and for all k in the
range 2 ≤ k ≤ j. For every neigborhood U1 of N ∩ V0 in V0 there exists a neighborhood V1
of N in U1 having the following property: for every neighborhood U2 of N there exists a
neighborhood V2 of N ∩ V0 in U2 such that if f : K j → V1 − N is a map of a j-dimensional
polyhedron into V1 − N , then f is homotopic in U1 − N to a map of K into U2 − N .
Furthermore, if L is a subpolyhedron of K and f (L) ⊂ V2 − N , then the homotopy can be
chosen to keep L fixed.
Proof. Again we first consider the case j = 1. Given U1 , we choose V1 in such a way
that any loop in V1 − N that is null-homologous in U1 − N is null-homotopic in U1 − N
(Lemma 2.3). Given U2 , we choose V2 to be any connected neighborhood of N contained
in U2 . Consider a map f : K 1 → V1 − N. It is easy to pull the 0-skeleton in, so we may
assume that f (K (0) ) ⊂ V2 . For each 1-simplex σ 1 in K, we choose an arc α in V2 − N
joining the two points f (∂σ 1 ) and then add a loop to it, if necessary, so that α ∪ f(σ 1 ) is
null-homologous in U1 − N . By the choice of V1 , this loop is null-homotopic in U1 − N , so
there is a homotopy from f (σ 1 ) to α that keeps the endpoints fixed.
Now consider the case j = 2. Let U1 be given. Use Lemma 2.3 to choose U 0 ⊂ U1
such that any singular 2-sphere in U 0 − N bounds a singular 3-cell in U1 − N. Then
use the previous case of this lemma to choose V1 ⊂ U 0 . Let U2 be given. Use Lemma
2.3 to choose V 0 ⊂ U2 so that any loop in V 0 − N that is null-homologous in U2 − N is
null-homotopic in U2 − N . Finally choose V1 ⊂ V 0 using the previous case of this lemma.
Suppose f : K 2 → V1 − N. By the previous case of this lemma, we can pull the image of
the 1-skeleton into V 0 − N so we assume that f(K (1) ) ⊂ V 0 − N . For any 2-simplex σ 2
in K, the choice of V 0 allows us to find a singular 2-cell C ⊂ U2 − N whose boundary is
f (∂σ 2 ). But the choice of U 0 says that the singular 2-sphere f(σ 2 ) ∪ C bounds in U1 − N ,
so there is a homotopy of f(σ 2 ) to C that keeps ∂σ 2 fixed.
The higher dimensional cases are similar. ¤
3. Homology properties of codimension two embeddings
We begin this section by fixing some notation which will be assumed for the remainder
6
GERARD A. VENEMA
of this paper. For the remainder of this paper it will be assumed that N ⊂ M satisfies the
hypotheses of Theorem 1.3, even though this will not be explicitly stated in the lemmas
which follow. Specifically, we assume the following.
Hypothesis. N n−2 is a topological (n − 2)-submanifold of the n-manifold M n such that
N is locally 1-alg at x for every x ∈ N and N is k-LCC at x for every x ∈ N and for
2 ≤ k < n/2.
Consider one point of x ∈ N and fix a neighborhood U0 of x. Let U1 ⊂ U0 be a
second neighborhood of x such that U1 ∩ N ∼
= Rn−2 and U1 satisfies the 1-alg condition
relative to U0 . We will denote the infinite cyclic group im[π1 (U1 − N) → π1 (U0 − N )]
by J. We use t to denote a fixed generator of J and write elements of J multiplicatively;
f → W denote the
thus J = {ti | i = 0, ±1, ±2, . . . }. Let W = U1 − N and let p : W
infinite cyclic cover determined by the natural homomorphism of π1 (W ) to J. We will
f
use the same symbol t to denote the generator of the group of deck transformations of W
f ; we
which corresponds to t. If P is any subset of W , we use P ∗ to denote p−1 (P ) ⊂ W
∗
use the notations Hi (P ; Z) and Hi (P ; Z[J]) interchangeably. From now on we will work
with W as our ambient manifold; the reader should be certain to note that W is a deleted
neighborhood of N and does not include any points of N . The reason for doing this is so
that we can take the J-cover of W .
We are assuming that N is k-LCC at each point for 2 ≤ k < n/2. We wish to conclude
that N is k-LCC at x for k ≥ n/2. Fix x ∈ N and let B2 ⊂ int B1 be two concentric
n-balls centered at x. If we can show that it is possible to choose B1 and B2 in such a way
that πk (B2 − N ) → πk (B1 − N ) is the trivial homomorphism for k ≥ n/2, then the proof
will be complete. Since each map of S k into B2 − N has compact image, it suffices to show
that there exist arbitrarily close neighborhoods W1 ⊂ W2 of N such that if Pi = Bi − Wi ,
then πk (P2 ) → πk (P1 ) is trivial. By the Eventual Hurewicz Theorem ([Fe, Proposition 3.1]
or [Q1, Theorem 5.2]), it suffices to prove this in homology with local coefficients. In other
words, we need to prove the following Proposition.
Proposition 3.1. Fix x ∈ N . For each sufficiently small n-ball B1 centered at x and
contained in V1 there exists a concentric n-ball B2 such that there are arbitrarily close
neighborhoods W1 ⊂ W2 of N for which the inclusion-induced homomorphism Hk (B2 −
W2 ; Z[J]) → Hk (B1 − W1 ; Z[J]) is trivial for k ≥ n/2.
The proof of Proposition 3.1 is broken into three parts. First, in the remainder of
this section we prove some lemmas about related homology inclusions that will be used
later. Second, in §4 we will prove that there are arbitrarily small neighborhoods of x
having cell decompositions with no cells above the middle dimension. This leaves only
the middle dimensional case to be completed. In §5 we prove the middle dimensional case
using coefficients in a field. Finally, we observe in the last section that the results of the
preceding two sections suffice to complete the proof of Proposition 1.3.
Fix an open set V0 such that V0 ∩ N = U1 ∩ N and there is a strong deformation
retraction of V0 to V0 ∩ N in U1 . The let V1 be a neighborhood of N ∩ U1 such that
polyhedra in V1 − N of dimension less than n/2 can be pulled near N via homotopies
whose tracks lie in V0 − N (Lemma 2.4).
LOCAL HOMOTOPY PROPERTIES
7
Lemma 3.2. Let B1 be an n-ball centered at x and contained in V1 , let C be an (n − 2)ball in the interior of N ∩ B1 , and let B2 be a second n-ball centered at x such that B2 ∩ N
is contained in the interior of C. There exist arbitrarily close neighborhoods W1 ⊂ W2 of
N such that if Pi = Bi − Wi , then
ker[Hk (W − P1 ; Z[J]) → Hk (W ; Z[J])] ⊂ ker[Hk (W − P1 ; Z[J]) → Hk (W − P2 ; Z[J])]
for each k < n/2.
In other words, if a k-dimensional class in W − P1 is null-homologous in W , then it is
null-homologous missing P2 .
Proof. Let W2 be any connected open neighborhood of N in V1 . By Lemma 2.3 there
exists a neighborhood W10 of N in W2 such that πk (p−1 (W10 − N)) → πk (p−1 (W2 − N ))
is trivial for every k < n/2. Choose ² to be a positive number that is smaller than either
1
1
00
3 dist(C, W − B1 ) or 3 dist(N − C, B2 ). Then use ² as input to Lemma 2.2 to choose W1
00
so that homotopies in W1 can be ²-pushed off N. Finally W1 should be chosen so that
there is an ²-homotopy deformation of W1 to N that stays inside W100 .
Now consider γ ∈ ker[Hk (W − P1 ; Z[J]) → Hk (W ; Z[J])]. We can represent γ with a
f, W
f − P1∗ ), where K is a compact (k + 1)-dimensional polyhedron
map g : (K, L) → (W
and L is a k-dimensional subpolyhedron. We trim away the part of g(K) that is outside
of p−1 (V1 − N ) so that p(g(K)) ⊂ V1 − N .
We first wish to push p(g(L)) out of B2 . The choice of W1 allows us to find an ²homotopy of p(g(L)) that pushes the part of p(g(L)) in the ²-neighborhood of C into N
and is the identity outside the 2²-neighborhood of C. The image of L under that homotopy
can be further pushed along N out of the (n − 2)-cell C because dim L < dim C. We then
use Lemma 2.2 to approximate by a homotopy in the complement of N. This last homotopy
f , so it allows us to assume that p ◦ g : (K, L) → (V1 − N, V1 − (N ∪ B2 )).
may be lifted to W
We next cut p(g(K)) off on ∂B2 . In other words, put p ◦ g in general position relative to
∂B2 . Then (p ◦ g)−1 (B2 ) is a subpolyhedron of K. So p ◦ g|(p ◦ g)−1 (∂B2 ) can be extended
(using the Tietze Extension Theorem) to a map of (p ◦ g)−1 (B2 ) into ∂B2 . Thus there is
a map g 0 : K → (V1 − (N ∪ B2 )) ∪ ∂B2 that agrees with p ◦ g outside (p ◦ g)−1 (B2 ).
By Lemma 2.4 and the choice of V1 , there exists a neighborhood W20 of N such that
k-dimensional polyhedra can be pulled into W10 keeping W20 fixed. Let Q be a compact
subpolyhedron of K such that g0 (Q) ⊂ W20 , g 0 (K − Q) ⊂ V1 − (N ∪ B2 ), and Q ∩ L = ∅. We
may assume that Q does not separate K. (If it does, adjust g 0 slightly so that arcs in Q
which connect up components of K − Q are moved off N and then remove neighborhoods
of the arcs from Q.) Then there exists a k-dimensional subpolyhedron R of K such that
K & Q ∪ R. By the choice of W20 there is a homotopy that pulls g0 (R) into W10 , keeping
g 0 (R ∩ Q) fixed. This defines a new map g00 : Q ∪ R → W10 such that g 00 |Q = g0 |Q and
g 00 (R) ∩ N = ∅. Let S denote a close neighborhood of Q ∪ R in K and extend g 00 to S.
Then p(g(L)) is homologous to g00 (∂S) since the difference bounds g 0 (K − Q) ∪ g 00 (S − Q).
All those homologies are supported in W . Furthermore, the fact that g0 ' g00 in W allows
f.
us to lift the homologies to W
00
Let γ denote the homology class represented by a lift of g 00 (∂S). We have that γ 00 ∼ γ
f − P ∗ and γ 00 is represented by a map into p−1 (W 0 − N ). The choice of W 0 now gives
in W
2
1
1
γ 00 ∼ 0 in p−1 (W2 − N ), so the proof is complete. ¤
8
GERARD A. VENEMA
Note: The entire construction described in the proof above takes place in V0 , so we may
assume that the homology classes we have constructed are supported in p−1 (V0 − N). This
observation will be important in the proof of Lemma 3.4, below.
Lemma 3.3. If P1 and P2 are as in Lemma 3.2, then
Hk (W, W − P1 ; Z[J]) → Hk (W, W − P2 ; Z[J])
is the trivial homomorphism for k < n/2.
Proof. Suppose (c, ∂c) is a class in Hk (W, W − P1 ; Z[J]). We use the notation |c| to denote
f − P ∗ . But
the support of a representative of c. Lemma 3.2 says ∂c = ∂d, where |d| ⊂ W
2
Lemma 2.4 says that c − d is homologous to e where |e| ⊂ W2 . Hence c is homologous to
f − P ∗. ¤
d + e and the proof is complete because |d| ∪ |e| ⊂ W
2
Downstairs we can improve the dimension by 1. The reason for this is the fact that the
local homology of the complement downstairs is always trivial.
Lemma 3.4. If P1 and P2 are as in Lemma 3.2 and P1 is sufficiently small, then
Hk (W, W − P1 ; Z) → Hk (W, W − P2 ; Z)
is the trivial homomorphism for k ≤ n/2.
Proof. (The reader should probably review the definitions of V0 and V1 at this point; those
definitions can be found just above the statement of Lemma 3.2) By Alexander Duality,
Hk+1 (U1 , U1 − N) ∼
= H̄cn−k−1 (U1 ∩ N ) and H̄cn−k−1 (U1 ∩ N ) ∼
= H̄cn−k−1 (Rn−2 ) = 0 since
k ≥ 2. The following diagram with exact rows shows that Hk (V0 − N) → Hk (U1 − N) is
trivial.
Hk+1 (V0 , V0 − N )


y
−−−−→ Hk (V0 − N ) −−−−→ Hk (V0 )




y
y0
0 = Hk+1 (U1 , U1 − N ) −−−−→ Hk (U1 − N ) −−−−→ Hk (U1 )
Now suppose (c, ∂c) is a class in Hk (W, W − P1 ; Z). We can trim off part of |c| so that
|c| ⊂ V1 . Then the proof of Lemma 3.2 shows that ∂c = ∂d, where |d| ⊂ V1 − P2 . Since
∂(c − d) = 0, c − d represents a class in Hk (V0 − N ) and c − d is null-homologous in W by
the paragraph above. Thus c = d in Hk (W, W − P1 ; Z) and the proof is complete. ¤
4. The structure of neighborhoods
In this section we use the homology results of the previous section to construct neighborhoods of x having improved homotopy properties. Before we can do that we need a
result about π1 .
LOCAL HOMOTOPY PROPERTIES
9
Lemma 4.1. Fix x ∈ N and suppose n ≥ 5. For each sufficiently small n-ball B1 centered
at x there exists a concentric n-ball B2 in B1 such that for sufficiently small n-balls B3 ⊂ B2
there are arbitrarily small neighborhoods W1 ⊂ W2 ⊂ W3 of N such that if Pi = Bi − Wi ,
then the inclusion-induced homomorphism π1 (∂P2∗ ) → π1 (P1∗ − P3∗ ) is trivial.
Proof. The outline of the proof is as follows. Step 1 consists of choosing B2 and B3 .
Step 2 is the construction of W2 . Step 3 is the proof that π1 (∂P2∗ ) → π1 ((B1 − N )∗ − P3∗ )
is trivial. The final step is the observation that W1 can be chosen so that the conclusion
of the Lemma holds.
Let B1 be given. We first explain how to choose B2 and B3 . Begin by choosing an
(n − 2)-cell C1 in N such that x ∈ int C1 and C1 ⊂ int B1 . Choose B2 to be an n-cell
centered at x such that B2 ∩ N ⊂ int C1 and the inclusion B2 − N ,→ B1 − N satisfies the
1-alg hypothesis. Let C2 be an (n − 2)-cell in N such that x ∈ int C2 and C2 ⊂ int B2 .
Let U be a neighborhood of C1 − C2 in B1 whose closure misses x. For each y ∈ C1 − C2 ,
choose a neighborhood Vy such that any loop in Vy − N which is null-homologous in W
is null-homotopic in U − N. Let ² be a Lesbegue number for this cover. Also choose ² so
that ² < 31 min{dist(B2 , N − C1 ), dist(∂B2 , C2 )}. Use this ² and Lemma 2.2 to determine
a neighborhood V of N . Choose another neighborhood V 0 such that V 0 ²-deforms to N in
V . Let U 0 be the preimage of C1 − C2 under this deformation. Define Q = ∂B2 − V 0 and
0
∂Q = Q ∩ V . We may assume that both Q and ∂Q are finite polyhedra. Let `1 , `2 , . . . , `q
be a set of generators for π1 (Q, ∂Q). For each arc `i there exists an arc mi in U 0 − N such
that `i ∪ mi is null-homologous in W . Hence `i ∪ mi bounds a singular 2-cell ∆i ⊂ B1 − N
by the choice of B1 . Choose B3 to be an n-ball centered at x which misses (∪qi=1 ∆i ) ∪ U .
We next explain how to choose W2 . Let W3 be any connected neighborhood of N
which is contained in V 0 . Use Lemma 2.2 to choose a neighborhood W20 of N such that
homotopies in W20 can be pushed off N staying in W3 . Then choose W2 so that W2 deforms
to N in W20 .
Define P2 = B2 − W2 and P3 = B3 − W3 . Our next goal is to show that π1 (∂P2∗ ) →
π1 ((B1 − N)∗ − P3∗ ) is trivial. Let γ be a loop in ∂P2∗ and define γ 0 = p(γ). We show
f to obtain the
that γ 0 is null-homotopic in (B1 − N ) − P3 and then lift this homotopy to W
0
desired result. We first use the choice of W2 to push γ to the boundary of C2 and out of
B3 in exactly the same way we pushed out of B2 in the proof of Lemma 3.2. Next put γ 0 in
general position with respect to ∂Q. Then γ 0 ∩ Q consists of a finite number γ1 , γ2 , . . . , γm
of subarcs. The choice of B3 allows us to find an arc µi in U 0 joining the endpoints of γi
such that γi + µi is null-homotopic missing B3 ∪ N. This means that γ 0 may be replaced
by a loop γ 00 in U 0 − N . There exists an ²-deformation of γ 00 to C1 − C2 . Since n ≥ 5,
C1 − C2 is simply connected. So γ 00 is homotopic to a point in the ² neighborhood of
C1 − C2 . We can use the choice of ² and V along with Lemma 2.2 to ²-push this homotopy
into the complement of N. This means that γ 00 is homotopic to γ 000 with diam(γ 000 ) < ².
The choice of ² then shows us that γ 00 is null-homotopic in U . So γ 0 is null-homotopic in
(B1 − N) − B3 .
Let g1 , g2 , . . . , gn , t be a set of generators for π1 (∂P2 ). We may assume that each gi lifts
to ∂P2∗ . (If not, replace gi with gi t` for some `.) Then each element of π1 (∂P2∗ ) can be
written in terms of {tj g̃i }i,j . We fix a homotopy shrinking each gi in (B1 − N ) − P3 and
choose W1 so close to N that it misses the tracks of all these homotopies. Then we will
10
GERARD A. VENEMA
have π1 (∂P2∗ ) → π1 (P1∗ − P3∗ ) trivial. ¤
Lemma 4.2. Fix x ∈ N and suppose n ≥ 5. For each q > 0 there exists a sequence B1 ⊃
B2 ⊃ · · · ⊃ Bq of n-balls centered at x such that there are arbitrarily close neighborhoods
W1 ⊂ W2 ⊂ · · · ⊂ Wq so that if Pi = Bi − Wi then the inclusion induced homomorphisms
∗
∗
∗
∗
π1 (Pi∗ ) → π1 (Pi−1
) and π1 (Pi∗ − Pi+1
) → π1 (Pi−1
− Pi+2
) are trivial.
Proof. Construct the sequence B1 ⊃ B2 ⊃ · · · ⊃ Bq inductively, using Lemma 4.1 to
construct Bi+1 from Bi . Using the 1-alg hypothesis we can also arrange that π1 (p−1 (Bi −
N )) → π1 (p−1 (Bi−1 −N)) is trivial. Choose W1 ⊂ W2 ⊂ · · · ⊂ Wq to satisfy the conclusion
of Lemma 4.1 as well.
Consider a loop ` ⊂ Pi∗ . There exists a map f : ∆2 → p−1 (Bi−1 − N ) such that
f |∂∆2 = `. Put f in general position with respect to ∂Pi∗ . Then f −1 (∂∆2 ) consists of a
∗
∗
finite collection of circles in int ∆2 . Each outermost circle shrinks in Pi−1
−Pi+1
by Lemma
∗
∗
4.1. Thus we can redefine f to map the interior of this circle into Pi−1 − Pi+1 . We do this
∗
for each outermost circle. The result is a map f 0 : ∆2 → Pi−1
such that f 0 |∂∆2 = `. So
∗
π1 (Pi∗ ) → π1 (Pi−1
) is the trivial homomorphism.
∗
The proof of the second part of the Lemma is similar. We start with a loop in Pi∗ −Pi+1
.
−1
It bounds a singular disk in p (Bi−1 − N). We use Lemma 4.1 to cut this disk off on
∗
∗
∗
∂Pi∗ ∪ ∂Pi+1
. The result is a singular disk in Pi−1
− Pi+2
. ¤
Proposition 4.3. Fix x ∈ N. For each sufficiently small n-ball B1 centered at x there
exists a concentric n-ball B2 and arbitrarily small neighborhoods W1 ⊂ W2 of N such that
there exists a compact PL manifold P with B2 − W2 ⊂ P ⊂ B1 − W1 and P has a handle
decomposition with no handles of index greater than n/2.
Proof. First suppose n ≥ 5. Fix q À 2n . Let P1 ⊃ P2 ⊃ · · · ⊃ Pq be a sequence as in
Lemma 4.2. We may also assume that each Pi+1 ⊂ Pi satisfies the hypotheses of Lemma
3.2. We may then apply the Relative Eventual Hurewicz Theorem [Q1, Theorem 5.2] to the
inclusions (Pi , Pi − Pi+j ) ,→ (Pi−1 , Pi−1 − Pi+j+1 ) since Lemma 4.2 gives us exactly the π1 conditions we need and Lemma 3.3 gives us the higher homology hypotheses we need. Thus
there is an integer s such that for all k < n/2, πk (Pi , Pi − Pi+1 ) → πk (Pi−s , Pi−s − Pi+s+1 )
is trivial. (The number s is independent of the particular Pi used; it depends only on k. In
fact s = 2k+2 will work; see [Q1, Theorem 5.2].) This means that relative k-dimensional
polyhedra in (Pi , Pi − Pi+1 ) can be pushed off Pi+s+1 with a homotopy and hence (using
engulfing) can be pushed off some Pi+t with an ambient isotopy whose support lies in the
interior of Pi .
The PL manifold P can now be constructed as follows. Start with P1 and form a handle
decomposition of P1 . Each j-dimensional handle has an (n − j)-dimensional cocore. If
j > n/2, then n − j < n/2, so the paragraph above implies that the cocore of each jhandle can be isotoped off Pi+t+1 keeping P1 − Pi fixed. We construct P by starting with
P1 , pushing the cocore of each j-handle off Pq for j > n/2, and then removing a small
regular neighborhood of the cocore of each of the j-handles, j > n/2. The remaining
manifold has only handles of index n/2 or less.
In case n = 4 the proof is much easier. Each Pi has a handle decomposition with
handles of dimensions ≤ 3, so we need to eliminate the 3-handles. It is enough to show
that π1 (Pi , Pi − Pi−1 ) → π1 (Pi+1 , Pi+1 ) is trivial. The proof proceeds as follows: Given
LOCAL HOMOTOPY PROPERTIES
11
a path α in Pi with endpoints in Pi − Pi−1 , use the fact that Pi+1 does not separate to
join the endpoints with a path β in Pi − Pi−1 . Add a loop, if necessary, so that α ∪ β is
null-homotopic in Pi+1 . ¤
Corollary 4.4. Proposition 3.1 holds for k > n/2.
Proof. This is clear since the inclusion factors through a space with trivial homology in
dimensions k > n/2. ¤
5. A duality principle for homology with coefficients in a field
In this section we prove a weak version of Proposition 3.1. It is weaker than the version
we need in that it only works for coefficients in a field. It uses a duality principle for
infinite cyclic covers that is similar to the one used by Milnor in [Mi]. Let F be a field
and let Λ denote the group algebra F [J]. Notice that for any U ⊂ W the homology group
f , U ∗ ; F ) can be thought of either as a vector space over F or as a module over Λ.
Hk ( W
It is important to distinguish between “finitely generated over F ” and “finitely generated
f , U ∗ ; F ).
over Λ.” We use Hk (W, U ; Λ) as a shorthand for Hk (W
Fix a sequence of three PL manifolds P3 ⊂ P2 ⊂ P1 such that each of the inclusions
P3 ⊂ P2 and P2 ⊂ P1 satisfies the hypotheses of Lemmas 3.2 and 3.4. Fix k ≤ n/2 and let
αi : Hk (W, W −Pi ; Λ) → Hk (W, W −Pi+1 ; Λ) denote the inclusion-induced homomorphism.
Lemma 5.1. im(α2 ) = im(α2 ◦ α1 ).
Proof. Consider the following commutative diagram in which each row is exact.
β1
Hk (W ; Λ) −−−−→ Hk (W, W − P1 ; Λ) −−−−→ Hk−1 (W − P1 ; Λ)
°


°


α1 y
°
y
β2
γ
Hk (W ; Λ) −−−−→ Hk (W, W − P2 ; Λ) −−−−→ Hk−1 (W − P2 ; Λ)
°


°


α
2
°
y
yδ
β3
Hk (W ; Λ) −−−−→ Hk (W, W − P3 ; Λ) −−−−→ Hk−1 (W − P3 ; Λ)
By Lemma 3.2, we have δ ◦γ = 0, so im(α2 ) ⊂ im(β3 ). On the other hand, im(α2 ) ⊃ im(β3 )
is obvious so we have im(α2 ) = im(β3 ). Now the inclusion P1 ⊂ P3 has all the same
properties that the inclusion P2 ⊂ P3 has, so we have im(α2 ◦ α1 ) = im(β3 ) as well. ¤
Lemma 5.2. im(α2 ) is finite dimensional over F .
Proof. For each i the exact sequence
t−1
p∗
0−
→ C∗ (W, W − Pi ; Λ) −−→ C∗ (W, W − Pi ; Λ) −→ C∗ (W, W − Pi ; F ) −
→0
of chain complexes gives rise to a long exact sequence
t−1
p∗
··· −
→ Hk (W, W − Pi ; Λ) −−→ Hk (W, W − Pi ; Λ) −→ Hk (W, W − Pi ; F ) −
→ ···
12
GERARD A. VENEMA
of homology groups. Therefore we have the following commutative diagram with exact
rows.
t−1
p∗
t−1
p∗
t−1
p∗
Hk (W, W − P1 ; Λ) −−−−→ Hk (W, W − P1 ; Λ) −−−−→ Hk (W, W − P1 ; F )




α

α1 y
y 1
y0
Hk (W, W − P2 ; Λ) −−−−→ Hk (W, W − P2 ; Λ) −−−−→ Hk (W, W − P2 ; F )




α

α2 y
y 2
y0
Hk (W, W − P3 ; Λ) −−−−→ Hk (W, W − P3 ; Λ) −−−−→ Hk (W, W − P3 ; F )
The vertical arrows on the right are 0 by Lemma 3.4. The upper half of the diagram shows
that im(α1 ) ⊂ (t − 1)(Hk (W, W − P2 ; Λ)) = im(t − 1). Thus im(α2 ) = im(α2 ◦ α1 ) ⊂
im(α2 ◦ (t − 1)) = (t − 1) im(α2 ). It follows that im(α2 ) = (t − 1) im(α2 ). Now im(α2 ) is a
subspace of Hk (W, W − P3 ; Λ) which is finitely generated over Λ. Since im(α2 ) is finitely
generated over Λ and im(α2 ) = (t − 1) im(α2 ), im(α2 ) is a finitely generated Λ-module
whose order ideal in non-trivial. It follows that im(α2 ) is finite dimensional over F . (See
the proof of Assertion 5, page 118, of [Mi]. This is the point in the proof at which it is
important to be working over a field.) ¤
f such that p(K) = W and for each compact
Let K be a connected subpolyhedron of W
f,
subpolyhedron L of W , p−1 (L) ∩ K is compact. Notice that, for each compact C ⊂ W
i
C ∩ t (K) = ∅ for all but finitely many i. Define
Nq =
[
ti (K),
and
i≥q
Nq0 =
[
ti (K).
i≤−q
∗
Lemma 5.3. im[Hk−1 (N0 , N0 − Pi∗ ; F ) → Hk−1 (N0 , N0 − Pi+1
; F )] is finite dimensional
over F for i = 1, 2.
Proof. Define N = N0 ∩ N00 . Consider the following commutative diagram in which each
row is part of an exact Mayer-Vietoris sequence. (Coefficients in F are understood.)
Hk−1 (N, N − Pi∗ ) −→


y
Hk−1 (N0 , N0 − Pi∗ ) ⊕ Hk−1 (N00 , N00 − Pi∗ )

γ
y
f ,W
f − P ∗)
−→ Hk−1 (W
i


yβ
∗ ) −
∗ )⊕H
0
0
∗
f ,W
f − P∗ )
Hk−1 (N, N − Pi+1
→ Hk−1 (N0 , N0 − Pi+1
→ Hk−1 (W
k−1 (N0 , N0 − Pi+1 ) −
i+1
Since Hk−1 (N, N − Pi∗ ; F ) is finite dimensional (by excision and the choice of K) and the
image of β is finite dimensional, it follows that the image of γ is finite dimensional. ¤
LOCAL HOMOTOPY PROPERTIES
13
f , N0 ∪ (W
f − P ∗ ); F ) → Hk (W
f , N0 ∪ (W
f − P ∗ ); F )] is finite dimenLemma 5.4. im[Hk (W
1
3
sional over F .
Proof. Consider the following diagram in which each row is part of the long exact sequence
of a triple.
f, W
f − P ∗; F ) −
f , N0 ∪ (W
f − P ∗ ); F ) −
f − P ∗ ), W
f − P ∗; F )
Hk ( W
→ Hk ( W
→ Hk−1 (N0 ∪ (W
1
1
1
1




δ
β
y
y1
y 1
f, W
f − P ∗; F ) −
f , N0 ∪ (W
f − P ∗ ); F ) −
f − P ∗ ), W
f − P ∗; F )
Hk ( W
→ Hk ( W
→ Hk−1 (N0 ∪ (W
2
2
2
2






α2 y
yδ2
y
f, W
f − P ∗; F ) −
f , N0 ∪ (W
f − P ∗ ); F ) −
f − P ∗ ), W
f − P ∗; F )
Hk ( W
→ Hk ( W
→ Hk−1 (N0 ∪ (W
3
3
3
3
Since both β1 and α2 have finite dimensional images (by Lemmas 5.2 and 5.3), an elementary linear algebra argument shows that δ2 ◦ δ1 has finite dimensional image as well. ¤
Lemma 5.5. There exists an integer s ≥ 0 such that
f , N s ∪ (W
f − P ∗ ); F ) → Hk (W
f, N0 ∪ (W
f − P ∗ ); F )
Hk (W
1
3
is the trivial homomorphism.
f , N0 ∪ (W
f − P ∗ ); F ) in Hk (W
f , N0 ∪ (W
f − P ∗ ); F )
Proof. By Lemma 5.4, the image of Hk (W
1
3
is finitely generated. We choose s large enough so that N−s contains a representative
f , N0 ∪ (W
f − P ∗ ); F ) →
of each of these generators. Then the homomorphism Hk (W
1
f , N−s ∪ (W
f − P ∗ ); F ) is trivial. We shift everything by ts in order to reach the
Hk ( W
3
desired conclusion. ¤
We end this section with a proof of a weak version of Proposition 3.1. It assumes
coefficients in a field, but is valid in the crucial middle dimension.
Proposition 5.6. If the sequence of PL manifolds P5 ⊂ P4 ⊂ P3 ⊂ P2 ⊂ P1 is such that
each of the inclusions Pi+1 ⊂ Pi satisfies the hypotheses of Lemmas 3.2 and 3.4, then the
inclusion-induced homomorphism Hk (P1 ; Λ) → Hk (P5 ; Λ) is trivial for k ≤ n/2
Proof. The cases k < n/2 are covered by Corollary 4.4, so we focus on the case k = n/2.
By Poincaré Duality, Hk (Pi ; Λ) ∼
= H k (Pi , ∂Pi ; Λ) and, by excision, H k (Pi , ∂Pi ; Λ) ∼
=
k
H (W, W − Pi ; Λ). Furthermore, the inclusion induced homomorphism Hk (P1 ; Λ) →
Hk (P3 ; Λ) is dual to H k (W, W − P1 ; Λ) → H k (W, W − P3 ; Λ), so we show that this last
homomorphism is trivial.
f, W
f − P ∗ ; F ) and that
Note that H k (W, W − Pi ; Λ) = Hck (W
i
f, W
f − P ∗ ; F ) = lim H k (W
f, (W
f − Pi ) ∪ Nq ∪ N 0 ; F ).
Hck (W
i
q
q→∞
14
GERARD A. VENEMA
Let s be the integer given by Lemma 5.5 and consider the following diagram. (Each row
is part of an exact Mayer-Vietoris sequence; coefficients in Λ are understood; Eq = Nq ∪Nq0 .)
f ,W
f − P ∗) −
H k−1 (W
→
1


0y
f , (W
f − P ∗ ) ∪ Eq )
H k (W
1
f ,W
f − P ∗) −
f , (W
f − P ∗ ) ∪ Eq+s ) −
H k−1 (W
→ H k (W
→
3
3


0y
f , (W
f − P ∗ ) ∪ N q ) ⊕ H k (W
f , (W
f − P ∗) ∪ N 0 )
H k (W
q
1
1
−→


y


y


y0
f , (W
f − P ∗ ) ∪ Nq+s ) ⊕ H k (W
f , (W
f − P ∗) ∪ N 0
H k (W
3
3
q+s )


y0
f ,W
f − P ∗) −
f , (W
f − P ∗ ) ∪ Eq+2s ) −
f , (W
f − P ∗ ) ∪ Nq+2s ) ⊕ H k (W
f , (W
f − P ∗) ∪ N 0
H k−1 (W
→ H k (W
→ H k (W
5
5
5
5
q+2s )
The vertical arrows on the right are 0 by Lemma 5.5 (translated by tq and tq+s ). The
vertical arrows on the left are 0 by Lemma 3.3. It follows that the composite vertical
arrow in the center is 0 as well. Taking a limit as q → ∞ gives the desired conclusion. ¤
6. Proof of Theorem 1.3
As observed in §3, it suffices to prove Proposition 3.1. Most cases of Proposition 3.1
were proved in §4. The only remaining case is that in which n is even and k = n/2.
Use Propositions 4.3 and 5.6 to choose PL manifolds P2 ⊂ P1 satisfying the hypotheses
of Lemmas 3.2 and 3.4 such that
(1) P1 has a cell decomposition with no cells of dimension greater than k, and
(2) Hk (P2∗ ; Q) → Hk (P1∗ ; Q) is trivial.
Since H k+1 (P1∗ ; Z) = 0, the Universal Coefficient Theorem for Cohomology shows that
Ext(Hk (P1∗ ; Z)) = 0. Thus Hk (P1∗ ; Z) is torsion-free [HS, Theorem III.6.1]. Consider the
commutative diagram
α
Hk (P2∗ ; Z) −−−−→ Hk (P1∗ ; Z)


⊗Q
⊗Q
y
y
0
Hk (P2∗ ; Q) −−−−→ Hk (P1∗ ; Q).
Let x ∈ Hk (P2∗ ; Z). If α(x) ∈ Hk (P1∗ ; Z) is not 0, then its image in Hk (P1∗ ; Q) is not 0
either by [HS, Exercise III.7.5]. But the bottom arrow is is the trivial homomorphism by
the choice of P1 and P2 . Hence α = 0 and the proof is complete.
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C a lv in C o l l e g e , G r a n d R a p id s , M ic h ig a n 4 9 5 4 6 U S A
Current address: University of Michigan, Ann Arbor, Michigan 48109-1003
E-mail address: [email protected]