POINCARE DUALITY EMBEDDINGS
AND
FIBERWISE HOMOTOPY THEORY
JOHN R. KLEIN
Abstract. We prove an embedding theorem for maps from a fi
nite complex into a Poincare duality space. The proof uses fiber
wise homotopy theory.
1. Introduction
The choice of a closed regular neighborhood of a finite polyhedron
embedded in a manifold enables one to write the ambient space as a
union of two manifolds, the neighborhood and its complement, glued
along a common boundary component. Poincare duality embeddings
are a homotopy theoretic version of this in which the manifolds are to
replaced by Poincare spaces and the gluing is now done with respect
to a homotopy equivalence.
To put it another way, a map f :K ! X from a finite complex
K into an ndimensional Poincare duality space X (or Poincare pair
(X; @X)) is said to Poincare embed if f extends to a homotopy equiv
alence
K [A C ' X
such that each piece of the decomposition satisfies Poincare duality.
This means that (K; A) and (C; A) (or (C; @X q A) when X has a
boundary) are Poincare pairs having the same dimension. Moreover,
the fundamental class in each case is induced from the fundamental
class of X (for the precise definition, see 2.2 below). Thus to specify
the Poincare embedding, we have to find the complement C and the
way which it is glued to K to give X.
There are obstructions to Poincare embedding. For example, for
such a decomposition of X to exist it is necessary that the homology of
K (with respect to any coefficient system) vanishes in degrees > n. Let
us write hodim K k if K is homotopy equivalent to a CW complex
of dimension k. We will be working with the codimension 3
hypothesis: k n 3.
Question. Given a map f :K ! X, when does it Poincare embed?
The problem may be broken up into two stages: first construct a
candidate for the complement of K, then, provided that the candidate
____________
Date: April 14, 1998.
1
2 JOHN R. KLEIN
has been correctly chosen, find the gluing data to build X. In this
paper we give a partial answer to the above question in terms of a
lower bound for the connectivity of f .
Theorem A. Let f :K ! X be rconnected. Then f Poincare embeds
provided k n 3 and
r 2k  n + 2 :
This is the Poincare analogue of a theorem of Wall which gives
criteria for a finite complex to embed up to homotopy in a smooth
manifold [Wa1 ]. Actually, Wall's result is one dimension sharper than
our Theorem A, and I do not know whether the bound in the Poincare
case can be improved to get the extra dimension. When r = 0, Theorem
A is a result of Levitt [Le ]. When r = 1, X is 1connected and K is
a Poincare space, it is a result of Hodgson [Ho ]. The proofs of the
theorems of Wall, Levitt and Hodgson are based on engulfing and the
Whitney trick.
In contrast, our approach will be homotopy theoretic. The technol
ogy developed here should be of interest to both homotopy theorists
and manifold topologists. For homotopy theorists, we introduce new
tools for studying cubical diagrams of spaces. Unbased fiberwise ho
motopy theory over a fixed space is discussed and a Freudenthaltype
desuspension theorem is proved (which is different from the correspond
ing based version that is to be found in the work of James [Ja , x9]; see
4.10 below). This fiberwise desuspension result enables us to construct
the complement.
For manifold theorists, we point out that Theorem A has applica
tions to embedding theory in codimension 3 via the surgery machine
(see e.g., Corollary A below).
We also mention that fiberwise technology is powerful enough so as
to classify embeddings in a range which is about twice as large as the
one appearing here (the classification is in terms of homotopy theoretic
data which are sometimes computable). There is also a companion
result to Theorem A which says that the embedding is unique `up to
isotopy' when the bound on connectivity is replaced by strict inequality.
The uniqueness result is a special case of a relative version of Theorem
A which applies to maps of pairs (K; L) ! (X; @X) whose source is
a relative CW pair and whose target is a Poincare pair, where it is
already assumed that a Poincare embedding has been specified for the
restriction L ! @X. The issue of uniqueness, the relative case and
classification will be addressed in another paper.
By combining Theorem A with the BrowderCassonSullivanWall
theorem [Wa5 , 11.3.2] (and some benign manipulations with White
head torsion which we omit) we obtain the following, originally due to
Haefliger [Ha ]:
POINCARE DUALITY EMBEDDINGS 3
Corollary A. An rconnected map f :V k ! Nn of smooth mani
folds (with V closed) is homotopic to a smooth embedding provided
that 3(k+ 1) 2n, n 6 and r 2k n+ 2.
Applying Theorem A to the diagonal map X ! Xx X of a closed
Poincare duality space, we settle in the affirmative an old conjecture
in the 2connected case, which concerns the existence of the unstable
homotopy tangent bundle for Poincare spaces:
Corollary B. If X is 2connected, then the diagonal X ! Xx X
Poincare embeds.
To provide some further context for Theorem A, we now make a few
remarks about Poincare surgery (however, we do not pursue the issue
of Poincare surgery in this article). Suppose that one is given a normal
map g :V n ! Xn of Poincare spaces, i.e., a degree one map which is
covered by a map of Spivak normal fibrations. A basic problem of this
subject is to decide when the map is cobordant through normal maps to
a homotopy equivalence. A reasonable way to go about this would be to
do surgery on framed Poincare embedded spheres in V to improve the
connectivity of g. If say, X is 1connected and n 5, then Theorem
A allows one to perform a sequence of surgeries on framed Poincare
embedded spheres to obtain an [n=2]connected normal map. Thus
Theorem A has application to surgery below the middle dimension.
Here is the scheme of the proof of Theorem A. Consider f :K ! X
followed by the inclusion X Xx Dj. For large j we show that the
composite map Poincare embeds. The next step is to compress this
Poincare embedding down into X. By a downward induction on codi
mension, it suffices to consider the case j = 1. Let W denote the
Poincare complement to K in Xx I. The first obstruction to com
pressing down into X is given by the existence of a map C ! X and a
fiberwise weak equivalence X C ' W , i.e., the complement W needs
to fiberwise desuspend over X. It turns out that our bound for the
connectivity is sufficient to guarantee that such a fiberwise desuspen
sion exists. The space C will be our candidate for the complement of
K in X. Let A0 be the space along which K and W are glued to make
Xx I. The final step of the proof is to show that A0 fiberwise desus
pends over K in a way compatible with the fiber desuspension we chose
for W (the resulting desuspension will be our candidate for gluing K
to C to build X). Identify A0 with K A for some map A ! K, and
identify W with X C. We show there exists a map A ! C such that
the resulting composite K A ! X A ! X C coincides with the given
map A0 !W via the identifications. Then the resulting data amount
to a Poincare embedding of f :K ! X.
Outline. This article is organized as follows: x2 sets forth the basic def
initions and conventions; most of the material here is wellknown. In
4 JOHN R. KLEIN
x3 we establish the existence of Poincare embeddings in the stable case:
we show that a map f :K ! X followed by the inclusion X Xx Dj
will Poincare embed when j is sufficiently large. In x4 we prove the
Truncation Lemma 4.1, which is the main technical tool for deducing
the Cocartesian Replacement Theorem 4.2 and the Desuspension The
orem 4.7. In x5 we prove the Face Theorem 5.1, which is an excision
statement about cubical diagrams of spaces. In x6 we prove Theorem
A.
Acknowledgements. My viewpoint on fiberwise homotopy theory has
been strongly influenced by Tom Goodwillie. I am also indebted to
Goodwillie for spotting a mistake in my original proof of the Face The
orem 5.1; the (corrected) proof appearing below is due to him. Andrew
Ranicki pointed out that results of [Wa2 ] are often easier to use in the
reformulation of [Wa3 ]; he also suggested some improvements in the in
troduction. My original argument for constructing the gluing space in
the proof of Theorem A made use of the calculus of homotopy functors.
The proof provided below, which is quite different, was first discovered
by Bill Richter, and then rediscovered by me. I am also indebted to
Richter for introducing me to the homotopy theory approach to em
bedding. Stefan Schwede was instrumental in convincing me to write
this paper using the language of model categories. I am grateful to
Jeff Smith for reading the paper and for suggesting improvements in
the exposition. Thanks to Bruce Williams for inspiring discussions in
Bielefeld during the summer of 1992 which got this project started.
2. Preliminaries
We will be using the Quillen model category structure on the cat
egory Top of compactly generated topological spaces [Qu ]. In this
model category, the weak equivalences are the weak homotopy equiv
alences, the fibrations are the Serre fibrations and the cofibrations are
the `Serre cofibrations', i.e., inclusion maps given by a sequence of cell
attachments (i.e., relative cellular inclusions.) or retracts thereof. In
particular every object is fibrant and the cofibrant objects are retracts
of cellular objects. Fibrations are specified as `i', cofibrations as `ae'
and weak equivalences as `~!'.
Each morphism of Top can be functorially factored in two ways as:
(1) a cofibration followed by a fibration which is also a weak equiva
lence, or as: (2) a cofibration which is a weak equivalence followed by
a fibration. Applying the first of these options to the map ; ! Y , we
~
obtain for each object a cofibrant replacement Y ci Y .
We will be working for the most part with cofibrant objects. Unless
otherwise specified, throughout this paper the term `space' will mean
a cofibrant object. A space is called homotopy finite if it is homo
topy equivalent to a finite complex. A map A ! B of spaces (with B
POINCARE DUALITY EMBEDDINGS 5
nonempty) is rconnected if for any choice of basepoint in B, the homo
topy fiber with respect to this choice of basepoint is an (r 1)connected
space (by convention, a nonempty space is at least (1)connected).
In particular, any map A ! B is (1)connected. A weak equivalence
is an 1connected map.
If no confusion arises, the following slightly ambiguous notation will
often be used: if f :A ! B is a map, we let the pair denoted (B ; A)
consist of the mapping cylinder B = B [f Ax [0; 1] together with the
inclusion of Ax 0.
The machinery of homotopy limits and colimits will be used through
out (cf. [BK ]). We also assume that the reader is familiar with homo
topy excision, i.e., the BlakersMassey theorem and its (dual) general
ization to cubical diagrams of spaces. A basic reference for the latter
is [Go , x2].
Lastly, a warning about terminology: suppose we are given a com
mutative square of spaces
A  ! C
? ?
? ?
y y
B  ! D
such that the induced map Bx 0 [ Ax [0; 1] [ Cx 1 ! D is a homotopy
equivalence. One usually says in this instance that the square is homo
topy cocartesian (or a homotopy pushout). However, we will instead
follow Goodwillie's conventions and say that the square is cocartesian.
Similarly, the square is jcocartesian if the map is jconnected (thus
1cocartesian is the same thing as cocartesian). Analogous terminol
ogy will be used in the cartesian case. We also use this terminology for
cubical diagrams of spaces.
2.1. Poincare duality spaces. Let X be a homotopy finite space
equipped with a local coefficient system L (i.e., a functor from the
fundamental groupoid of X to the category of abelian groups) which
is pointwise free abelian of rank one. Let [X] 2 Hn(X; L) be a class.
The data (X; L; [X]) equip X with the structure of a Poincare duality
space of dimension n if cap product induces an isomorphism
~=
\[X] :H*(X; M) ! Hn* (X; L M)
for every local system M. When L and [X] are understood, we will
simply refer to X as a PD space.
If ssx denotes the fundamental group at x 2 X, then the local system
which assigns to x the integral group ring Z[ssx] is denoted by . It is
a fact that \[X] defines an isomorphism for all local systems M if and
only if it does for (cf. [Wa4 , 1.1]).
A cofibration pair (X; @X) consisting of homotopy finite spaces to
gether with L and a class [X] 2 Hn(X; @X; L) will be called a Poincare
6 JOHN R. KLEIN
pair of dimension n if, similarly, cap product induces an isomorphism
~=
\[X] :H*(X; M) ! Hn* (X; @X; L M)
for all M and moreover, the restriction of L to @X together with the
image of [X] under the boundary homomorphism Hn(X; @X; L) !
Hn1 (@X; L) equips @X with the structure of a PD space. Again, it
is enough to check these conditions in the case when M is . We will
refer to Poincare pairs as PD pairs.
There is often redundancy (compare [Br , 2.2.3]).
Lemma 2.1. If (X; @X) is 2connected and \[X] is an isomorphism,
then (X; @X) is a PD pair.
Proof. I learned of the following argument from A. Ranicki: let [@X]
denote the image of [X] with respect to the boundary homomorphism.
Since ss1(X) ~=ss1(@X), it will be enough to check that
\[@X] :H*(@X; ) ! Hn*1 (@X; L )
is an isomorphism.
Consider the commutative diagram
C*(X; @X; ) _______________C*(X;w) ________________wC*(@X; )
  
  
  
  
u u u
Cn* (X; L ) ________Cn*w(X; @X; L ) ______wCn*1 (@X; L )
whose horizontal maps induce long exact sequences in homology. The
middle and right vertical maps are given by chain level versions of \[X]
and \[@X] respectively. The left vertical arrow is also given by a cap
product with [X]. By hypothesis, the middle vertical map induces a
homology isomorphism. Therefore, by the five lemma, it is sufficient
to show that the left vertical map induces a homology isomorphism.
The left vertical map can also be obtained as follows: by hypothesis,
we have a chain homotopy equivalence
\[X] :Cn* (X; L ) '!C*(X; @X; ) :
For a left (right) module P , let P # = hom (P; ) denote its dual
right (left) module given by taking module homomorphisms into .
Dualize the map \[X] to get another chain homotopy equivalence,
(\[X])# .
Since C*(X; @X; ) is (up to homotopy) a chain complex of finitely
generated free modules, C*(X; @X; )# is identified with C*(X; @X; )
(because a finitely generated free module is canonically isomorphic
to its double dual). Similarly, Cn* (X; L )# is identified with
Cn* (X; L). With respect to these identifications, the map (\[X])#
is the left vertical map of the diagram (this follows from the way cap
POINCARE DUALITY EMBEDDINGS 7
products are constructed). Consequently, the left vertical map of the
diagram is a chain homotopy equivalence. ___
2.2. Poincare duality embeddings.
Definition 2.2. Let f :K ! X denote a map from a connected homo
topy finite space K to a PD space X or PD pair (X; @X) of dimension
n. A PD embedding for f is a commutative square of spaces
A  ! C
? ?
? ?
y y j
K  ! X
f
j
and a choice of factorization of @X X as @X ! C ! X, such that:
o The square is cocartesian.
o The spaces A and C are homotopy finite.
o The image of the fundamental class [X] under the composite
Hn(X; @X) ~=Hn(X ; @X) ! Hn(X ; C) ~=Hn(K ; A)
equips (K ; A) with the structure of a PD pair (here we are sup
pressing the local systems in the notation). Similarly, the image of
[X] with respect to the evident map Hn(X; @X) ! Hn(C ; @X qA)
equips (C ; @X q A) with the structure of a Poincare pair.
o If hodim K k, then A ! K is (n k 1)connected.
The space C is called the complement, and A is called the gluing
space. If there exists a PD embedding for f , then we say that f PD
embeds. If hodim K k, then we say that the codimension of the
embedding is n k.
Again, there is often some redundancy:
Lemma 2.3. If hodim K n 3, and all of the conditions of the defi
nition are known to hold except perhaps duality for the pair (C ; @XqA),
then the diagram is a PD embedding.
Proof. This is essentially proved in [Wa5 , 2.7ii,11.1], but with a missing
hypothesis. We need to establish that (C ; @X q A) is a Poincare pair.
For this, it will be enough to check that
\[C] :H*(C) ! Hn* (C ; @X q A)
is an isomorphism, where [C] 2 Hn(C ; @X q A) is obtained from [X]
as indicated in the definition. By the cohomology exact sequence
. . .!H*(C ; @X q A) ! H*(C ; A) ! H*(@X) ! . . .
and the naturality of cap product, it suffices to show that
\[C] :H*(C ; A) ! Hn* (C ; @X)
is an isomorphism (since @X is a PD space).
8 JOHN R. KLEIN
The exact sequence
* * *
. . .!H (C ; A) ! H (X) ! H (K) ! . . .
associated with the cocartesian square, the naturality of cap product,
and the fact that (X; @X) and (K ; A) are PD pairs implies that \[C] __
is an isomorphism. __
Remarks 2.4. (1). Assume that K is a PD space of dimension k with
k n 3. Then the map A ! K has an (n k 1)spherical homotopy
fiber over any point (by [Sp , 4.4], [Br , I.4.3]). In this situation A ! K
plays the role of normal bundle for K in X. This particular kind of
PD embedding is discussed in [Wa5 , Chap. 11].
(2). PD embeddings arise from manifold embeddings in the follow
ing way. Suppose that V is a closed regular neighborhood of a k
dimensional finite connected polyhedron embedded in the interior of a
compact nmanifold N. Let C be the closure of N \ V , and let A be
the boundary of V . Then N = V [A C, and the data determine a PD
embedding of V in N.
(3). If k is an integer such that hodim K k n 3, then to check
that A ! K is (n k 1)connected, it is sufficient to know that it is 2
connected, once we know that (K ; A) satisfies ndimensional Poincare
duality. The reason this is true is that duality implies the homology of
(K ; A) (with respect to any coefficient system) will vanish in degrees
n k 1. The relative Hurewicz theorem [Wh , 7.2] then shows that
the relative homotopy groups will also vanish in this range when A ! K
is 2connected.
In any case, the assumption that A ! K is (n k 1)connected
arises from geometry: if the PD embedding arises from a manifold
embedding as in (2) above, then this connectivity is a consequence of
transversality.
The next lemma concerns the extent to which the notion of PD
embedding is homotopy invariant.
Lemma 2.5. Suppose that f :K ! X PD embeds. Then
(1). If g is homotopic to f , then g PD embeds.
(2). Let ae :L ~! K be a homotopy equivalence. Then f O ae :L ! X
PD embeds.
(3). Let h :(X; @X) ~! (Y; @Y ) be a homotopy equivalence. Then
h O f :K ! Y PD embeds.
Proof. Let
A  ! C
? ?
i?y ?y
K  ! X
f
POINCARE DUALITY EMBEDDINGS 9
be a PD embedding.
(1). Replace f by g and C by the mapping cylinder C of A q @X ! C.
A choice of homotopy from f to g induces a map C ! X which defines
the desired PD embedding for g.
(2). Let ae1 :K ! L be a choice of homotopy inverse for ae. Then the
diagram
A  ! C
? ?
ae1Oi?y ?y
L  ! X
fOae
is homotopy commutative. As in the first part, replace C by a suitable
mapping cylinder to get the desired PD embedding of f O ae.
(3). By taking an appropriate mapping cylinder, we can assume that
the map A q @X ! C is a cofibration. Let C0 denote the space
(A q @Y ) [Aq@X C :
Then there is an evident PD embedding
A   ! C0
? ?
? ?
y y :
K   ! Y
hOf
__
__
2.3. Stabilization. Let S() ! X denote a (j 1)spherical fibration
with S() not necessarily cofibrant. Even if S() were cofibrant, the
restriction S(Z) of S() along a cofibration Z ae X needn't be.
For this reason, we introduce the following technical innovation: let
[ :Top ! Top be the functor which maps a space to the geometric
realization of its total singular complex. Then [ is pointwise equivalent
to the identity. Furthermore, [ applied to a monomorphism gives a
cofibration. If F :J ! Top denotes a finite diagram, let hocolim [F
denote the effect of first applying [ pointwise and then taking the re
sulting homotopy colimit.
With respect to this convention, let @D() be defined as the hocolim [
of the diagram
@X S(@X) ! S() :
Similarly, Let D() be defined as hocolim [of the diagram
X S() =!S()
(equivalently, the mapping cylinder of the map [(S()) ! [(X)).
Then (D(); @D()) is a PD pair of dimension n+ j (the orientation
and fundamental class are induced from the ones on (X; @X) via the
10 JOHN R. KLEIN
Thom isomorphism). This construction is the Poincare analogue of
replacing an nmanifold with boundary by the total space of a jdisk
bundle which lies over it.
Given a PD embedding
A   ! C
? ?
? ?
y y ;
K   ! X
f
we shall construct another PD embedding whose target is D(). Let
C be the space
hocolim [(C S(C) ! S())
and similarly, let K A be the space
hocolim [(A S(A) ! S(K)) :
Then these assemble to a PD embedding
K A  ! C
? ?
? ?
y y
D(K)  ! D()
where D(K) means hocolim [of the diagram K S(K) ! S(K) .
In particular, K and D(K) are canonically homotopy equivalent, so
by 2.5(2), we obtain a PD embedding
K A ______w C

 
 
  :
 
 
u u
K ________wD()
A special case of this construction occurs when S() ! X is the
trivial fibration with fiber S0. If this is the case, the pair (D(); @D())
identifies with (Xx I; @(Xx I)), and the new PD embedding is called
the decompression. It increases the codimension by one, and is the
Poincare analogue of the standard way of passing from an embedding
in a manifold to one in the product of the manifold with the unit
interval.
In this instance, C is a variant of the fiberwise suspension of
C ! X. This fiberwise suspension, denoted X C, is given by the
double mapping cylinder
Xx 0 [ Cx [0; 1] [ Xx 1 :
the map @(Xx I) ! Xx I factors canonically through X C. Note that
@(Xx I) is just X @X.
POINCARE DUALITY EMBEDDINGS 11
With respect to this variant of the construction, the PD embedding
becomes
K A ______wX C
 
 
  :
 
 
u u
K ________XxwI
Notes 2.6. The basic reference for much of the material in this section
is Wall's foundational paper [Wa4 ]. For a recent survey about Poincare
duality spaces, see [Kl2 ]. The definition of PD embedding given here
is very similar to the one proposed by Levitt [Le , 2.3].
3. Existence of stable PD embeddings
Given a PD pair (X; @X) of dimension n, the cartesian product with
a disk Dj yields a PD pair (Xx Dj; @(Xx Dj)) of dimension n+j, where
@(Xx Dj) is the amalgamated union Xx Sj1 [ (@X)x Dj.
Given a map f :K ! X, we let f also denote the composition
f j
K ! X Xx D
where the second of these maps is given by identifying X with Xx 0
by means of the identity.
Lemma 3.1. There exists a positive integer j such that f :K ! Xx Dj
PD embeds.
Proof. Suppose first that (X; @X) has the homotopy type of a PL
manifold with boundary. If this holds, we may further assume that
(X; @X) is actually a PL manifold, by 2.5(3). Take the cartesian prod
uct with a suitably large disk Dj, and use general position to replace
f :K ! Xx Dj by an embedding up to homotopy (see e.g. [Wa1 ]).
Thus there is a codimension zero compact submanifold N in the in
terior of Xx Dj and a homotopy equivalence K ' N such that the
composite K ' N Xx Dj coincides with f up to homotopy. Apply
ing 2.5(1), obtain the desired PD embedding.
Now assume that (X; @X) is general. By regular neighborhood the
ory [RS , Chap. 3], for some t n there exists a compact PL man
ifold Nn+t Rn+t equipped with a decomposition of its boundary
@N = @ N [@0N @+ N, and a homotopy equivalence of pairs (X; @X) '
(N; @ N). Then the homotopy fiber of the map @+ N ! N is an
(n+ t 1)sphere (see [Sp , 4.4], [Br , I.4.1]).
By the previous case, we can assume that the composite
K ! X ~!N
PD embeds.
12 JOHN R. KLEIN
Choose a fiber homotopy inverse S() ! N for @+ N ! N in the
reduced Grothendieck group of spherical fibrations over N. Suppose
that the fiber of S() ! N is S`1.
Stabilizing with respect to S() ! X, we obtain a PD embedding
K U ______w C

 
 
  :
 
 
u u
K ________wD()
It is straightforward to check that there is a homotopy equivalence of
pairs
(D(); @D()) ' (Xx Dt+`; @(Xx Dt+`))
in such a way that the map K ! D() corresponds to f up to homo
topy. Applying 2.5(1) completes the proof.
__
__
Notes 3.2. This is the only argument of the paper which uses mani
folds. However, there is an alternative proof which is entirely homo
topy theoretic. The alternative argument requires the technology of
fiberwise/equivariant duality. For reasons of space we relegate this to
another paper.
We have implicitly used the Spivak normal fibration of (X; @X) in
the proof of 3.1. For a homotopy theoretic proof of the existence of the
Spivak fibration, see [Kl1 ], [Kl2 ].
4. Truncation, cocartesian replacement, and fiberwise
desuspension
4.1. The Truncation Lemma. Let ss be a group. Let P be a based
connected space with fundamental group ss. Let U ! P be a map of
spaces. Let K* be a chain complex of projective Z[ss]modules. Lastly,
let C*(P; U) ! K* be a Z[ss]linear chain map, where C*(P; U) is the
free chain complex of Z[ss]modules which computes the relative ho
mology of U ! P .
Assume that dim K* n in the sense that its cohomology (for any
coefficient module) vanishes in degrees > n. Assume that the chain
map C*(P; U) ! K* is nconnected.
Lemma 4.1. (Truncation). If n 2 there exists a factorization
U ! A ! P
such that:
o C*(A; U) ! K* is a chain homotopy equivalence.
o U ! A is a relative CW complex of dim n.
o A ! P is (n 1)connected.
POINCARE DUALITY EMBEDDINGS 13
Proof. For this proof, chain complexes and homology are understood
to be taken with respect to the coefficient module Z[ss].
Factor U ! P as U ae W ! P where (W; U) is a relative CW
complex of dimension n 1 and the map W ! P is (n 1)connected.
The chain map C*(W; U) ! K* is (n 1)connected. The cohomology
of its mapping cone with respect to any coefficient module vanishes
in degrees > n, (since dim K* n and dim C*(W; U) n 1). The
homology of the mapping cone vanishes in degrees < n. Therefore, by
an observation of Wall, homology of the mapping cone in degree n is a
projective Z[ss]module (see [Wa5 , 2.3] or the proof of [Wa2 , 2.1]). Call
it Q.
Case (1): Q is free. Choose a basis for Q. The long exact sequence
gives a surjection Hn(P ; W ) ! Q. The relative Hurewicz theorem
gives a surjection ssn(P ; W ) ! Hn(P ; W ). Choose a lift for each basis
element of Q and attach ncells to W corresponding to these lifts. Call
the resulting space A. Then A gives the desired factorization.
Case (2): Q is arbitrary. At the cost of adding cells we can make Q free
as follows: let Q0be such that Q Q0is free. Let F be the free module
Q0 Q Q0 . ...Then Q F ~= F , by the Eilenberg Swindle. Attach
(n 1)cells to W in a trivial way indexed by a basis for F . Let W 0be
the result of this procedure. Extend W ! P to W 0by mapping the
new cells to the basepoint of P . The relative of homology of W 0! P is
again concentrated in degree in n and it is isomorphic to Q F . Case __
(1) now applies. __
We next give some applications.
4.2. Cocartesian Replacement. Let
X;   ! X2
? ?
? ?
y y
X1   ! X12
be a commutative square of connected based spaces. We shall provide
criteria for deciding when it is possible to replace X; by another space
such that the new square is cocartesian.
Let ss = ss1(X12). Assume that
(1). The square is jcocartesian for some j 3.
(2). The homomorphism ss1(X;) ! ss is an isomorphism.
(3). The relative cohomology (with any Z[ss]module coefficients) of
the map X1 _ X2 ! X12 vanishes in degrees > j.
14 JOHN R. KLEIN
Theorem 4.2. Under the above assumptions, there exists a based space
A and a based map A ! X; such that the resulting diagram
A   ! X2
? ?
? ?
y y
X1   ! X12
is cocartesian. Furthermore, the map A ! X; can be chosen as (j 2)
connected.
Proof. Let K* denote the Z[ss]module chain complex given by
holim (C*(X1) ! C*(X12) C*(X2))
(equivalently, the desuspension of the mapping cone of C*(X1)C*(X2) !
C*(X12)). Then the map
C*(X;) ! K*
is (j 1)connected. Furthermore, the cohomology of K* with respect
to any coefficient system vanishes in degrees > j 1. Applying the
Truncation Lemma 4.1 (with U = ;), we obtain a map A ! X; with
the desired properties. (To check that the resulting square is cocarte
sian, use the Whitehead theorem in conjunction with the fact that the
map hocolim (X1 A ! X2) ! X12 induces an isomorphism of fun
damental groups and an isomorphism in cohomology with respect to __
any coefficient module.) __
We also have a relative version:
Addendum 4.3. With a square satisfying conditions (1) and (2) above
(cf. before 4.2), let Z ! X; be a map of spaces (where Z is not neces
sarily based). Suppose instead of (3) that the map
hocolim (X1 Z ! X2) ! X12
has vanishing relative cohomology in degrees > j (with respect to any
Z[ss]module coefficients).
Then there exists a space A and a factorization Z ! A ! X; such
that the square given by replacing X; with A is cocartesian. Further
more A ! X; can be chosen as (j 2)connected.
Clearly, this specializes to 4.2 by taking Z to be a point.
Proof. Let K* be the chain complex given by taking the mapping cone
of the map
C*(Z) ! holim (C*(X1) ! C*(X12) C*(X2)) :
Apply 4.1 to the evident map
C*(X; ; Z) ! K* :
POINCARE DUALITY EMBEDDINGS 15
This gives a factorization Z ! A ! X; such that the composite
C*(A ; Z) ! C*(X; ; Z) ! K*
is a chain equivalence. For this choice of A, the new square is cocarte_
sian. __
4.3. Fiberwise Desuspension. Let f :A ! X be a map of spaces.
Let Top f be the category in which an object is specified by a factor
ization A ! Y ! X. A morphism is a map Y ! Z which preserves
factorizations. A morphism is a weak equivalence, fibration or cofibra
tion if it respectively is so when considered in Top by means of the
forgetful functor Top f ! Top .
Lemma 4.4. With respect to the above conventions, Top fis a model
category.
Proof. For any model category C, Quillen [Qu , II.2.8] shows that the
over category C=X and the opposite category Cop are also model cate
gories. The weak equivalences and fibrations of C=X are defined via the
forgetful functor C=X ! C. The weak equivalences and the cofibrations
of Cop correspond to the weak equivalences and fibrations of C.
The map f :A ! X defines an object of Top =X. Denote it by [f ].
Then Top f is isomorphic to
((Top =X)op=[f]op)op
__
and the result follows by remarks in the previous paragraph. __
Remark 4.5. An object Y 2 Top f is fibrant when the structure map
Y ! X is a Serre fibration. It is cofibrant when the structure map
A ! Y is a Serre cofibration.
We will assume in what follows that X is a connected space. Using
the above conventions, we shall regard fiberwise suspension as a functor
X : Top =X ! Top r ;
where r :X q X ! X is the fold map. It is straightforward to check
that X maps cofibrant objects to cofibrant objects.
Definition 4.6. An object Y 2 Top r is jconnected if the structure
map Y !X is a (j+ 1)connected map of topological spaces. We will
say that Y has dimension n if the structure map X q X ! Y has
the property that its relative cohomology (with respect to the pullback
of any local system on X along Y ! X) vanishes in degrees > n.
Theorem 4.7. (Desuspension). Let Y 2 Top r be a fibrant and cofi
brant object which is jconnected and has dimension 2j+ 1, for some
integer j 1. Then there exists an object A 2 Top =X and a weak
equivalence
X A ~!Y :
Moreover, the map A ! X can be chosen as jconnected.
16 JOHN R. KLEIN
Proof. Let i :X ! Y be the maps obtained by restricting the struc
ture map X q X ! Y to each summand. Let X be the effect of
~
factorizing the map i as X ae X i Y and let X+ be defined simi
larly using i+ . We have a cartesian square
B   ! X+
? ?
? ?
y y ;
X   ! Y
where B := X xY X+ denotes the fiber product of i and i+ . Each
map in this square is at least 2connected. Furthermore, the square is
(2j+ 1)cocartesian by the dual BlakersMassey excision theorem [Go ,
p. 309].
The map X qX+ ! Y has vanishing relative cohomology in degrees
> 2j+ 1, so we may apply 4.2 to conclude that there exists a (2j 1)
connected map of spaces A ! B such that the square
A  ! X+
? ?
? ?
y y
X  ! Y
is cocartesian. Make A an object of Top =X by means of the composite
A ! Y ! X.
There is an evident chain of weak equivalences of Top r given by
X A = Xx 0[Ax [0; 1][Xx 1 ~ X x0[Ax [0; 1][X+ x1 ~! Y :
The proof is completed using a wellknown general fact about model
categories: an isomorphism in the homotopy category from a cofibrant __
object to a fibrant object always lifts to a weak equivalence. __
Remark 4.8. Theorem 4.7 (and its relative version 4.9 below) will be
used to construct the complement for the Poincare embedding in the
proof of Theorem A. Richter has pointed out that one can get by with
slightly less. Namely, the above cocartesian square involving A, X ,
X+ and Y can be inserted into the proof of Theorem A instead of
the choice of fiberwise desuspension. This replacement would be one
way of removing the fiberwise homotopy theory in this paper, but for
aesthetic reasons we refrain from doing so.
Here is the relative version of 4.7:
Addendum 4.9. Let X Z ae Y be a cofibration of Top r for some
cofibrant object Z 2 Top =X . Assume that
o the relative cohomology of the underlying map vanishes in degrees
> 2j+ 1 for j 1 (for all coefficient systems).
o The object Y is jconnected.
POINCARE DUALITY EMBEDDINGS 17
Then there exists a cofibrant object A 2 Top =X , a morphism Z ! A,
and a weak equivalence
X A ~!Y
which is relative to X Z. Moreover, the map A ! X can be chosen as
jconnected.
__
Proof. Follow the proof of 4.7, but work relative to Z and use 4.3. __
Notes 4.10. Special cases of the Truncation Lemma 4.1 are to be found
in the literature. The first result in this direction that I know of is in
a paper by Berstein and Hilton [BH , Thm. 2.1], who in effect prove a
version of 4.1 when ss is trivial. Richter [Ri2 ] had a version of 4.2 when
X is simply connected.
The Desuspension Theorem 4.7 reduces to the usual Freudenthal
suspension theorem when X is a point. On the other hand, 4.7 isn't
the kind of fiberwise desuspension result that has appeared in the fiber
wise topology literature. The latter falls under the rubric of based fiber
wise homotopy theory, and concerns the extent to which the reduced
suspension functor
X : Top idX! Top idX
is surjective on the level of homotopy categories.
Incidentally, our relative version 4.9 contains both the based and
unbased variants as extreme cases, where we desuspend relative to
either the initial or terminal object of Top =X. Taking Z to be the
empty space, we obtain 4.7. When Z = X, we obtain the based result.
Although there are two different forgetful functors Top r ! Top idX,
the based and unbased suspensions are generally very different. For ex
ample, take X = S1. Consider the nontrivial bundle ! S1 with
fiber S1, where is the Klein bottle. Then = S1S1, where we
fiberwise suspend the multiplication by 2 map S1 ! S1. But the mul
tiplication by 2 map doesn't admit a section. This example shows that
there are objects of Top r which fiberwise desuspend in the unbased
sense but which fail to do so in the based sense.
5. The Face Theorem
We now prove a technical result which concerns the degree to which
the faces in a cartesian 3cube are cocartesian. The result will be crucial
in the proof of Theorem A.
18 JOHN R. KLEIN
Let
X; _______________wX3
ae  ae 
ae  ae 
aeAE  aeaeAE 
X1 _______________wX13 
 
   
   
   
 u  u
 X2 _______ ______wX23
 
 ae  ae
 
 ae  ae
uaeaeAE uaeaeAE
X12 ______________wX123
be a commutative 3cube of spaces.
Theorem 5.1. Suppose that the 3cube is cartesian and that
o the spaces XS are connected for each nonempty S {1; 2; 3};
o each two dimensional face which meets X123 is cocartesian;
o the maps Xj ! Xij are kiconnected and the maps Xi ! Xij are
kjconnected, for 1 i < j 3.
Then each of the squares
X;  ! Xj
? ?
? ?
y y
Xi  ! Xij
is (k1+ k2+ k3)cocartesian for 1 i < j 3. Furthermore, if k1+ k2+ k3
1, then X; is nonempty. If two of the integers ki are 1, then X; is
connected.
Remark 5.2. Here is the why the result is true on the level of ordinary
homology. Call the 3cube Xo and rewrite it as a map of squares
Yo ! Zo. Let H*(Xo) mean the reduced homology of the iterated
homotopy cofiber of Xo. This measures the extent to which Xo fails to
be cocartesian on the level of homology.
For general reasons, there is a long exact sequence
. ..!H*(Yo) ! H*(Zo) ! H*(Xo) ! . .:.
By hypothesis, H*(Zo) is trivial. The dual BlakersMassey theorem
for 3cubes [Go , Thm 2.6] implies that H*(Xo) vanishes in degrees
k1+ k2+ k3+ 1. This shows that H*(Yo) vanishes in degrees k1+ k2+ k3,
which is what the theorem asserts on the level of homology.
Proof of 5.1. We can map the 3cube Xo to another 3cube by a point
wise weak equivalence such that every map in the new cube is a fibra
tion. So without loss in generality, we will assume that the maps of Xo
are all fibrations.
POINCARE DUALITY EMBEDDINGS 19
If k1+ k2+ k3 1, then remark 5.2 shows that H*(Yo) vanishes in
degrees 1. A straightforward argument involving the MayerVietoris
sequence implies that X; is nonempty.
If two of the ki, say k1; k2 1, then the map H*(X;) ! H*(X1) is
an isomorphism in degree 0, because X2 ! X12 is 1connected and the
square containing X;; X1; X2 and X12 is homologically 2cocartesian
(again by 5.2). It follows that X; is connected (since X1 is connected).
We now prove the part of the statement concerning the degree to
which the 2faces meeting X; are cocartesian. Choose any 2face of Xo
which meets X;, say
X;  ! X1
? ?
? ?
y y :
X2  ! X12
It will be enough to show that this square is (k1+ k2+ k3)cocartesian.
Without loss in generality, we can assume that ki 0. Call this square
Yo. We consider two cases.
Case (1). k3 = 0.
In this instance we are asking whether Yo is (k1+ k2)cocartesian.
The 2face opposite to Yo in the 3cube Xo is cocartesian (the one
involving X2; X3; X23 and X123), so by the BlakersMassey theorem for
squares [Go , p. 309] the latter 2face is (k1+ k2 1)cartesian.
But this implies that Yo is also (k1+ k2 1)cartesian, since Xo is
cartesian (compare [Go , Prop. 1.6]). Applying the dual BlakersMassey
theorem, [Go , Th. 2.6] we infer that Yo is (k1+ k2)cocartesian. This
completes case (1).
Case (2). k3 1.
The square Yo factorizes into four squares:
X; _____________X1wxX13 X3 _____________X1w
  
  
  
u u u
X2 xX23 X3 _______wX12 xX123 X3 ______wX12 xX123 X13
  
  
  
u u u
X2 ___________wX12 xX123 X23 ___________wX12
where the new spaces introduced are all fiber products. Let us give
each of the newer squares a name: the square on the upper left will be
denoted (I), the one on the upper right (II), lower left (III) and lower
20 JOHN R. KLEIN
right (IV). It will be enough to show that each of the squares (I)(IV)
is (k1+ k2+ k3)cocartesian.
Claim 5.3. The square (IV) is 1cocartesian.
The square (IV) is obtained from the 1cocartesian square
X3   ! X13
? ?
? ?
y y
X23   ! X123
by taking the pullback of its spaces along the map X12 ! X123. This
procedure preserves the degree to which a square is cocartesian, so 5.3
follows.
Claim 5.4. The squares (II) and (III) are (k1+ k2 + k3)cocartesian.
First note a general fact: let Ao ! Co and Bo ! Co be morphisms of
squares of spaces such that each of the squares Ao; Bo and Co is carte
sian. and a pointwise fibration (i.e., AS ! CS is a weak equivalence
and fibration for all S {0; 1}). Then the square Ao xCo Bo (given by
AS xCS BS) is also cartesian.
A straightforward check (which we omit) shows that the square (II)
is obtained in this fashion. Consequently, (II) is cartesian. In particu
lar, the map X1 xX13 X3 ! X1 is k1connected (since it is opposes the
map X3 ! X13 in (II)). Similarly, the map X1 xX13 X3 ! X12xX123X3
is (k2+ k3 1)connected, since its connectivity may be identified that of
X1 ! X12xX123X13, and the latter map is (k2+ k3 1)connected, by the
BlakersMassey theorem applied to the square involving X1; X12; X13
and X123. Claim 5.4 for (II) now follows by applying the dual Blakers
Massey theorem. The argument for the square (III) is similar, and will
therefore be omitted.
Claim 5.5. The square (I) is (k1 + k2 + k3)cocartesian.
The square (I) is cartesian, since Xo is (the homotopy limit of Xo
with X; deleted coincides with the homotopy limit of (I) with X;
deleted). As the map X1 xX13 X3 ! X12 xX123 X3 is (k2+ k3 1)
connected (see 5.4 above), we infer (using the cartesianness of (I))
that the map X; ! X2 xX23 X3 is also (k2+ k3 1)connected.
The map X; ! X1 xX13 X3 is (k1+ k3 1)connected (this can be
seen as follows: The squares (I) and (III) taken together are cartesian,
and the parallel map X2 ! X12 xX123 X23 is (k1+ k3 1)connected,
by the BlakersMassey theorem for the cocartesian square involving
X2; X12; X23 and X123).
It follows by the dual BlakersMassey theorem that the square (I) is
(k2+ k3 1) + (k1+ k3 1) + 1 = k1 + k2 + 2k3  1
POINCARE DUALITY EMBEDDINGS 21
cocartesian. By assumption k3 1, so the displayed integer is at least
k1+ k2+ k3. This establishes 5.5, and completes the proof of 5.1 ___
The following lemma will be used in the next section.
Lemma 5.6. Let
X;   ! X2
? ?
? ?
y y
X1   ! X12
be a commutative square of spaces.
(1). If the diagram is cocartesian and the map X; ! X2 is rconnected,
then the map X1 ! X12 is also rconnected.
(2). Assume that the diagram is cocartesian. If the map X; ! X1
is 2connected and the map X1 ! X12 is sconnected, then the map
X; ! X2 is also sconnected.
Proof. (1). By homotopy invariance, we can assume that X2 is obtained
from X; by attaching cells of dimension > r. Then up to homotopy,
X12 obtained from X1 by attaching cells of dimension > r. Hence
X1 ! X12 is rconnected.
(2). The assertion is trivial if s 1. We now argue by induction.
Suppose that the result holds for some s 1, and let X1 ! X12 be
(s+ 1)connected. It follows by the induction hypothesis that the map
X; ! X2 is sconnected. Let r be the connectivity of the map X; !
X1. The BlakersMassey excision theorem implies that the diagram
is (r + s  1)cartesian. Since r 2, we infer that the diagram is
(s + 1)cartesian. Consequently, X; ! X2 is also (s + 1)connected. __
This completes the inductive step. __
Corollary 5.7. Let
A  ! B  ! C
? ? ?
? ? ?
y y y
X  ! Y  ! Z
be a commutative diagram of connected spaces. Assume that
o the outer square is jcocartesian for some j 0,
o the righthand square is cocartesian, and
o B ! C is 2connected.
Then the lefthand square is also jcocartesian.
Proof. Assume without loss in generality that A ! B and B ! C are
cofibrations. For formal reasons, if the righthand square is cocartesian
22 JOHN R. KLEIN
then so is the square
X [A B   ! Y
? ?
? ?
y y :
X [A C   ! Z
Since B ! C is 2connected we can apply 5.6(1) to infer that the left__
vertical map is also 2connected. Now use 5.6(2). __
Notes 5.8. Richter [Ri1 ] had the first proof of 5.1 under the assumption
that all spaces of X; are simply connected (see 5.2 for the proof in this
instance). My original proof of the face lemma required each of the ki
to be 2. The above proof, due to Goodwillie, places no constraints
on the ki.
6. Proof of Theorem A
We recall the setup of the introduction. Let K be a connected
homotopy finite space with hodim K k. Let (X; @X) be a PD pair
of dimension n. Let f :K ! X be an rconnected map. Recall the
statement of Theorem A:
Theorem 6.1. If k n 3 and r 2k n+ 2, then f PD embeds.
Proof. By 3.1 there exists a nonnegative integer j such that the com
posite
f j
K ! X Xx D
PD embeds. By a downward induction on codimension, we may assume
that j = 1. The strategy will be to recognize the PD embedding of
K ! Xx I as a decompression of a PD embedding of f :K ! X (cf.
2.3).
Let
A0 ________Ww
 
 
 
 
u u
K ______wXx I
f
be a PD embedding of K ! X Xx I (in codimension n k+ 1).
Recall that there is a factorization @(Xx I) ! W ! Xx I. The space
@(Xx I) is just X @X. In particular W is an object of Top r and
X @X ! W is a morphism of Top r .
Using functorial factorization, we may assume that W is fibrant.
Using the projection Xx I ! X, we will from now on be considering
the square given by replacing Xx I by X. We can also assume that
the map f :K ! X is a fibration.
POINCARE DUALITY EMBEDDINGS 23
Claim 6.2. The object W desuspends relative to X @X, i.e., there
is a cofibrant object C 2 Top =X, a cofibration @X ae C and a weak
equivalence
X C ~!W
which is relative to X @X. Moreover, the map C ! X can be chosen
as (n k 1)connected.
The proof of 6.2 will use the Desuspension Theorem 4.9. Since
W ! Xx I opposes A0! K in a cocartesian square, 5.6(1) shows that
the object W 2 Top r has connectivity one less than the connectivity
of the map A0! K. Since (K ; A0) is a PD pair of dimension n+ 1, this
connectivity is just n k. Consequently, W is an (n k 1)connected
object. In particular, the codimension 3 hypothesis says that X and
W have isomorphic fundamental groups, so every local system on W
arises by pullback from one on X.
Furthermore, there are isomorphisms
\[W] excision
H*(W ; X @X) ~= Hn+1* (W ; A0) ~= Hn+1* (X ; K)
for any local system on X. Hence, the fact that f is rconnected
implies these groups vanish whenever * n r+ 1. Therefore the map
X @X ! W has vanishing relative cohomology in these degrees.
Applying 4.9, we see that W desuspends relative to X @X provided
that n r 2(n k 1) + 1. This will happen if r 2k n+ 1, so we
have one dimension to spare. This establishes 6.2.
Let K qK ! K be the fold map. Let K qK ! W be the composite
fqf
K q K ! X q X W :
These maps are compatible with projection to X, and therefore define
a map
K q K ! K xX W :
where the target denotes the fiber product of K with W along X (re
call we have arranged it so that f :K ! X is a fibration, so the fiber
product has the correct homotopy type). By the BlakersMassey the
orem, the map A0 ! K xX W is (r+ n k 1)connected. But K q K
has hodim k, so obstruction theory gives us a factorization up to
homotopy
K q K ! A0 ! K xX W
provided r 2k n+ 1, so we again have one dimension to spare. By
functorial factorization, we can assume that the map A0 ! K xX W
is a fibration. But then the homotopy lifting property gives us a fac
torization on the nose.
24 JOHN R. KLEIN
The data constructed thus far may be displayed as the following
commutative 3dimensional punctured cube:
K

ae 
ae 
aeAE 
K ______________wA0 
  
  
   :
  u
 C ______ _____wX
 
 ae  ae
 ae  ae
uaeAE uaeAE
X ______________wW
The bottom 2face of this cube is the cocartesian square associated with
the weak equivalence X C ~! W ; the space X denotes the mapping
cylinder of C ! X.
Let B be homotopy inverse limit of the punctured cube. Then the
resulting 3cube of spaces
B ______________wK
ae  ae 
ae  ae 
aeAE  aeAE 
K ______________wA0 
   
   
   
 u  u
 C ______ ______Xw
 
 ae  ae
 ae  ae
uaeAE uaeAE
X ______________wW
is commutative up to canonical homotopy.
It will be more convenient to work with a commutative version of
this cube. One way to do this is as follows: map the original punctured
cube to a new punctured cube by a pointwise weak equivalence, in such
a way that the limit of the new punctured cube is the homotopy limit
of the original punctured cube. The new punctured cube together with
its limit gives the desired strictly commutative cube. In what follows,
we will be working with the commutative cube. However, to avoid a
notational clutter, we will keep the notation of the old cube to designate
the spaces in the new one.
Consider next the top 2face of the 3cube.
POINCARE DUALITY EMBEDDINGS 25
Claim 6.3. The top 2face
B  ! K
? ?
? ?
y y
K  ! A0
is (2(n k 1) + r)cocartesian. Moreover, the space B is connected.
We wish to apply the Face Theorem 5.1. We therefore need to verify
its hypotheses.
All spaces of the 3cube with the exception of perhaps B are con
nected. It is straightforward to check that each 2face meeting W is
cocartesian. The maps labeled K ! A0 and C ! X are (n k 1)
connected. The maps labeled K ! X are rconnected. With the
notation as in the Face Theorem, this means k1 = k3 = n k 1 and
k2 = r. Since k n 3, we infer that k1; k3 2. Consequently, we may
apply the Face Theorem to conclude that B is connected and that the
square is (2(n k 1) + r)cocartesian. This proves 6.3.
We continue to restrict our attention to the top face.
Claim 6.4. There exists a connected space A and a (2(n k 1)+ r 2)
connected map A ! B such that the square
A  ! K
? ?
? ?
y y
K  ! A0
(given by replacing B by A) , is cocartesian.
Choose a basepoint for B to equip the top 2face with the structure
of a square of based spaces. The map K_K ! A0has vanishing relative
cohomology (with respect to any local system on A0) in degrees n,
since A0 is a PD space of dimension n and k n 3. Thus if
n 2(n  k  1) + r ;
i.e., when r 2k n+ 2, we can apply 4.2 to obtain a space A and a
(2(n k 1)+ r 2)connected map A ! B which satisfies the statement
of the claim.
We note that this is the first (and only) time in the argument that
the sharp lower bound for the connectivity of f is used.
Now consider one of the other 2faces of the 3cube meeting B,
labeled
B   ! C
? ?
? ?
y y :
K   ! X
26 JOHN R. KLEIN
Recall that @X ! X comes equipped with a factorization @X ! C ! X .
Replace B by A, and replace X by X to obtain a new commutative
square
A   ! C
? ?
? ?
y y :
K   ! X
Claim 6.5. This square is cocartesian.
The see this, consider the diagram
A _______Cw _______wX

  
   :
  
u u u
K ______wX ______wW
The righthand square is clearly cocartesian. The outer one is also
cocartesian because it factors as a pair of cocartesian squares
A _______Kw _______wX
 
  
   :
  
u u u
K ______wA0 ______Ww
The map C ! X is 2connected by construction. Then 6.5 follows by
application of 5.7 to the previous diagram.
Claim 6.6. For the cocartesian square of 6.5, we have
(1). The map A ! K is (n k 1)connected (in particular, it is 2
connected).
(2). The spaces A and C are homotopy finite.
(3). The pair (K ; A) is a PD pair of dimension n with fundamental
class induced from [X].
To prove (1), we return to the cocartesian square of 6.4. The map
A0 ! K which makes (K ; A0) a PD pair is (n k)connected (since it
is part of a PD embedding). The maps K ! A0 of the square are
coretractions to A0 ! K. Hence, the maps K ! A0 are (n k 1)
connected. Applying 5.6(2), we see that A ! K is also (n k 1)
connected, since k n 3.
To prove that A is homotopy finite, recall that a connected based
space Y is homotopy finite if and only if ss1(Y ) is finitely presented and
the associated Z[ss1(Y )]module chain complex C*(Y ) is chain homo
topy finite in the sense that it is equivalent to a bounded above chain
complex which is degreewise finitely generated and free (see [Wa3 ,
2.2]).
POINCARE DUALITY EMBEDDINGS 27
Since ss1(A) = ss1(K) and K is homotopy finite, we infer that ss1(A)
is finitely presented. Now use the homotopy cofiber sequence
C*(A) ! C*(K) ! C*(A0; K)
and the fact that A0and K are homotopy finite to conclude that C*(A)
is chain homotopy finite. A similar argument shows that C is homotopy
finite. This establishes (2).
Assertion (3) follows from the isomorphism
H*(K ; A) ~=H*+1 (K ; A0)
(induced by the cocartesian square of 6.4) with respect to any coefficient
system on K, using the fact that the fundamental class for (K ; A0) is
induced from [Xx I].
This completes the proof of 6.6.
From the above it follows that
A  ! C
? ?
? ?
y y
K  ! X
is a PD embedding. However, recall that we chose to replace the orig
inal homotopy commutative 3cube by a strictly commutative one (cf.
before 6.3). In doing so, the space K got replaced by something else
homotopy equivalent to it (although we didn't change the notation).__
The proof of Theorem A is completed by invoking 2.5(2). __
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Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33615 Bielefeld,
Germany
Email address: klein@mathematik.unibielefeld.de