EMBEDDING, COMPRESSION AND FIBERWISE
HOMOTOPY THEORY
JOHN R. KLEIN
Abstract. Given Poincare spaces M and X, we study the pos-
sibility of compressing embeddings of Mx I in Xx I down to em-
beddings of M in X. This results in a new approach to embedding
in the metastable range both in the smooth and Poincare duality
categories.
1. Introduction
Let M and X be compact n-manifolds. The word compression of the
title refers to a situation in which one is presented with an embedding
of Mx I in the interior of Xx I and then tries to decide whether it
arises from an embedding of M in X, up to isotopy. If so, then the
original embedding compresses. One aim of the present paper is to
decide when this is possible.
The compression problem is mirrored in the Poincare duality cate-
gory. From now on, let M and X be Poincare duality spaces of dimen-
sion n. One says that M (Poincare) embeds in X with complement C
if there exists a decomposition X ' M [@M C in which @M q @X is
identified with a Poincare duality boundary for C (we also assume a
compatibility of fundamental classes_see 2.4 below.)
It will be convenient to have separate notation for intervals of dif-
ferent lengths. Let I = [0; 1] and J = [1=3; 2=3]. For a subspace S I
set MS := Mx S. We start with the following data: an embedding of
the (n+ 1)-dimensional Poincare space MJ in XI with complement W .
This gives us a map :M ! W by taking the composition
M1=3 @MJ ! W :
Let R(X) denote the category of retractive spaces over X. An object
of R(X) is a space Y equipped with maps sY :X ! Y and rY :Y ! X
(called respectively inclusion and retraction) such that rY O sY is the
identity (objects are usually specified without reference to their struc-
ture maps). A morphism Y ! Z is a map of spaces which is compatible
with the structure maps. According to Quillen [Qu ], R(X) is a model
____________
Date: October 29, 1998.
1
2 JOHN R. KLEIN
category in which a weak equivalence is a morphism Y ! Z which
when considered as a map of spaces is a weak homotopy equivalence
(for the remaining structure, see 2.1 below). Hence, it makes sense to
speak of its homotopy category hoR(X).
The inclusion X0 W and the composite W ! XI ! X equip the
space W with the structure of an object of R(X). Let M+ denote the
object of R(X) given by taking the disjoint union of M with X; the
inclusion X ! M+ is evident and the retraction M+ ! X is defined
to be the composite
project
M q X = M1=3q X0 MJ q X0 -! XI --- ! X :
With respect to these conventions, the map :M ! W induces a
morphism
+ :M+ ! W
of R(X). Then + determines a fiberwise homotopy class
[+ ] 2 [M+ ; W ]X :
Remark 1.1. This will be the primary obstruction to compression. In-
formally, it should be thought of as measuring the self-linking of M in
XI. Several authors have studied non-fiberwise versions of this con-
struction (see Hirsch [Hi ], Levitt [Le2 ] and Williams [Wi1 ]).
Following Goodwillie [Go2 ], the homotopy codimension of M is q,
if
o M is homotopy equivalent to a CW complex of dim n- q, and
o the inclusion @M -! M is (q- 1)-connected.
In what follows, we write codim M q. By a result of Wall [Wa1 ], the
first condition is a consequence of the second whenever q 3.
Examples 1.2. (1). If M is regular neighborhood of p-dimensional com-
plex in an n-dimensional manifold, then codim M n- p.
(2). Let V pbe a closed Poincare space of dimension p equipped with an
(n- p- 1)-spherical fibration :S() -! V . Let D() be the mapping
cylinder of . Then (D(); S()) is a Poincare pair of dimension n with
codim D() n- p.
We now state the main result.
Theorem A. Assume codim M n- p 3 and 3p+ 4 2n. Then
there exists an embedding of M in X which induces the given an embed-
ding of MJ in XI (up to "concordance") if and only if [+ ] 2 [M+ ; W ]X
is trivial.
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 3
We remark that this is valid in both the smooth and Poincare cases
(the smooth case follows by application of the surgery machine_see
below). In the special case X = Dn is a disk, Theorem A reduces to
a non-fiberwise result which is implicit in the work of Williams [Wi2 ].
In fact, our proof of Theorem A is a fiberwisation of one of Williams'
arguments.
With respect to the numerical assumptions of Theorem A, we have
Addendum 1.3. The map of fiberwise homotopy classes
X :[M+ ; W ]X -! [X M+ ; X W ]X
is an isomorphism, where X denotes fiberwise suspension. Conse-
quently, the obstruction to compression [+ ] is stable.
This is proved in x7 using the Freudenthal suspension theorem for
hoR(X) (cf. 2.3 below).
1.1. Unstable fiberwise normal invariants. Let M and X be n-
dimensional Poincare spaces, and let f :M -! X be a map. These data
define an object
M==@M 2 R(X)
whose underlying space is X [f|@M M (note: collapsing X to a point
gives the quotient M=@M). Similarly, we have X==@X 2 R(X) which
turns out to be the double X [@X X (which gives X+ if @X is empty.)
If f :M -! X is the underlying map of an embedding of M in X
with complement C, then there is an associated fiberwise homotopy
class
fff 2 [X==@X; M==@M]X
defined by taking
X [@X X- ' X [@X (C [@M M) -! X [X (X [@M M) = M==@M :
This is the fiberwise (Thom-Pontryagin) collapse of the embedding.
By analogy with Smale-Hirsch theory, a map f :M ! X is said
to (Poincare) immerse if there exists an integer j 0 such that
fx id: Mx Dj ! Xx Dj is the underlying map of some embedding.
Remark 1.4. A fact we won't need, but which is nevertheless true, is
that f Poincare immerses if and only if there is a stable fiber homotopy
equivalence f*X ' M , where X and M denote the Spivak normal
fibrations of X and M respectively. For a proof of this, see [Kl3 ].
Taking the fiberwise collapse of the embedding Mx Dj -! Xx Dj
enables us to associate a fiberwise stable homotopy class
ffstf2 {X==@X; M==@M}X
4 JOHN R. KLEIN
called the fiberwise (stable) normal invariant of the immersion (this is
independent of the choice of embedding.)
Obviously, a necessary obstruction to compressing the given embed-
ding to an embedding of M in X is that ffstfshould desuspend to an
element fff 2 [X==@X; M==@M]X . Call any such desuspension a fiber-
wise unstable normal invariant of the immersion.
Theorem B. Assume f :M -! X immerses. Again, suppose that
codim M n- p 3 and 3p+ 4 2n. Then f embeds (inducing
the given immersion) if and only if there exists a fiberwise unstable
normal invariant fff. Moreover, the embedding can be chosen so that
its collapse induces fff.
In the case @X = ;, Richter has also proved Theorem B using
fiberwise Hopf invariants and fiberwise S-duality. By contrast, we will
deduce Theorem B from Theorem A (in fact, the theorems are equiv-
alent).
A consequence of the above is a Whitney embedding theorem for
immersions in the Poincare duality category:
Corollary C. Assume f :Mp -! Xn immerses, where codim M
n- p 3 and 2p+ 1 n. Then f embeds (inducing the given immersion
up to concordance).
This follows because the fiberwise stable normal invariant destabi-
lizes by 2.3.
1.2. A Levine style embedding theorem. When X is `highly' con-
nected, Theorem B simplifies to a non-fiberwise statement. Here is its
formulation: given an immersion of f :M -! X as above, there is an
associated stable (Thom-Pontryagin) collapse
fist2 {X=@X; M=@M} :
Any homotopy class
fi 2 [X=@X; M=@M]
which suspends to fist is called an unstable normal invariant.
Theorem D. Assume codim M n- p 3, X is [p=2]-connected
and 3p+ 4 2n. Then there exists an embedding inducing the given
immersion of M in X if and only if there exists an unstable normal
invariant fi. Moreover, the embedding can be chosen so that its collapse
coincides with fi.
For example, if we take X = Dn then we recover the Williams-
Richter embedding theorem [Wi2 ], [Ri ]. Levine's embedding theorem
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 5
[Le1 , Thm. 4] amounts to the case when X is a smooth n-manifold and
M = D() is the unit disk bundle of a vector bundle over a smooth
p-manifold V .
1.3. Embedding spheres in the middle dimension. In applica-
tions to surgery on Poincare spaces, one of the main issues is whether
or not homotopy classes in the middle dimension are represented by
`framed' embedded spheres.
Let Xn be a Poincare space, and suppose that n = 2p. Set
P := Spx Dp ;
and suppose that f :P -! X is a map which immerses. Let Xe be the
universal cover of X, and let ss be the group of deck transformations.
A map Y -! X then induces a ss-covering of Y which we denote by eY.
Note that Xe=@Xe is a based ss-space, which is free in the based sense.
The immersion f gives rise to an equivariant stable homotopy class
efist2 {Xe=@Xe; eP=@Pe}ss;
called the equivariant stable collapse. This is constructed as follows:
choose a representative embedding for fx idDj :P xDj -! Xx Dj. The
diagram for this embedding can then be pulled-back along Xe. The
Thom-Pontryagin collapse of the resulting diagram of ss-spaces then
yields efist.
Theorem E. Assume p > 2. An immersion f :P -! X is represented
by an embedding if and only if the equivariant stable collapse desuspends
to an element efi2 [Xe=@Xe; eP=@Pe]ss. Furthermore, the embedding can
be chosen so that its equivariant collapse is efi.
1.4. Embedded thickenings. Up until now, we have discussed em-
bedding theorems between Poincare spaces having the same dimension.
In a previous paper [Kl1 ], we studied the following related problem:
suppose that K is the homotopy type of a finite complex, Xn is a
Poincare space, and f :K -! X is a map. Does there exist a `Poincare
boundary' for K, say A -! K, such that f :K -! X embeds? (More
precisely, we should really replace K by the mapping cylinder of the
map A -! K to get a Poincare pair.) Additionally, one assumes a codi-
mensional restriction: k n- 3, where k is the homotopy dimension of
K (an integer such that K is homotopy equivalent to a CW complex
of that dimension).
This is the notion of Poincare embedding in which the `normal data'
are not a priori chosen. In [Kl1 ] we termed this notion a PD embedding.
6 JOHN R. KLEIN
In this paper, we will call it an embedded thickening, since the choice
of Poincare boundary is a `Poincare thickening' of K.
An important special case of this concept is when K itself is a closed
Poincare space. In this instance, the homotopy fiber of the map A -! K
is a sphere, and one recovers the notion of Poincare embedding used
by Wall [Wa3 , Chap. 11].
In [Kl1 ], we proved that f :Kk -! Xn embedded thickens whenever
f is (2k- n+ 2)-connected. It was was expected that this is not the
sharpest result, for in the smooth case, this result can be improved
by one dimension. We show that the result can be improved by one
dimension in the range of Theorem A:
Theorem F. Assume f :K -! X is (2k- n+ 1)-connected, k n- 3
and 3k+ 4 2n. Then there exists an embedded thickening of f.
Note that this immediately implies the Poincare versions of the
`easy' and `hard' Whitney embedding theorems: let f :K -! X be
a map with k n- 3.
Corollary G. (1). If 2k+ 1 n, then f embedded thickens.
(2). If 2k n and and f is 1-connected, then f embedded thickens
with the possible exception of the case k= 3 and n= 6.
Remark 1.5. The first part of the corollary settles an issue raised by
Levitt [Le2 , p. 402].
Another application yields an extension of [Kl1 , Cor. C], which
concerns the existence of the unstable homotopy tangent bundle for
Poincare spaces:
Corollary H. Let Xn be a 1-connected closed Poincare space. Then
the diagonal X -! Xx X has an embedded thickening.
This follows by Theorem F if n 4, and is trivial if n < 4.
1.5. Smooth embeddings. If M and X are compact smooth mani-
folds, then the Browder-Casson-Sullivan-Wall theorem [Wa3 , Chap. 11]
shows that all of the above results imply smooth embedding results,
(some new, some known). We leave it to the reader to make sense of
this translation.
The inequality 3p+ 4 2n can be improved to 3p+ 3 2n in the
smooth case: in proving Theorem A we make use of the relative em-
bedding theorem of [Kl2 ], which is the Poincare variant of a result of
Hodgson [Ho ] with a loss of one dimension. In the smooth case, Hodg-
son's result may be directly substituted in the appropriate part of the
proof of Theorem A to yield the sharper result.
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 7
1.6. History. The concept of Poincare embedding surfaced in an at-
tempt to understand smooth embeddings within the framework of surgery
theory. The Browder-Casson-Sullivan-Wall theorem asserts that the
smooth embedding problem of Mn in Xn is equivalent to the corre-
sponding Poincare embedding problem as long as n 6 and codim M
3. Consequently, the problem of smooth embedding is reduced to ho-
motopy theory.
The inequality 3p+ 3 2n is called the metastable range. Roughly, it
is the place where triple point obstructions don't arise for dimensional
reasons. From 1960-1975 there emerged (at least) three different strate-
gies to (smooth) embedding in the metastable range. Firstly, there was
the school of Haefliger, which reduced the problem to a question about
isovariant maps Mx 2-! Xx 2(an equivariant map such that the inverse
image of the diagonal of X coincides with the diagonal of M). Secondly,
there was the bordism theoretic approach, as seen in the papers of Dax
[Da ] and Hatcher-Quinn [H-Q ]. Both of these schools relied heavily on
the Whitney trick and/or engulfing methods.
Lastly, there was the surgery school_most notably the works of
Browder [Br1 ], [Br2 ] and Wall [Wa3 ]_which reduced the problem of
smooth embedding to that of Poincare embedding. Furthermore, Levine
[Le1 ], using surgery, constructed embeddings from unstable normal in-
variants when the source M is the total space of a disk bundle over
a smooth manifold and the ambient space X is an n-sphere, or more
generally when X is a sufficiently highly connected manifold. Here, the
role of the normal bundle is prominent.
Somewhat later, Williams [Wi2 ], [Wi1 ], Rigdon-Williams [R-W ] and
Richter [Ri ], extended Levine's work to the case when M is a Poincare
space and X = Dn. The work of Williams et. al. used smooth manifold
techniques to deduce results about Poincare embeddings. Richter gave
the first manifold-free proof of Williams' results using homotopy theory.
It was only recently observed [Kl1 ] that fiberwise homotopy theory
technology was to play a role in extending the surgery approach to an
arbitrary ambient Poincare space X. This connection was discovered
by Shmuel Weinberger and the author (independently). The present
work is an attempt to complete the thread begun by the surgery school.
1.7. Outline. x2 is mostly language; the reader should be familiar
with the majority of material in this section. In x3 we show that
the existence of a fiberwise normal invariant is sufficient to give an
embedding of MJ in XI whose obstruction to compression is trivial,
so Theorem A implies the first part of Theorem B. x4 concerns the
proof of Theorems D and E, which are a consequence of Theorem B
8 JOHN R. KLEIN
and Milgram's EHP sequence. In x5 we prove Theorem A. The main
tool in the proof is the relative embedded thickening theorem of [Kl2 ].
In x6 we show that the embedding constructed in x3 has the correct
collapse, thereby completing the proof of Theorem B. In x7 we prove
the stability of the obstruction [+ ]. In x8 we prove Theorem F.
1.8. Acknowledgements. This paper could not have been written
were it not for discussions I had with Tom Goodwillie and Bill Richter.
The proof of Theorem A was in part motivated by techniques employed
by Goodwillie to study the stability map in relative pseudoistopy the-
ory. As I mentioned above, the first proof of Theorem B is due to
Richter. Also, the idea of the proof of 8.2 was aided by interaction
with Richter. Thanks to Andrew Ranicki for improvements in the ex-
position. Lastly, I've benefited from the papers of Bruce Williams.
2. Preliminaries
Our ground category is Top, the category of compactly generated
Hausdorff spaces. This comes equipped with the structure of a Quillen
model category:
o The weak equivalences are the weak homotopy equivalences (i.e.,
maps X ! Y such that the associated realization of its singular
map |S.X| -! |S.Y | is a homotopy equivalence). Weak equiva-
lences are denoted -~!.
o The fibrations, denoted i, are the Serre fibrations.
o The cofibrations, denoted ae, are the `Serre cofibrations', i.e., in-
clusion maps given by a sequence of cell attachments (i.e., relative
cellular inclusions) or retracts thereof.
Every object is fibrant. The cofibrant objects are the retracts of
iterated cell attachments built up from the empty space. Every object
~
Y comes equipped with a functorial cofibrant approximation Y ci Y .
A non-empty space is always (-1)-connected. A connected space is
0-connected, and is r-connected for some r > 0 if its homotopy groups
vanish up through degree r, for any choice of basepoint. A map of non-
empty spaces X -! Y is called r-connected if its homotopy fiber with
respect to any choice of basepoint in Y is an (r- 1)-connected space.
An 1-connected map is a weak equivalence.
A space is homotopy finite if it is homotopy equivalent to a finite
CW complex.
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 9
A commutative square of cofibrant spaces
A --- ! B
? ?
? ?
y y
C --- ! D
is r-cocartesian if the evident map C0 [ A[0;1][ B1 -! D (whose source
is a double mapping cylinder) is r-connected. More generally, a square
of not necessarily cofibrant spaces is r-cocartesian if it is after apply-
ing cofibrant approximation. An 1-cocartesian square is cocartesian.
Dually, the square is r-cartesian if the map A -! C0xD[0;1]B1 is r-
connected (the target is a homotopy pullback: DS means the function
space of maps S ! D, for S [0; 1]), and an 1-cartesian square is
cartesian.
We introduce one last non-standard notation: given a map of spaces
A -! B, if no confusion arises we will often let (B ; A) denote the pair
given by the mapping cylinder B0 [ AI with the inclusion of A1.
2.1. Fiberwise spaces. For X 2 Top an object, R(X) will denote
the category of retractive spaces, as in the introduction (in another
notation, not to be used here, X\Top =X). We will assume in what
follows that X is a cofibrant object of Top.
According to Quillen [Qu ], R(X) inherits a model category structure
arising from the one on Top. Weak equivalences and fibrations are
defined using the forgetful functor R(X) -! Top. Cofibrations are
those maps satisfying the left lifting property with respect to the acyclic
fibrations (the word `acyclic' is synonymous with weak equivalence).
Any object Y 2 R(X) comes equipped with a functorial cofibrant
~ ~
approximation Y ci Y and a functorial fibrant approximation Y ae
Y f.
Given an object Y 2 R(X), define its fiberwise suspension X Y
to be the object whose underlying space is obtained by collapsing the
subspace XI X Y to X (via the first factor projection) in the double
mapping cylinder X0[YI[X1. If Y is cofibrant, then so is its fiberwise
suspension. We use the notation jXY to denote the j-fold iterated
application of X to Y .
The homotopy category of R(X), denoted hoR(X), is the category
whose objects are those of R(X) and in which the hom-set from an
object Y to an object Z is given by homotopy classes of morphisms
Y c! Zf. This is denoted [Y; Z]X ; it is a pointed set. The correspond-
ing stable hom-set is {Y; Z}X := limj[jXY; jXZ]X .
10 JOHN R. KLEIN
Obstruction theory in Top gives rise to an obstruction theory in
R(X). Let Z 2 R(X) be an object. A commutative diagram
Sj-1 --- ! Z
? ?
? ?
y y
Dj --- ! X
defines another object Z [ DjX, whose underlying space is Z [Sj-1 Dj.
This operation is called attaching a j-cell to Z.
Definition 2.1. An object P 2 R(X) has dimension s if its fibrant
approximation admits a factorization X ae P 0-~!P fsuch that P 0is
obtained from X by attaching cells of dimension s.
A morphism Y -! Z is r-connected if it is r-connected as a map of
spaces. In particular, an object Y is r-connected if its structure map
X -! Y is.
Lemma 2.2. Let Y -! Z be r-connected morphism of R(X) and sup-
pose that P has dimension r. Then the induced map of homotopy
sets
[P; Y ]X -! [P; Z]X
is surjective. It is also injective if P has dimension r- 1.
This is essentially [Ja , 9.2].
2.2. The stable range. The Freudenthal theorem measures the ex-
tent to which fiberwise suspension is an isomorphism on the level of
fiberwise homotopy classes.
Lemma 2.3. (James [Ja , 9.3]). If Y; Z 2 R(X) cofibrant objects such
that Z r-connected and Y has dimension 2r+ 1, then fiberwise sus-
pension gives a surjection of pointed sets
[Y; Z]X -! [X Y; X Z]X :
This surjection is an isomorphism whenever Y has dimension 2r.
2.3. Poincare spaces. In this paper, a Poincare space X of dimension
n is a pair (X; @X) such that X and @X are cofibrant and homotopy
finite, @X ! X is a cofibration, and X satisfies Poincare duality:
o there exists a local system of abelian groups L of rank one defined
on X, and a fundamental class [X] 2 Hn(X; @X; L) such that the
cap product homomorphisms
\[X] :H*(X; M) -! Hn- *(X; @X; L M)
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 11
and
*
\[@X] :H (@X; N) -! Hn- *- 1(@X; L|@X N)
are isomorphisms, where [@X] 2 Hn- 1(@X; L|@X) is the image of
[X] under the connecting homomorphism in the homology exact
sequence of the pair (X; @X), and M (N) is any local system on
X (resp. on @X) (compare [Kl1 ], [Wa2 ]).
If (X; @X) is a pair such that @X -! X is 2-connected, then the
first duality isomorphism implies the second one (cf. [Kl1 , 2.1]). In
these circumstances, X is n-dimensional Poincare if and only if XI is
(n+ 1)-dimensional Poincare.
2.4. Embeddings. Let M and X a Poincare spaces of dimension n,
where X is connected. An embedding of M in X is a commutative
cocartesian square of cofibrant homotopy finite spaces
@M -- - ! C
? ?
incl.?y ?yg
M -- - ! X
f
together with a factorization of the inclusion @X -! C -! X, such that
(M; @M) and (C ; @M q @X) satisfy Poincare duality with respect to
the fundamental classes obtained by taking the image of a fundamental
class for X under the homomorphisms
Hn(X; @X; L) -! Hn(X ; C; L) ~=Hn(M; @M; f*L)
and
Hn(X; @X; L) -! Hn(X; M q @X; L) ~=Hn(C ; @M q @X; g*L) :
If codim M 3 then one only need verify the compatibility of funda-
mental classes for M (see [Kl1 , 2.3]).
The space C is called the complement, and f :M -! X is the un-
derlying map of the embedding.
The decompression of an embedding of M in X is the embedding of
MI in XI defined by the diagram
@MI -- - ! W
? ?
? ?
y y
MI -- - ! XI
where W = X0 [ CI [ X1 is (unreduced) fiberwise suspension, and the
factorization @XI -! W -! XI is evident.
12 JOHN R. KLEIN
Two embeddings from M to X with complements C0 and C1 are
elementary concordant if there exists a diagram of pairs
(@MI; @M0 q @M1) ______(W;wC0 [ (@X)I [ C1)
| |
| |
| |
| |
|u |u
(MI; M0 q M1) ______________(XI;w@XI)
in which each associated diagram of spaces
______w
@MI --- ! W @M0 q @M1 C0 [ (@X)I [ C1
? ? | |
? ? | |
y y and | |
| |
|u |u
MI --- ! XI M ______________
0 q M1 @XIw
is cocartesian (the latter of these is obtained from the disjoint union
of the embedding diagrams using the inclusion @X0 q @X1 @XI).
Moreover, the maps Ci -! W are required to be weak equivalences.
More generally, concordance is the equivalence relation generated by
elementary concordance.
2.5. Embedded thickenings. Suppose that K is a cofibrant space
which is homotopy equivalent to a finite connected CW complex of
dimension k. Let f :K -! X be a map, where Xn is a connected
Poincare space of dimension n. A cocartesian square
A --- ! C
? ?
? ?
y y
f
K --- ! X
(in which A and C are cofibrant and homotopy finite), together with a
factorization @X -! C -! X is called an embedded thickening of f if
o (K ; A) gives an n-dimensional Poincare space such that codim K
n- k, and
o Replacing K by K in the diagram yields an embedding in the
sense of 2.4.
An embedded thickening is what was called a PD embedding in the
terminology of [Kl1 ]. In order to avoid confusion, we have changed the
name to distinguish between the embeddings appearing in this paper
(where the boundary data are a priori given) and the ones of [Kl1 ]
(embeddings of complexes in Poincare spaces).
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 13
3. Proof of Theorem B (first part)
We show how Theorem A can be used to construct an embedding
of M in X from an unstable fiberwise normal invariant.
Let fff 2 [X==@X; M==@M]X be an unstable fiberwise normal invari-
ant associated to an immersion of f :M -! X. Based on a construction
of Browder [Br2 ] we will associate a Poincare embedding of MJ in XI.
For this section only, let us agree that M==@M now means the object
of R(X) whose underlying space is X0 [ (@M)I [ M1 (the formulation
provided in the introduction differs from this description by a canonical
weak equivalence). Similarly, let X==@X now mean X0 [ (@X)I [ X1.
Let h :J -! I be the homeomorphism t 7! 3t- 1.
Then there is a commutative diagram of spaces
@MJ ______wM==@M
| |
| |
| |
| |
| |
| |
|u |u
MJ _________XIw
in which the top arrow is defined by
f[id[id
M1=3[ (@M)J [ M2=3-idxh-!M0[ (@M)I[ M1 --- - ! X0[ (@M)I[ M1 ;
the bottom arrow is fx h, and the vertical arrows are evident. This
diagram is cocartesian. In what follows, we must replace M==@M in
the diagram with its fibrant approximation (M==@M)f. assume that
this has been done.
The Poincare boundary for XI is X==@X; it factors through (M==@M)f
via a representative for fff. This defines the embedding of MJ in XI.
In particular, the complement of this embedding is (M==@M)f.
Applying Theorem A, we see that the given embedding compresses
to a embedding of M in X if and only if [+ ] 2 [M+ ; M==@M]X is the
trivial element. But by construction, [+ ] is the fiberwise homotopy
class determined by making the composite (fiberwise) map
M1=3-! M1=3[ (@M)J [ M2=3-! X0 [ (@M)I [ M1
"based" (i.e., add on a disjoint copy of X to M1=3). The composite
clearly factors through the "basepoint" X0 X0[ (@M)I[ M1, so [+ ]
is the trivial element.
It remains to check that the collapse of the embedding of M in X
equals fff. This is not a formal consequence of Theorem A, but rather,
a consequence of the construction of the particular embedding in the
14 JOHN R. KLEIN
proof of Theorem A contained in x5 below. For this reason, we defer
the proof of this until x6.
4. Proof of Theorems D and E
Proof of Theorem D. We first explain the idea of the proof while ignor-
ing technical details. There is a commutative diagram of R(X)
M==@M _________w(M=@M)x X
| |
| |
| |
|u |u
QX M==@M ______w(QM=@M)x X
in which
o (M=@M)x X has structure maps given by the second factor pro-
jection and the inclusion *x X (M=@M)x X.
o The morphism M==@M -! (M=@M)x X is given by the quotient
map M==@M -! M=@M together with the retraction M==@M -!
X.
o QX means the fiberwise version of stable homotopy, and the bot-
tom map of the diagram is defined in a way similar to the top
map.
o The vertical maps are defined by means of the natural transfor-
mation from the identity to (fiberwise) stable homotopy.
Ignoring for the moment the issue of homotopy invariance of the terms
in the diagram, it will follow by an argument sketched below that
the diagram is n-cartesian. Assuming this the argument proceeds as
follows:
The fiberwise stable homotopy class ffst is represented by a mor-
phism X==@X -! QX M==@M and the homotopy class fi is represented
by a morphism X==@X -! (M=@M)x X. Up to fiberwise homotopy
the maps are compatible with the diagram. By 2.2 applied to the
n-connected morphism
M==@M -! holim (QX M==@M -! (QM=@M)x X- (M=@M)x X)
there is an unstable fiberwise normal invariant ff 2 [X==@X; M==@M]X .
Theorem D now follows by application of Theorem B.
We now proceed to establish the degree to which the square is carte-
sian. First of all, we replace the square by an equivalent one which is
homotopy invariant (for the extent to which QX is a homotopy invari-
ant functor is unclear, even for objects which are fibrant and cofibrant).
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 15
Choose a basepoint for X. Since X is a connected cofibrant space,
there is a homotopy equivalence X ' BG where G is the geometric
realization of the simplicial set given which is the Kan loop group of
the total singular complex of X. Here, we think of G as a topological
group object in Top. In what follows, we will assume X is BG.
Let RG (*) denote the category of based G-spaces. This admits the
structure of a model category in which a morphism Y -!Z is a weak
equivalence if (and only if) it is a weak homotopy equivalence of spaces.
Every object is fibrant and the cofibrant objects are the retracts of free
based G-CW complexes. In fact, the homotopy categories of RG (*)
and R(BG) are equivalent (but we will not require this.)
Let M~ denote the pullback of M -! BG- EG. Then M~ =@M~
is an object of RG (*). We recover M==@M 2 R(BG) up to weak equiv-
alence by taking the Borel construction (M~ =@M~ )x GEG. We re-
cover M=@M as the homotopy orbits (i.e., reduced Borel construction)
(M~ =@M~ )hG := (M~ =@M~ ) ^G EG+ . In its homotopy invariant for-
mulation, the square is now given by the diagram of morphisms of
R(BG)
(M~ =@M~ )cx GEG ________(M~w=@M~ )chGxBG
| |
| |
| | (1)
| |
|u |u
(QM~ =@M~ )cx GEG ______(QM~w=@M~ )chGxBG
(here, for an object Y 2 RG (*), the object Y c denotes its cofibrant
approximation).
Finally, we calculate the degree to which the square is cartesian.
In what follows, set N := M~ =@M~ , and note that N is (n- p- 1)-
connected. The homotopy fiber of the left vertical map is the same
thing as the homotopy fiber of the map N -! QN. Denote this fiber
by F1. By Milgram's EHP-sequence [Mi , 1.11], there is a (3n- 3p- 3)-
connected map (N ^ N)hZ=2 -!F1. On the other hand the homotopy
fiber of the right vertical map is the same as the homotopy fiber of
the map NhG -! Q(NhG ). If we denote this homotopy fiber by F2, it
16 JOHN R. KLEIN
again follows by Milgram's EHP-sequence that there is a (3n- 3p- 3)-
connected map (NhG ^ NhG )hZ=2 -!F2. Moreover, the square
(N ^ N)hZ=2 ______(NhGw^ NhG )hZ=2
| |
| |
| |
| |
|u |u
F1 ____________________F2w
is commutative. The top map of the latter square is induced by the
evident map N ^ N -! (N ^ N)hGx G. This last map is easily checked to
be (2n- 2p+ [p=2])-connected. Assembling this information, it follows
that the map F1 -! F2 is min (3n- 3p- 4; 2n- 2p+ [p=2]- 1)-connected.
By hypothesis, 3p+ 4 n, so this connectivity is at least n. Conse- __
quently, the square (1) is n-cartesian, as claimed. |__|
Proof of Theorem E. The proof is similar to the proof of Theorem D
(where here P plays the role of M). Therefore, we will only sketch the
argument and leave it to the reader to fill in the details.
As above, there is a diagram
eP==@Pe_________w(Pe=@Pe)x X"
| |
| |
| |
| |
| |
|u |u
QXeeP==@Pe ______w(QPe=@Pe)x X"
which one checks (by essentially the same argument) to be (2p)-cartesian.
The fiberwise stable normal invariant can be lifted to a fiberwise equi-
variant map Xe=@Xe -! QXeeP==@Pe. The rest of the argument follows as
in the proof of Theorem B, substituting obstruction theory by equi-
variant obstruction theory, and using the fact that the equivariant ho-
__
motopy dimension of Xe=@Xe is 2p. |__|
5. Proof of Theorem A
Our main tool will be the relative embedded thickening theorem of
[Kl2 ] (see also [Kl1 ] for the absolute version). The statement of this
result will require some preparation.
Let (K; L) be a cofibration pair in Top. We assume for simplicity
that K and L are cofibrant spaces which are homotopy finite. Write
dim (K; L) k
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 17
if there exists a factorization L ! K0 ! K in which K0 is obtained
from L by attaching cells of dimension k and the map K0 ! K is a
weak equivalence.
Let X be an n-dimensional Poincare space.
By a relative embedded thickening of (K; L) in (X; @X) we mean a
commutative diagram of cofibration pairs
(AK ; AL) --- ! (CK ; CL)
? ?
? ?
y y
(K; L) --- ! (X; @X)
having the following properties.
o Each space appearing in the diagram is cofibrant and homotopy
finite.
o Each of the diagrams of spaces
AK -- - ! CK AL --- ! CL
? ? ? ?
? ? ? ?
y y and y y
K -- - ! X L --- ! @X
is cocartesian and the latter of these diagrams is a embedded
thickening of L in @X.
o The image of the fundamental class of X with respect to the
composite
Hn(X; @X) -! Hn(X ; @X [CL CK ) ~=Hn(K ; L [AL AK )
gives (K ; L [AL AK ) the structure of an n-dimensional Poincare
space (here, coefficients are given by pulling back the orientation
bundle for (X; @X)). Similarly, (CK ; CL [AL AK ) has the structure
of a Poincare space with fundamental class induced from X.
o The map AK -! K is (n- k- 1)-connected.
The decomposition of (X; @X) is depicted in figure 1 below.
Fig. 1.
18 JOHN R. KLEIN
Now let f :(K; L) -! (X; @X) be a map with dim (K; L) k and
suppose that the restriction f|L: L -! @X embedded thickens. The
main theorem of [Kl2 ] is
Theorem 5.1. Assume k n- 3 and f :K -! X is (2k- n+ 2)-connected.
Then there exists a relative embedded thickening of f :(K; L) -! (X; @X)
extending the given embedded thickening of f|L: L -! @X.
Remark 5.2. The above is the Poincare version of the relative embed-
ded thickening theorem of Hodgson [Ho ], with a loss of one dimension.
We now begin the proof of Theorem A. Assume [+ ] 2 [M+ ; W ]X is
trivial, where W is the complement of an embedding of MJ in XI. We
may also assume without loss in generality that W 2 R(X) is fibrant.
A choice of fiberwise null-homotopy may be thought of as a family of
maps t: Mt -! W for t 2 [0; 1=3] which commute with projection to
X such that = 1=3 and 0 factors through X0 ! W .
This null-homotopy gives rise to a map of pairs
(X0 [ M[0;1=3]; X0 q M1=3) -! (W; @W )
in which X0 [ M[0;1=3]is the mapping cylinder of the map M1=3 -!X.
These circumstances are depicted in figure 2.
Fig. 2.
The restricted map of spaces
X q M -! @W
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 19
is already embedded thickened (here, @W = @XI q @MJ). This em-
bedded thickening is given by the cocartesian square
@X0 q @M1=3 ______w((@X)I [ X1) q ((@M)J [ M2=3)
| |
| |
| |
|u |u
X0 q M1=3 _____________________w@W :
The map
X0 [ M[0;1=3]-!W
is (n- p- 1)-connected (since it, followed by the map W -! XI is a
weak equivalence, and the latter map is (n- p)-connected). Moreover,
the pair (X0 [ M[0;1=3]; X q M) has relative dimension p+ 1.
Since n+ 1 2(p+ 1) - (n- p- 1) + 2 if and only if 2n 3p+ 4, by
5.1 there exists a relative embedded thickening of
(X0 [ M[0;1=3]; X q M) -! (W; @W )
which extends the given embedded thickening of X q M -! @W . Thus
we have a diagram of pairs (cf. fig. 3)
(A; @X0 q @M1=3) ______(C;w((@X)I [ X1) q ((@M)J [ M2=3))
| |
| |
| |
| |
|u |u
(X0 [ M[0;1=3]; X q M) _________________w(W; @W ) :
Fig. 3.
20 JOHN R. KLEIN
Consider the associated commutative diagram
@M -- - ! A
? ?
? ?
y y (2)
M -- - ! X
and note that there is an evident factorization of @X ! X through the
map A -! X.
To complete the proof of Theorem A, it suffices to show
Claim 5.3. The square (2) is an embedding of M in X. It induces the
given embedding of MJ in XI after decompression.
To establish the claim, we first need to show that the square is
cocartesian. According to the definitions X0 [ M[0;1=3]' X has an
n-dimensional Poincare boundary given by X0 [@X0 (M1=3[@M1=3 A).
Application of Poincare-Lefshetz duality gives an isomorphism
H*(X ; M [@M A) ~=Hn+ 1-*(X ; X1) = 0 :
in all degrees, for any bundle of coefficients on X. Moreover, the map
M [@M A -! X induces an isomorphism on fundamental groups (since
A -! X and @M -! M are 2-connected), so the square is cocartesian
by application of Whitehead's theorem.
Secondly, a straightforward argument which we omit shows that the
inclusion X1 C is a weak equivalence. Consequently, the composite
C -! W -! X is also a weak equivalence. Using this, we have a chain
of weak equivalences
X A- ~ X [A C -~!W
which is compatible with projection to XI and is relative to @XI. We
infer that the decompression of (2) yields the embedding of MJ in MI
up to concordance. Compatibility of fundamental classes is a conse-
quence of the remarks at the end of 2.3 and 2.4. This completes the__
proof of Theorem A. |__|
6. Theorem B: completion of the proof
Given an unstable fiberwise unstable normal invariant
fff 2 [X==@X; M==@M]X ;
we constructed in x3 an embedding of M in X by first associating
an embedding of MJ in XI and then applying Theorem A (using the
observation that the compression obstruction of the latter embedding
is trivial).
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 21
It remains to show that the collapse of this embedding coincides
with fff. We will give the argument in the case when @X = ;. The
general case, which is straightforward, will be left to the reader.
Returning to the proof of Theorem A and in particular fig. 3 above,
note that the collapse of the embedding of M in X is the fiberwise
homotopy class of the map
X0 q (A [@M M) -! X [@M M (-~! X [@M M) ;
whose restriction to X0 is given by the inclusion X0 ! X and the
restriction to A[@M M is given by the amalgamation of the map A -! X
with the identity map of M.
Using fig. 3, we rewrite this as follows: consider the amalgamated
union
M0 := (@M)J [@M2=3 M2=3:
and write X0 := A [@M1=3 M0 (so X0 is identified with X up to weak
equivalence). Then the fiberwise homotopy class of the composite
X0 q X0 -!X [@M1=3 M0 ~-!W
represents the collapse of the embedding (recall that W is M==@M made
fibrant). Note there is an evident factorization X0qX0 -!X0qC -! W .
On the other hand, the composite
X0 q X1 -! X0 q C -! W
induces fff.
Consequently, the restrictions of the map X q C -! W to X0 q X0
and X0 q X1 induce respectively the collapse map of the embedding
and fff.
But the maps X1 -! C and X0 -! C are weak homotopy equiva-
lences. Consequently, the map X0 q C -! W induces both the collapse
of the embedding of M in X and fff on fiberwise homotopy. Thus
fff coincides with the collapse. This completes the proof of Theorem__
B. |__|
7. Stability of the obstruction
To prove 1.3, we apply 2.3 to the homotopy set [M+ ; W ]X . Since
M is homotopy equivalent to a complex of dimension p, we infer
that the object M+ 2 R(X) has dimension p. On the other hand,
the connectivity of W 2 R(X) is one less than the connectivity of
the map W -! XI, which in turn, is at least the connectivity of the
map @MJ -! MJ since the former is the cobase change of the latter.
But codim MJ n- p+ 1, so @MJ -! MJ is (n- p)-connected. Hence
W 2 R(X) is an (n- p- 1)-connected object.
22 JOHN R. KLEIN
Consequently, 2.3 implies that
+ +
[M ; W ]X -! [X M ; X W ]X
is an isomorphism whenever p 2(n- p- 1), or equivalently, whenever
3p+ 2 2n. Thus, the obstruction to compression is stable in the range_
of Theorem A (with two dimensions to spare). |__|
8. Proof of Theorem F
In this section we show how Theorem A implies a partial improve-
ment of the main result of [Kl1 ]. Let K be a cofibrant space which is
homotopy equivalent to a connected CW complex of dimension k.
Let X be a connected n-dimensional Poincare space.
The main result of [Kl1 ] is
Theorem 8.1. Assume that f :K -! X is (2k- n+ 2)-connected and
k n- 3. Then there exists an embedded thickening of f.
Now we have the statement of Theorem F, which is an improvement
of 8.1 in the metastable range:
Theorem 8.2. Assume f :K -! X is (2k- n+ 1)-connected, k n- 3
and 3k+ 4 2n. Then there exists an embedded thickening of f.
Proof. By 8.1, there exists an embedded thickening of the composite
fI: K -! X = X0 -! XI. Let this be denoted
A0 --- ! W
? ?
? ?
y y
K --- ! XI :
fI
Without loss in generality, we may take A0 -! K to be a fibration.
By straightforward application of the Blakers-Massey theorem [Go1 ,
p. 309], this square is k-cartesian. Let P = K0 x(XI)IW 1denote the
homotopy pullback of the diagram given by deleting A0. Then the
evident map A0-! P is k-connected.
f
The maps id:K -! K and K -! X = X0 W are compatible up to
homotopy when followed by the given maps to XI. Consequently, there
is an induced map K -! P . As A0 -! P is k-connected, we obtain a
factorization K -! A0-! P . Since A0-! K is a fibration, the homotopy
lifting property plus the factorization yield a section i :K -! A0. By
construction, the composite
iqidX 0
K+ = K q X -- - ! A q X0 -! W (3)
EMBEDDING, COMPRESSION AND FIBERWISE HOMOTOPY THEORY 23
is fiberwise null homotopic.
The map i :K -! A0 is (n- k- 1)-connected. By 8.1, it embedded
thickens since n- k- 1 2k- n+ 2 is equivalent to 3k+ 3 2n. Let
A --- ! C
? ?
? ?
y y
K --- ! A0
be such an embedded thickening. We claim that the composite C -!
A0 -! K is a weak equivalence. To see this, use the naturality of
cap product with the fundamental class to identify the exact sequence
. .-.!H*(K ; A) -! H*(K ; C) -! H*(K ; A0) -! . . .with the exact se-
quence . . .-!Hn- *(K) -! Hn+ 1- *(K; K) -! Hn+ 1- *(K) -! . . .(we
are suppressing coefficients here). From this we infer that C -! K is a
homology equivalence (with respect to any coefficient bundle). More-
over, C -! K is 2-connected, since it is the composite of the (n- p- 1)-
connected map C -! A0 with the (n- p)-connected map A0 -! K. So
C -! K is a weak homotopy equivalence by the Whitehead theorem.
Let (M; @M) denote the pair (K ; A). Then the argument of the
last paragraph implies that (MI; @MI) coincides with (K ; A0) up to
homotopy. Furthermore, with respect to this homotopy equivalence,
the inclusion M0 @MI corresponds to i :K ! A0.
Assembling these data, we have an embedding of MI in XI whose
obstruction [+ ] vanishes by (3). Applying Theorem A yields an em-__
bedded thickening of f :K -! X. |__|
References
[Br1] Browder, W.: Embedding 1-connected manifolds. Bull. Amer. Math. Soc.72,
225-231 (1966)
[Br2] Browder, W.: Embedding smooth manifolds. Proc. I.C.M. , 712-719 (1968).
Moscow 1966
[Da] Dax, J. P.: Etude homotopique des espaces de plongements. Ann. Sci. Ecole
Norm. Sup. 5, 303-377 (1972)
[Go1] Goodwillie, T. G.: Calculus II: analytic functors. K-theory 5, 295-332 (1*
*992)
[Go2] Goodwillie, T. G.: Multiple disjunction for spaces of homotopy equivalenc*
*es.
Preprint, November 29, 1995
[H-Q] Hatcher, A., Quinn, F.: Bordism invariants of intersections of submanifol*
*ds.
Trans. AMS 200, 327-344 (1974)
[Hi] Hirsch, M.: Embeddings and compressions of smooth manifolds and poly-
hedra. Topology 4, 361-369 (1966)
[Ho] Hodgson, J. P. E.: Obstructions to concordance for thickenings. Invent.
Math. 5, 292-316 (1970)
[Ja] James, I. M.: Fibrewise homotopy theory. Handbook of Algebraic Topology
North Holland, 169-194 (1995)
24 JOHN R. KLEIN
[Kl1] Klein, J. R.: Poincare embeddings and fiberwise homotopy theory. To ap-
pear in Topology
[Kl2] Klein, J. R.: Poincare embeddings and fiberwise homotopy theory, II. In
preparation
[Kl3] Klein, J. R.: Poincare immersions. Preprint, 1998
[Le1] Levine, J.: On differentiable imbeddings of simply-connected manifolds.
Bull. AMS 69, 806-809 (1963)
[Le2] Levitt, N.: Normal fibrations for complexes. Illinois J. Math. 14, 385-4*
*08
(1970)
[Mi] Milgram, R. J.: Unstable homotopy from the stable point of view. (LNM,
Vol. 368). Springer 1974
[Qu] Quillen, D.: Homotopical Algebra. (LNM, Vol. 43). Springer 1967
[R-W] Rigdon, R., Williams, B.: Embeddings and immersions of manifolds (Proc.
Conf., Evanston, Ill., 1977), I. LNM 657 , 423-454 (1978)
[Ri] Richter, W.: A homotopy theoretic proof of Williams's Poincare embedding
theorem. Duke Math. J. 88, 435-447 (1997)
[Wa1] Wall, C. T. C.: Finiteness conditions for CW -complexes. Ann. of Math. 81,
55-69 (1965)
[Wa2] Wall, C. T. C.: Poincare complexes I. Ann. of Math. 86, 213-245 (1967)
[Wa3] Wall, C. T. C.: Surgery on Compact Manifolds. Academic Press 1970
[Wi1] Williams, B.: Applications of unstable normal invariants I. Comp. Math.
38, 55-66 (1979)
[Wi2] Williams, B.: Hopf invariants, localizations, and embeddings of Poincare
complexes. Pacific J. Math. 84, 217-224 (1979)
Wayne State University, Detroit, MI 48202
E-mail address: klein@math.wayne.edu