ON EMBEDDINGS IN THE SPHERE
JOHN R. KLEIN
Abstract. We consider embeddings of a finite complex in a sphere.
We give a homotopy theoretic classification such embeddings in a
wide range.
1. Introduction
Let K be a finite complex. An embedding up to homotopy of K in
Sn consists of a a pair
(M, h)
where Mn is compact codimension zero PL submanifold of Sn and
h :K ! M is a homotopy equivalence. Two such pairs (M0, h0) and
(M1, h1) are said to be concordant if there is an embedded h-cobordism
W in Snx [0, 1] from M0 to M1 together with a homotopy equivalence
H :Kx [0, 1] ! W extending both h0 and h1. Let
E(K, Sn)
denote the set of concordance classes of embeddings up to homotopy of
K in Sn. (Note: if K is 1-connected and dim K n- 3, the existence
of a concordance implies the existence of an ambient isotopy.) Unless
confusion arises, we refer to embeddings up to homotopy as embeddings.
Constraints. We fix throughout integers
k, n, r
satisfying
0 k n- 3, r 1, and n 6 .
If n 7, we also assume k - r 2.
In addition to these constraints, we consider the inequalities
1
(1) r max (__(2k- n), 3k- 2n+ 2)
2
____________
Date: October 15, 2003.
The author is partially supported by NSF Grant DMS-0201695.
2000 MSC. Primary: 55.P25; Secondary 57Q35.
1
2 JOHN R. KLEIN
1
(2) r max (__(2k- n+ 1), 3k- 2n+ 3)
2
The inequalities can be interpreted as follows: the integer r will be
the connectivity of the space to be embedded. Consider maps from
manifolds of dimension k to Sn. Then roughly, the inequalities rep-
resent the demand that the connectivity r exceeds both the generic
dimension of the triple point set and also, one half the generic dimen-
sion of the double point set.
Main results. To formulate our main results requires some prepara-
tion. Let
ff :Z2 ! GL1(R)
denote the sign representation. If s, t 0 are is a integers, let Stff+s
denote the one point compactification of the direct sum of t copies of ff
with s-copies of the trivial representation. This is a sphere of dimension
t + s having a based action of Z2.
If X and Y are based spaces, we let F st(X, Y ) denote the spectrum
of stable maps from X to Y (the j-th space of this spectrum is the
function space of maps from X to Q( jY ).
If X and Y are based Z2-spaces, then F st(X, Y ) comes equipped
the structure of a naive Z2-spectrum by conjugating functions. Let
F st(X, Y )hZ2 denote the associated homotopy orbit spectrum.
Choose a basepoint for K. We consider the case when X = K ^ K
with permutation action and Y = S(n- 1)ff+1^ K with the diagonal
action (where Z2 acts trivially on K).
We are now in a position to state our main results.
Theorem A (Existence). Let Assume K be r-connected and dim K
k. There is an obstruction
`K 2 ß0(F st(K ^ K, S(n- 1)ff+1^ K)hZ2)
(depending only on the homotopy type of K) whose vanishing is a nec-
essary condition for E(K, Sn) to be non-empty.
If the inequality (1) holds, then the vanishing of `K implies E(K, Sn)
is non-empty.
Remarks. When K is (2k- n)-connected, the obstruction group is triv-
ial, so there is an embedding of K in Sn. Thus we recover the Stallings-
Wall embedding theorem [Wa1 ].
ON EMBEDDINGS IN THE SPHERE 3
When K is (2k- n- 1)-connected, the obstruction group is isomor-
phic to
H2k(Kx K; ß2k-n(K))=(1 - T ) ,
where T is the involution on H2k(Kx K; ß2k-n(K)) given by tOE, where
E switches the factors of K x K, and t is the involution of ß2k-n(K)
given by multiplication by (-1)n- 1.
This abelian group appears in the work of Habegger [Ha ], who gave
necessary and sufficient conditions for finding embeddings in the fringe
dimension beyond the Stallings-Wall range. Habegger defined his ob-
struction using PL intersection theory.
Theorem B (Enumeration). Let K be as above. Fix a basepoint in
E(K, Sn). Then there is a function
OEK :E(K, Sn) ! ß0(F st(K ^ K, S(n- 1)ff^ K)hZ2)
which is onto if inequality (1) holds. If inequality (2) holds, then OEK
is also one to one.
Corollary C (Group Structure). Assume E(K, Sn) is non-empty. If
inequality (2) holds, then E(K, Sn) has the structure of an abelian
group.
The above results have corollaries which are too numerous to de-
scribe in this introduction (see x5-7). For example, a consequence of
Theorem B is that, in the range of inequality (2), an r-connected closed
PL manifold Mk with trivial betti number b2k- n+(1M) admits only
finitely many locally flat embeddings in Sn up to isotopy.
Outline. In x2 we recall the statement of the Connolly-Williams Classi-
fication Theorem. In x3 we prove Theorem B. In x4 we prove Theorem
A by modifying the proof of Theorem B. x5 contains applications to
embeddings of complexes with 2-cells (these applications are already in
the literature in some form). In x6 we give applications to embeddings
of Poincar'e spaces and manifolds (many of the results in this section
are new to the literature). In x7 we show that the obstructions to
embedding in the range of inequality (1) are 2-local.
4 JOHN R. KLEIN
Conventions. We work within the category of compactly generated
(based) spaces. Products are to be re-topologized using the compactly
generated topology. A space is homotopy finite if it is the retract of a
finite cell complex.
A non-empty space X is r-connected if its homotopy vanishes in
degrees r for every choice of basepoint. Note that every non-empty
space is (- 1)-connected. By convention, the empty space is (- 2)-
connected.
A map X ! Y (with Y non-empty) is r-connected if it's homotopy
fiber at every choice of basepoint is (r- 1)-connected. A weak equiv-
alence is a map which is r-connected for every r. By convention, the
unique map of empty spaces is a weak equivalence.
We write dim X n if X is weak equivalent to a space that is built
up from the empty set by attaching cells of dimension n.
Acknowledgements. I wish to thank Bill Richter for introducing me
to the notion of Poincar'e embedding. Bill also gave me a copy of
Habegger's thesis to read when I was an undergraduate in the early
1980s. I am very much indebted to Bruce Williams for introducing me
to his papers on embeddings. I am also grateful to Bruce for the wealth
of mathematical discussion I've partaken with him.
2. The Connolly-Williams classification theorem
We recall an important (but little known) result of Connolly and
Williams which relates E(K, Sn) to a desuspension question.
For a 1-connected homotopy finite space K, consider the set of pairs
(C, ff) where C is a 1-connected homotopy finite space and
ff :Sn ! K * C
(the join) induces, via the slant product, an isomorphism in reduced
singular homology H~*(K) ~= ~Hn- *-(1C). Introduce an equivalence
relation on such pairs by declaring that (C, ff) ~ (C0, ff0) if and only
if there is a homotopy equivalence of spaces g :C ! C0 satisfying
(idK *g)Off ' ff0. Call the resulting set of equivalence classes SWn(K).
There is an evident map of sets
E(K, Sn) ! SWn(K)
which assigns to an embedding (M, h) of K the complement of a choice
regular neighborhood of M together its Spanier-Whitehead duality
pairing.
ON EMBEDDINGS IN THE SPHERE 5
Theorem 2.1 (Connolly-Williams [C-W ]). Assume that K is r-connected
(r 1) and dim K k. Furthermore, assume k n- 3, n 6 and
2(k - r) n; if n 7 assume k - r 2. Then
E(K, Sn) ! SWn(K)
is onto. If in addition, 2(k - r) n - 1. The map is one to one.
Remarks. On the face of it, this result doesn't provide a "classification"
of embeddings. Indeed, it isn't clear whether SWn(K) is non-empty.
The remainder of this paper will be concerned with the problem of
determining SWn(K) when additional constraints are present.
The Connolly-Williams result requires n 6 because surgery the-
ory is used in the proof. A Poincar'e embedding version of this result
also holds without the requirement 6 or additional conditions in di-
mensions 7. The Poincar'e version can be proved with the fiberwise
homotopy theoretic techniques appearing in [Kl2 ]. I intend to give a
proof of the Poincar'e version in a future paper.
A variant. We next describe a variant of SWn(K) which is more con-
venient to work with. Assume that K is equipped with a basepoint.
Let Dn- 1(K) be defined as follows: consider the set of pairs (W, ff)
such that W is a based space and ff :Sn-1 ! K ^ W is a stable
S-duality map. Define equivalence relation by (W, ff) ~ (W 0, ff0) if
and only if there is an (unstable, based) map g :W ! W 0such that
(idK ^ g) O ff ' ff0.
Lemma 2.2. Assume that K is r-connected (r 1), dim K k and
k n- 3. Then there is a function
OE :SWn(K) ! Dn- 1(K)
which is onto if 2(k- r) n + 1. If 2(k- r) n, then OE is also one to
one.
Proof. Let (C, ff) be a representative of SWn(K). Choose a basepoint
for C. Then there is a natural weak equivalence
K * C !~ K ^ C
precomposing this weak equivalence with the map ff, we obtain a map
Sn ! K^C which we can arrange to be a based map by precomposing
with a suitable rotation. The associated stable map Sn- 1! K ^ C is
an S-duality. We leave it to the reader to check that OE is well-defined.
6 JOHN R. KLEIN
We now check that OE is onto. Let (W, ff) respresent an element of
Dn-1(K). Then ff :Sn- 1! K ^W is a stable S-duality map. It follows
that H~*(W ) ~=H~n- *- 1(K) = 0 if n- * - 1 > k. Thus W has vanishing
homology when * n- k- 2. In particular, as k n- 3, it follows that
H1(W ) = 0.
Let i :W ! W + natural map to the plus construction. Then W +
is 1-connected and we have
(W, ff) ~ (W +, idK ^ i) .
Using S-duality, it is also straightforward to check that W + is homo-
topy finite. Consequently, we are entitled to assume without loss in
generality that W is 1-connected and homotopy finite.
In fact, the above argument shows that W is (n- k- 2)-connected.
We infer that the smash product K ^ W is (n- k+ r)-connected. By
the Freudenthal suspension theorem, the stable map Sn- 1! K ^ W is
represented by an unstable map fi :Sn ! K ^W when 2(k- r) n+1
(unique up to homotopy if 2(k- r) n). This shows that the function
OE is onto if 2(k- r) n + 1. This argument also shows that OE is one to
one if 2(k- r) n.
Corollary 2.3. The statement of Theorem 2.1 holds when SWn(K) is
replaced by Dn- 1(K).
3. Proof of Theorem B
Theorem B will follow from an enumeration result for suspension
spectra appearing in [Kl1 ]. We first review the statement of this result.
Fix a 1-connected spectrum E. For technical reasons, we shall as-
sume that E is an -spectrum, and that spaces of the spectrum Ej are
cofibrant (i.e., retracts of cell complexes). Consider the set of pairs
(X, h)
such that X is a based space and h : 1 X ! E is a weak (homotopy)
equivalence. Define
(X, h) ~ (Y, g)
if there is a map of spaces f :X ! Y such that g O 1 f is homotopic
to h (in particular, f is a homology isomorphism). This generates an
equivalence relation. Let E denote the associated set of equivalence
classes.
We write dim E k if E can be obtained from the trivial spectrum
by attaching cells of dimension k. Recall that the second extended
power D2(E) is the homotopy orbit spectrum of Z2 acting on E^2.
ON EMBEDDINGS IN THE SPHERE 7
Theorem 3.1 (Klein [Kl1 ]). Assume E is nonempty and is equipped
with a choice of basepoint. Then there is a basepoint preserving function
OE : E ! [E, D2(E)] .
If E is r-connected, r 1 and dim E 3r + 2, Then OE is a surjection.
If in addition dim E 3r + 1, OE is a bijection.
3.1. Recall that
F st(K, Sn- 1)
is spectrum of stable maps from K to Sn- 1;
Lemma 3.2. There is a bijection
Fst(K,Sn-1)~=Dn-1(K)
Proof. An element of Sn-1^K* is represented by a pair (C, ff) where
C is a based space and a weak equivalence ff : 1 C ! F st(K, Sn- 1).
Taking the adjunction, this is the same as specifying a (stable) S-
duality map ff :C ^ K ! Sn-1 . As standard application of S-duality
then allows us to associate to ff an S-duality map ff*: Sn-1 ! K ^
C. The pair (C, ff*) then represents an element of Dn- 1(K). It is
straightforward to check that this procedure defines a bijection.
Lemma 3.3. Let E = F st(K, Sn-1 ). Then there is an isomorphism of
abelian groups
[E, D2(E)] ~=ß0(F st(K ^ K, S(n- 1)ff^ K)hZ2)
Proof. It will be convenient for us to rewrite E ' K* ^ Sn-1 , where
K* = F st(K, S0) is the S-dual of K. For spectra A and B, let F (A, B)
denote the associated function spectrum. Then ß0(F (A, B)) = [A, B].
The first step is to rewrite
F (E, D2(E)) ' F (E, E ^ E)hZ2
(the Z2-action on F (E, E ^ E) is induced by permutation action on the
smash product E ^ E.) To see this, note there is a natural map from
right to left. That this map is a weak equivalence can established by
induction on a cell structure for E, recalling that E is homotopy finite.
Substituting in the value of E into the above, we get
F (E, D2(E)) ' F (K* ^ Sn-1 , (K* ^ Sn-1 )^2)hZ2 .
Now, using the fact that Sn-1 ^ Sn-1 with permutation action is home-
omorphic to S(n- 1)ff^ Sn- 1with diagonal action, the right side of the
last display can be rewritten as
F (K*, S(n- 1)ff^ K* ^ K*)hZ2
8 JOHN R. KLEIN
For homotopy finite spectra A and B, it is well known that the trans-
pose map F (A, B) ! F (B*, A*) is a weak equivalence. Consequently,
there is a Z2-equivariant weak equivalence of spectra
F st(K ^ K, S(n- 1)ff^ K) ' F (K*, S(n- 1)ff^ K* ^ K*) .
given by the transpose map.
Taking homotopy orbits of this last equivalence, and assembling
the prior information we conclude that there is a weak equivalence of
spectra
F (E, D2(E)) ' F st(K ^ K, S(n- 1)ff^ K)hZ2 .
Applying ß0 to this last equivalence completes the proof.
To complete the proof of Theorem B, one just needs to apply Corol-
lary 2.3, Lemma 3.2, Lemma 3.3 and Theorem 3.1 in the stated order
(to apply the 3.1 use the fact that E = F st(K, Sn- 1) is (n- k- 2)-
connected and dim E n - r - 2). We leave it to the reader to check
that the inequalities listed in the statement of Theorem B suffice to
apply these results.
4. Proof of Theorem A
The proof of Theorem A is almost identical to the proof of Theorem
B. There are two essential differences: the first is that instead of using
Theorem 3.1, we need to use the following existence result for realizing
a spectrum as a suspension spectrum in the metastable range:
Theorem 4.1 (Klein [Kl1 ]). There is an obstruction
ffiE 2 [E, D2(E)] ,
(depending only on the homotopy type of E) which is trivial whenever
E has the homotopy type of a suspension spectrum.
Conversely, if E is r-connected, r 1 and dim E 3r+ 2, then E
has the homotopy type of a suspension spectrum if ffiE = 0.
The second essential difference is that when E = F st(K, Sn- 1), we
have an isomorphism of abelian groups
[E, D2(E)] ~=F st(K ^ K, S(n- 1)ff+1^ K)hZ2 .
The obstruction `K is defined so as to correspond to the obstruction
ffiE with respect to this isomorphism of abelian groups. We omit the
details.
ON EMBEDDINGS IN THE SPHERE 9
5. Applications to two cell complexes
Existence. It seems that case of embedding complexes with two cells
was first considered by Cooke [Co1 ] (see also [Co2 ]) and later by Con-
nolly and Williams [C-W , x5].
Let K = Sp[feq+1 be a two cell complex, where f :Sq ! Sp is some
map. Let E := F st(K, Sn- 1) denote the stable Spanier-Whitehead
(n - 1)-dual of K. Set p0= n- p- 2 and q0 = n- q- 2.
Then E is the homotopy cofiber of a stable umkehr map
0 q0
f* :Sp ! S .
As stable classes in ßstq-(pS0), we have
[f*] = [ f] .
Tracing through the definition of the umkehr0map, with slightly extra
care, the sign can be determined as (-1)qp.
In any case, E has the homotopy type of a suspension spectrum if
and only if f* is represented by an unstable map. In our range, this is
equivalent to demanding that the James-Hopf invariant
0
H2(f*) = ßstp0(D2(Sq )) .
be trivial.
Enumeration. Suppose K = Sp [f eq+ 1admits an embedding in Sn.
An analysis similar to the previous case shows that there is an isomor-
phism of based sets
0
E(K, Sn) ~=ßstp0+1(D2(Sq ))
At the prime 2, the stable homotopy groups appearing on the right
have been calculated by Mahowald in degrees p0 min (3q0- 3, 2q0+ 29)
(see Mahowald [Ma , table 4.1]).
For example, suppose that q0 1 mod 16. Then the first few groups
are
_______________________________________________________|||||||||
|______j_________|0___|_1__|2___|3__|4_|5___|____6_____|0
|ß 0 (D (Sq )) |Z |Z |Z |Z |0 |Z |Z Z |
|_2q_+j_2________|_2__|_2__|_8__|_2_|__|_2__|_16___2___|.
10 JOHN R. KLEIN
6.Embeddings of Poincar'e spaces
In this section we assume that K is a r-connected Poincar'e duality
space of formal dimension k.
Remarks. The Browder-Casson-Sullivan-Wall theorem ([Wa2 , Th. 12.1])
says that concordance classes of Poincar'e embeddings of K in Sn are
in one to one correspondence with embeddings up to homotopy of K
in Sn.
If K is a closed PL manifold, then [Wa2 , Th. 11.3.1] implies that
E(K, Sn) is in bijection with the isotopy classes of locally flat PL em-
beddings of K in Sn.
By [Wa2 , Lem. 2.8], we can find a homotopy equivalence K ' L[ek,
where L is a finite complex and dim L k- r- 1. In particular, we have
a cofibration sequence of Z2-spaces
L ^ K [L^L K ^ L ! K ^ K ! Sk ^ Sk .
The first term of this sequence has dimension 2k- r- 1, so we may
infer that the evident map
F st(Sk ^ Sk, S(n- 1)ff+^1K)hZ2 ! F st(K ^ K, S(n- 1)ff+^1K)hZ2
is (n- 2(k- r)+ 1)-connected. In particular, if n 2(k- r), we see that
this map induces an isomorphism on path components.
By elementary manipulations, there is an evident identification
F st(Sk ^ Sk, S(n- 1)ff+^1K)hZ2 ' F st(Sn-2 , K ^ D2(Sn- k- 1)) .
We conclude:
Theorem 6.1. Assume in addition n 2(k- r). Then the obstruction
`K is detected in the abelian group
ßstn-2(K ^ D2(Sn- k- 1)) .
Remark. Let be the Spivak normal fibration of K; we consider has
having fiber a stable (-k)-sphere. Let K denote the Thom spectrum
of . When K embeds in Sn, the fibration compresses down to an
unstable (n- k- 1)-spherical fibration. Conversely, when compresses,
an construction due to Browder gives an embedding of K in Sn+ 1(see
[B ]).
It is therefore tempting to try and relate `K to the obstruction
theoretic problem of finding a compression of . We do not as yet have
a solution to this.
ON EMBEDDINGS IN THE SPHERE 11
By essentially the same argument that proves 6.1, we have
Theorem 6.2. Assume n > 2(k- r). Then the function OEK can be
rewritten as
OEK :E(K, Sn) ! ßstn-(1K ^ D2(Sn- k- 1)) .
The remainder of this section is devoted to obtaining corollaries of
6.1 and 6.2. Our first result shows that OEK is homological in the fringe
dimension beyond the stable range.
Corollary 6.3 (Compare [H-H , Th. 2.3], [Ha ]). The obstruction OEK to
embedding K in S2k- r- 1lives in the abelian group
Hr+ 1(K; Zs) ,
where s = 1 + (-1)k- r+.1
Proof. The Hurewicz map
ßst2k- r-(3K ^ D2(Sk- r- 2))! H2k- r-(3K ^ D2(Sk- r- 2))
~= Hr+ 1(K) H2(k- r-(2)D2(Sk- r- 2))
~= Hr+ 1(K; Zs)
is an isomorphism in this degree. Now apply 6.1.
By a similar argument which we omit (use 6.2), we obtain
Corollary 6.4 (Compare [H-H , Th. 2.4], [Ha ]). The set of concordance
classes of embeddings of K in S2k- r+ 2is isomorphic to
Hr+ 1(K; Zs) ,
where s = 1 + (-1)k- r.
Our next pair of corollaries concern the outcome of tensoring with
the rationals.
Corollary 6.5. If n k mod 2 , then `K Q is trivial. Otherwise,
`K Q is detected in the vector space H2k- n(K; Q).
Proof. If n k mod 2 then ß*(D2(Sn-k-1 )) Q is trivial. We infer
that ß*(K ^ D2(Sn-k-1 )) Q is also trivial. The first part now follows
using 6.1.
For the second part, note that the transfer
D2(Sn- k- 1) ! (Sn- k- 1)^2
is, rationally, the inclusion of a wedge summand. Smashing with K and
applying rational homotopy, we infer that ßstn-(2K ^ D2(Sn- k- 1)) Q
12 JOHN R. KLEIN
is a summand of ßstn-(2K ^ (Sn- k- 1)^2) Q. Over the rationals, stable
homotopy coincides with homology. It follows that `K Q is detected
in H2k- n(K; Q).
Corollary 6.6. Assume K embeds in Sn. Assume inequality (2) holds.
Then E(K, Sn) is finitely generated.
If n k mod 2, then E(K, Sn) is finite. Otherwise, E(K, Sn) Q
is a direct summand of H2k- n+(1K; Q).
Proof of 6.6.The first part follows from 6.2 because ßstn-(1K^D2(Sn- k- 1))
is finitely generated. The second part is proved in a manner similar to
6.5. We omit the details.
A direct consequence of 6.6 is:
Corollary 6.7. Assume the inequality (2) holds. If the betti number
b2k- n+1(K) is trivial, then there are finitely many concordance classes
of embeddings of K in Sn.
7. Localization at 2
Let K and K0 be r-connected finite complexes with dim K, dim K0
k.
Theorem 7.1. Suppose that f :K ! K0 is a 2-local homotopy equiv-
alence. Assume that inequality (1) holds. Then K embeds in Sn if and
only if K0 does.
Remark. Rigdon [Ri ] and Williams [Wi ] prove a similar result in the
metastable range n 3=2(k+ 1). The difference between their result
and ours is that ours holds outside of the metastable range at the
expense of an additional connectivity hypothesis.
Proof of 7.1.The induced map of stable (n- 1)-duals
f* st n- 1
E0 := F st(K0, Sn- 1) ! F (K, S ) =: E
is clearly a 2-local equivalence. By [Kl1 , Th. D], E0 is a suspension
spectrum if and only if E is. The result now follows by applying lemmas
3.2, 2.2 and Theorem 2.1.
ON EMBEDDINGS IN THE SPHERE 13
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Wayne State University, Detroit, MI 48202
E-mail address: klein@math.wayne.edu