HOMOTOPICAL INTERSECTION THEORY, I.
JOHN R. KLEIN AND E. BRUCE WILLIAMS
Abstract. We give a new approach to intersection theory. Our
"cycles" are closed manifolds mapping into compact manifolds and
our "intersections" are elements of a homotopy group of a certain
Thom space. The results are then applied in various contexts, in-
cluding fixed point, linking and disjunction problems. Our main
theorems resemble those of Hatcher and Quinn [H-Q ], but our
proofs are fundamentally different.
Contents
1. Introduction 1
2. Language 6
3. The stable cohomotopy Euler class 8
4. The stable homotopy Euler characteristic 12
5. The complement formula 13
6. Proof of Theorem A 16
7. Proof of Theorem B 18
8. A symmetric description 18
9. Linking 20
10. Fixed point theory 21
11. Disjunction 26
12. An index theorem 27
References 30
1. Introduction
In this article an intersection theory on manifolds is developed us-
ing the techniques of algebraic topology. The "cycles" in our theory
will be maps between manifolds, whereas "intersections" will live in
____________
Date: December 20, 2005.
1991 Mathematics Subject Classification. Primary: 57R19; Secondary: 55N45.
Both authors are partially supported by the NSF.
1
2 JOHN R. KLEIN AND E. BRUCE WILLIAMS
the homotopy groups of a certain Thom space. In order to give the
obstructions a geometric interpretation, one must identify the homo-
topy of this Thom space with a suitable bordism group. Making this
identification requires transversality. Consequently, if one is willing
to forgo a geometric intepretation and work exclusively with Thom
spaces, transversality can be dispensed with altogether. We empha-
size this point for the following reason: although we will not pursue
the matter here, our methods straightforwardly extend to give an in-
tersection theory for Poincar'e duality spaces, even though the usual
transversality results are known to fail in this wider context.
Suppose N is an n-dimensional compact manifold equipped with a
closed submanifold
Q N
of dimension q. Given a map f :P ! N, where P is a closed manifold
of dimension p, we ask for necessary and sufficient conditions insuring
that f is homotopic to a map g whose image is disjoint from Q. We
call these data an intersection problem. The situation is depicted by
the diagram
N;-;Q_
____
_____|_
______ |
_____ fflffl|
P ___f___//N
where we seek to fill in the dotted arrow by a map making the dia-
gram homotopy commute. Note that transversality implies the above
problem always has a solution when p + q < n.
When f happens to be an embedding (or immersion), one typi-
cally requires a deformation of f through isotopies (resp. regular ho-
motopies). This version of the problem was studied by Hatcher and
Quinn [H-Q ], who approached it geometrically using the methods of
bordism theory (see also the related papers by Dax [Da ], Laudenbach
[Lau ] and Salomonsen [Sa ]). Since many of the key proofs in [H-Q ]
are just sketched, we feel it is useful to give independent homotopy
theoretic proofs of their results. Also, some of our steps, such as the
Complement Formula in Section 5, should be of independent interest.
We now summarize our approach. Let E ! P be the fibration
given by converting the inclusion N - Q ! N into a fibration and
then pulling the latter back along P . Then the desired lift exists if and
only if E ! P admits a section. Step one is to produce an obstruction
whose vanishing guarantees the existence of such a section.
HOMOTOPICAL INTERSECTION THEORY, I. 3
In a certain range of dimensions, it turns out that the complete
obstruction to finding a section has been known for at least 35 years:
it goes by the name of stable cohomotopy Euler class (see e.g. Crabb
[C , Ch. 2], who attributes the ideas to various people, notably Becker
[B1 ], [B2 ] and Larmore [Lar ]).
The second step is to equate the stable cohomotopy Euler class, a
`cohomological' invariant, with a stable homotopy Euler characteristic,
a `homological' one. This is achieved using a version of Poincar'e duality
which appeared in [K2 ]. The characteristic lives in a homotopy group
of a certain spectrum.
The third step of the program is to identify the spectrum in step
two as a Thom spectrum. The idea here, which we believe is new, is to
give an explicit homotopy theoretic model for the complement of the
inclusion Q N in a certain stable range. We exhibit this model in
Theorem 5.1 (this is the "Complement Formula" alluded to above).
The final step, which is optional, is to relate the Thom spectrum
of step three with a twisted bordism theory (cf. the next paragraph).
This is a standard application of the Thom transversality theorem. As
pointed out above, this step is omitted in the case of an intersection
problem involving Poincar'e duality spaces. This completes our outline
of the program. We now proceed to state our main results in the
manifold case. This will require some preparation.
We first make some well-known remarks about the relationship be-
tween bordism and Thom spectra. Suppose X is a space equipped with
a vector bundle , of rank k. Consider triples
(M, g, OE) ,
in which M is a closed smooth manifold of dimension n equipped with
map g :M ! X and OE is stable vector bundle isomorphism between
the normal bundle of M and the pullback g*,. The set of equivalence
classes of these under the relation of bordism defines an abelian group
n(X; ,) ,
in which addition is given by disjoint union. This is the bordism group
of X with coefficients in , in degree n.
The Thom space T (,) is the one point compactification of the total
space of ,. If fflj denotes the rank j trivial bundle over X, then there is
an evident map
T (, fflj) ! T (, fflj+1)
which gives the collection {T (, fflj)}j 0 the structure of a (pre-)spectrum.
Its associated -spectrum is called the Thom spectrum of ,, which we
4 JOHN R. KLEIN AND E. BRUCE WILLIAMS
denote by
X, .
The Thom-Pontryagin construction defines a homomorphism
n(X; ,) ! ssn+k (X,)
which is an isomorphism by transversality. These remarks apply equally
as well in the more general case when , is a virtual vector bundle of
rank k.
More generally, if , is a stable spherical fibration, the Thom-Pontryagin
homomorphism is still defined, but can fail to be an isomorphism. The
deviation from it being an isomorphism is detected by the surgery the-
ory L-groups of (ss1(X), w1(,)) (cf. Levitt [Le ], Quinn [Q ], Jones [J],
Hausmann and Vogell [H-V ]). In essence, the L-groups detect the fail-
ure of transversality.
We now return to our intersection problem. Let iQ :Q N denote
the inclusion. Define
E(f, iQ )
to be homotopy fiber product of f and iQ (also known as the homotopy
pullback). Explicitly, a point of E(f, iQ ) is a triple
(x, ~, y)
in which x 2 P , y 2 Q and ~ :[0, 1] ! N is a path such that ~(0) =
f(x) and ~(1) = y.
Define a virtual vector bundle , over E(f, iQ ) as follows: let
jP :E(f, iQ ) ! P and jQ :E(f, iQ ) ! Q
be the forgetful maps and let
jN :E(f, iQ ) ! N
be the map given by (p, ~, q) 7! ~(1=2). Then , is defined as the rank
n - p - q virtual bundle
(jN )*oN - (jP )*oP - (jQ )*oQ ,
where, for example, oN denotes the tangent bundle of N and (jN )*oN
is its pullback along jN .
Theorem A. Given an intersection problem, there is an obstruction
O(f, iQ ) 2 p+q-n (E(f, iQ ); ,) ,
which vanishes when f is homotopic to a map with image disjoint from
Q.
HOMOTOPICAL INTERSECTION THEORY, I. 5
Conversely, if p+2q +3 2n and O(f, iQ ) = 0, then the intersection
problem has a solution: there is a homotopy from f to a map with image
disjoint from Q.
Our second main result identifies the homotopy fibers of the map of
mapping spaces
map (P, N - Q) ! map (P, N)
in a range. Choose a basepoint f 2 map (P, N). Then we have a
homotopy fiber sequence
Ff ! map (P, N - Q) ! map (P, N) ,
where Ff denotes the homotopy fiber at f.
Theorem B. Assume O(f, iQ ) is trivial. Then there is a (2n- 2q- p- 3)-
connected map
Ff ! 1+ 1E(f, iQ ), ,
where the target is the loop space of the zeroth space of the Thom spec-
trum E(f, iQ ),.
Theorems A and B are the main results of this work. In x9-11,
we give applications of these results to fixed point theory, embedding
theory and linking problems.
Families. Theorem B yields an obtruction to removing intersections
in families. Let
F :P x Dj ! N
be a j-parameter family of maps whose restriction to P x Sj-1 has
disjoint image with Q, and whose restriction to P x * is denoted by f,
where * 2 Sj-1 is the basepoint. Assume j > 0.
The adjoint of F determines a based map of pairs
(Dj, Sj-1) ! (map (P, N), map (P, N - Q))
whose associated homotopy class gives rise to an element of ssj-1(Ff).
By Theorem B, this class is determined by its image in the abelian
group
ssj-1( 1+1 E(f, iQ ),) ~= j+p+q-n (E(f, iQ ); ,)
provided j + p + 2q + 3 2n. Denote this element by Ofam(F ).
Corollary C. Assume 0 < j 2n - 2q - p - 3. Then the family
F :P x Dj ! N is homotopic rel P x Sj-1 to a family having disjoint
image with Q if and only if Ofam(F ) is trivial.
6 JOHN R. KLEIN AND E. BRUCE WILLIAMS
Additional remarks. We will not deal with self intersection problems
here as that was in effect handled by the first author in [K3 ].
Some of the machinery developed below has recently been applied
by M. Aouina [A ] to identify the homotopy type of the moduli space
of thickenings of a finite complex in the metastable range.
We plan two sequels to this paper. One will consider a multirelative
version of the theory, in which we develop and obstruction theory for
deforming maps off of more than one submanifold. The other sequel
will develop tools to study periodic points of dynamical systems.
Acknowledgements. We are much indebted to Bill Dwyer for discussions
that motivated this work. We were also to a great extent inspired by
the ideas of the Hatcher-Quinn paper [H-Q ].
Outline. Section 2 sets forth language. Section 3 contains various re-
sults about section spaces, and we define the stable cohomotopy Euler
class. Most of this material is probably classical. In Section 4 we use a
version of Poincar'e duality to define the stable homotopy Euler class.
Section 5 contains a "Complement Formula" which identifies the homo-
topy type of the complement of a submanifold in a stable range. Section
6 contains the proof of Theorem A and Section 7 the proof of Theorem
B. Section 8 gives an alternative definition of the main invariant which
doesn't require iQ :Q ! N to be an embedding. Section 9 describes a
generalized linking invariant based on our intersection invariant. Sec-
tion 10 applies our results to fixed point problems. Sections 11-12
show how our results, in conjunction with a result of Goodwillie and
first author, can be used to deduce the intersection theory of Hatcher
and Quinn.
2. Language
Spaces. All spaces will be compactly generated, and products are to
be re-topologized using the compactly generated topology. Mapping
spaces are to be given the compactly generated, compact open topol-
ogy. A weak equivalence of spaces denotes a (chain of) weak homotopy
equivalence(s).
Some connectivity conventions: the empty space is (-2)-connected.
Every nonempty space is (-1)-connected. A nonempty space X is
r-connected for r 0 if ssj(X, *) is trivial for j r for all choices
of basepoint * 2 X. A map X ! Y of nonempty spaces is (-1)-
connected and is 0-connected if it is surjective on path components.
It is r-connected for r > 0 if all of its homotopy fibers are (r - 1)-
connected.
HOMOTOPICAL INTERSECTION THEORY, I. 7
When speaking of manifolds, we work exclusively in the smooth
(C1 ) category. However, all results of the paper hold equally well in
the PL and topological categories.
Spaces as classifying spaces. Suppose that Y is a connected based
space. The simplicial total singular complex of Y is a based simplicial
set. Take its Kan simplicial loop group. Define GY to be its geometric
realization. Then GY is a topological group and there is a functorial
weak equivalence
Y ' BGY ,
where BGY is the classifying space of GY . To emphasize that GY is a
group model for the loop space, we usually abuse notation and rename
Y := GY .
Thus, Y is identified with B Y .
The thick fiber of a map. Let
f :X ! B
be a map, where B = BG is the classifying space of a topological group
G which is the realization of a simplicial group.
The thick fiber of f is the space
F := pullback(EG ! B X)
where EG ! B is a universal principal G-bundle. Let G act on the
product EG x X by means of its action on the first factor. This ac-
tion leaves the subspace F EG x X setwise invariant, so F comes
equipped with a G-action.
Furthermore, as EG is contractible, F has the homotopy type of the
homotopy fiber of f. Observe that X has the weak homotopy type of
the Borel construction EG xG F and f :X ! B is then identified up
to homotopy as the fibration EG xG F ! B.
Naive equivariant spectra. We will be using a low tech version of
equivariant spectra, which are defined over any topological group.
Let G be as above. A (naive) G-spectrum E consists of based (left)
G-spaces Ei for i 0, and equivariant based maps Ei ! Ei+ 1(where
we let G act trivially on the suspension coordinate of Ei). A mor-
phism E ! E0 of G-spectra consists of maps of based spaces Ei ! E0i
which are compatible with the structure maps. A weak equivalence
of G-spectra is a map inducing an isomorphism on homotopy groups.
E is an -spectrum if the adjoint maps Ei ! Ei+ 1are weak equiv-
alences. We will for the most part assume that our spectra are -
spectra. If E isn't an -spectra, we can functorially approximate it
8 JOHN R. KLEIN AND E. BRUCE WILLIAMS
by one: E !~ E0, where E0iis the homotopy colimit of the diagram of
G-spaces { kEi+k}k 0. We use the notation 1 E for E00, and by slight
abuse of language, we call it the zeroth space of E.
If X is a based G-space, then its suspension spectrum 1 X is a G-
spectrum with j-th space Q(Sj ^ X), where Q = 1 1 is the stable
homotopy functor.
The homotopy orbit spectrum
EhG
of G acting on E is the spectrum whose jth space is the orbit space of
G acting diagonally on the smash product Ej ^ EG+ . The structure
maps in this case are evident.
3.The stable cohomotopy Euler class
Suppose p :E ! B is a Hurewicz fibration over a connected space
B. We seek a generalized cohomology theoretic obstruction to finding
a section.
The fiberwise suspension of E over B is the double mapping cylinder
SB E := B x 0 [ E x [0, 1] [ B x 1 .
This comes equipped with a map SB p :SB E ! B which is also a
fibration (cf. [St]). If Fb denotes a fiber of p at b 2 B, then the fiber of
SB p at b is SFb, the unreduced suspension of Fb.
Let
s- , s+ :B ! SB E
denote the sections given by the inclusions of B x 0, B x 1 into the
double mapping cylinder.
Proposition 3.1. If p :E ! B admits a section, then s- and s+ are
vertically homotopic.
Conversely, assume p :E ! B is (r+ 1)-connected and B homotopi-
cally a retract of a cell complex with cells in dimensions 2r + 1. If
s- and s+ are vertically homotopic, then p has a section.
Proof. Assume p :E ! B has a section s :B ! E. Apply the functor
SB to s to get a map
SB s :SB B ! SB E
and note SB B = B x [0, 1]. Then SB s is a vertical homotopy from s-
to s+ .
HOMOTOPICAL INTERSECTION THEORY, I. 9
To prove the converse, consider the square
p
E ______//_B
p|| s+||
fflffl| fflffl|
B __s-_//SB E
which is commutative up to preferred homotopy. The square is ho-
motopy cocartesian. Since the maps out of E are (r+ 1)-connected,
we infer via the Blakers-Massey theorem that the square is (2r+ 1)-
cartesian.
Let P be the homotopy pullback of s- and s+ . Then the map E ! P
is (2r+ 1)-connected. Furthermore, a choice of vertical homotopy from
s- to s+ yields a map B ! P. Using the dimensional constraints on B,
we can find a map s :B ! E which factorizes B ! P up to homotopy.
Then ps is homotopic to the identity. The homotopy lifting property
then enables us to deform s to an actual section of p.
Section spaces. For a fibration E ! B, let
sec(E ! B)
denote its space of sections. Proposition 3.1 gives criteria for decid-
ing when this space is non-empty. Another way to formulate it is to
consider sec(SB E ! B) as a space with basepoint s- . Then the ob-
struction of 3.1 is given by asking whether the homotopy class
[s+ ] 2 ss0(sec(SB E ! B))
is that of the basepoint.
Stabilization. As remarked above, s- equips the section space
sec(SB E ! B)
with a basepoint, and the fibers SFb of SB E ! B are based spaces
(with basepoint given by the south pole).
Let
QB SB E ! B
be the effect of applying the stable homotopy functor Q = 1 1 to
each fiber SFb of SB E ! B.
Lemma 3.2. Assume p :E ! B is (r+ 1)-connected and B homotopi-
cally the retract of a cell complex with cells in dimensions k. Then
the evident map
sec(SB E ! B) ! sec(QB SB E ! B)
10 JOHN R. KLEIN AND E. BRUCE WILLIAMS
is (2r- k+ 3)-connected.
Proof. For each b 2 B, the space SFb is (r+ 1)-connected. The Freuden-
thal suspension theorem implies that the map SFb ! QSFb is (2r+ 3)-
connected. From this we infer that the map SB E ! QB SB E is (2r+ 3)-
connected. The result now follows from elementary obstruction the-
ory.
We call sec(QB SB E ! B) the stable section space of SB E ! B and
we change its notation to
secst(SB E ! B) .
In fact, the stable section space is the zeroth space of a spectrum whose
j-th space is the stable section space of a fibration Ej ! B in which the
fiber at b 2 B is Q jSFb. In particular, the set of path components
of secst(SB E ! B) has the structure of an abelian group. In what
follows, we regard s- , s+ as points of secst(SB E ! B).
Definition 3.3. The stable cohomotopy Euler class of p :E ! B is
given by
e(p) := [s+ ] 2 ss0(secst(SB E ! B)) .
Corollary 3.4. If p has a section, then e(p) is trivial. Conversely, as-
sume p :E ! B is (r+ 1)-connected and B is homotopically the retract
of a cell complex with cells in dimensions 2r+ 1. If e(p) = 0, then p
has a section.
We will also require an alternative description of the homotopy type
of sec(E ! B) in a range. For a space X equipped with two points
-, + 2 X, let
X
be the space of paths ~ :[0, 1] ! X such that ~(0) = - and ~(1) = +.
When X = SY , and are the poles of SY , we obtain a natural map
Y ! QSY
which maps a point y to the path [0, 1] ! QSY by t 7! t ^ y, where
t ^ y 2 SY is considered as a point of QSY in the evident way.
Next, suppose that E ! B is a fibration. Then the associated
fibration
QB SB E ! B
has QSFb as its fiber at b 2 B. So we have a map
Fb ! QSFb.
Let
B QB SB E ! B
HOMOTOPICAL INTERSECTION THEORY, I. 11
be the fibration whose fiber at b is the space QSFb. Then the above
yields a map of section spaces
sec(E ! B) ! sec( B QB SB E ! B) .
Lemma 3.5. Assume E ! B is (r+ 1)-connected and B is homotopi-
cally the retract of a cell complex with cells in dimensions k. Then
the map
sec(E ! B) ! sec( B QB SB E ! B)
is (2r+ 1 - k)-connected.
Proof. For each b 2 B, the map of fibers
Fb ! QSFb
factors as a composite
Fb ! SFb ! QSFb.
The first map in the composite is (2r+ 1)-connected (by the Blakers-
Massey theorem) and the second map is is (2r+ 2)-connected (by Freuden-
thal's suspension theorem). Hence, the composed map is (2r+ 1)-
connected. We infer by the five lemma that the map E ! B QB SB E
is also (2r+ 1)-connected. Taking section spaces then reduces the con-
nectivity by k.
Lemma 3.6. Fix a section s of the fibration E ! B. Then with respect
to this choice, there is a preferred weak equivalence
sec( B QB SB E ! B) ' sec(QB SB E ! B) .
Proof. The fiber of ( B QB SB E ! B) at b 2 B is the space QSFb.
The hypothesis that E ! B is equipped with a section shows that
Fb is based and therefore QSFb is also based using the map Fb !
QSFb.
A point of QSFb is a path in QSFb having fixed endpoints. Given
another point of QSFb, we get another path having the same end-
points. Now form the loop which starts by traversing the first path
and returns by means of the second path. So we get a map
QSFb ! QSFb
which is a weak equivalence (an inverse weak equivalence is given by
mapping a loop in QSFb to the path given by concatenating the base
path with the given loop). This weak equivalence then induces a weak
equivalence
sec( B QB SB E ! B) ! sec( B QB SB E ! B)
12 JOHN R. KLEIN AND E. BRUCE WILLIAMS
where B is the fiberwise loop space functor. Now use the evident
homeomorphism
sec( B QB SB E ! B) ~= secst(SB E ! B)
to complete the proof.
Assembling the above lemmas, we conclude
Corollary 3.7. Let E ! B be a fibration equipped with section. As-
sume E ! B is (r+ 1)-connected and that B is homotopically the re-
tract of a cell complex whose cells have dimension k. Then the map
sec(E ! B) ! secst(SB E ! B)
is (2r + 1 - k)-connected.
4. The stable homotopy Euler characteristic
Let B be a connected based space. We identify B with the classi-
fying space of the topological group B described in x2. Let E ! B
be a fibration, and let F be its thick fiber (x2). Take its unreduced
suspension SF . Then SF is a based B-space.
Assume now that B is a closed manifold of dimension d. Let oB
be its tangent bundle, and let S(oB + ffl) be the fiberwise one point
compactification of oB . Define
SoB
to be its thick fiber. This is a based B-space.
Define
S-oB
to be the mapping spectrum map (SoB, S0), i.e., the spectrum whose jth
space consists of the stable based maps from SoB to Sj. Then SoB is an
B-spectrum whose underlying unequivariant homotopy type is that
of a (-d)-sphere.
Give the smash product S-oB ^ SF the diagonal action of B. Let
S-oB ^h B SF be its homotopy orbit spectrum.
Theorem 4.1 ("Poincar'e Duality"). There is a preferred weak equiv-
alence of infinite loop spaces
1 (S-oB ^h B SF ) ' secst(SB E ! B) .
In particular, there is an preferred isomorphism of abelian groups
ss0(S-oB ^h B SF ) ~= ss0(secst(SB E ! B)) .
HOMOTOPICAL INTERSECTION THEORY, I. 13
Remark 4.2. A form of this statement which resembles classical Poincar'e
duality is given by thinking of the right side as "cohomology" with co-
efficients in the "cosheaf" of spectra E over B whose stalk at b 2 B is
the spectrum 1 (SFb). Then symbolically, the result identifies coho-
mology with twisted homology:
Ho(B; S-oB E) ' Ho(B; E) ,
where we interpret the displayed tensor product as fiberwise smash
product.
Proof of Theorem 4.1. The theorem is actually a special case of the
main results of [K2 ]. There, for any B-spectrum W , we constructed
a weak natural transformation
S-oB ^h B W ! W h B
called the norm map, which was subsequently shown to be a weak
equivalence for every W (cf. th. D and cor. 5.1 of [K2 ]). The target of
the norm map is the homotopy fixed points of B acting on W . Recall
that when W is an -spectrum, W h B is the spectrum whose jth space
is the section space of the fibration E B x B Wj ! B.
Specializing the norm map to the B-spectrum W = 1 SF , source
of the norm map is identified with S-oB ^h B SF whereas its target
is identified with the spectrum whose associated infinite loop space is
secst(SB E ! B).
Definition 4.3. The stable homotopy Euler characteristic of p :E ! B
is the class
O(p) 2 ss0(S-oB ^h B SF )
which corresponds to the stable cohomotopy Euler class e(p) via the
isomorphism of Theorem 4.1. That is, O(p) is the Poincar'e dual of e(p).
Corollary 4.4. Assume p :E ! B is (r+ 1)-connected, B is a closed
manifold dimension d and d 2r + 1. Then p has a section if and only
if O(p) is trivial.
5. The complement formula
Suppose
iQ :Qq Nn
is the inclusion of a closed connected submanifold. We will also assume
that N is connected.
Choose a basepoint * 2 Q. Then N gets a basepoint. Fix once and
for all identifications
Q ' B Q N ' B N .
14 JOHN R. KLEIN AND E. BRUCE WILLIAMS
We also have a homomorphism Q ! N, such that application of
the classifying space functor yields iQ :Q ! N up to homotopy.
Let
F := thick fiber(N - Q ! N)
be the thick fiber of the inclusion N - Q ! N taken at the basepoint.
Then F is a N-space whose unreduced suspension SF is a based N-
space. Consequently, the suspension spectrum 1 SF has the structure
of an N-spectrum. We will identify the equivariant homotopy type
of this spectrum.
Let Q N denote the normal bundle of iQ :Q ! N. Form the
fiberwise one-point compactification of this vector bundle to obtain
a sphere bundle over Q equipped with a preferred section at 1. Let
S Q N
denote its thick fiber. This is, up to homotopy, a sphere whose dimen-
sion coincides with the rank of Q N . Then S Q N comes equipped
with an Q-action.
Another way to construct S Q N which emphasizes its dependence
only on the homotopy class of iQ is as follows: let SoN be the thick
fiber of the fiberwise one point compactification of the tangent bundle
of N. This is an N-spectrum, and therefore and Q-spectrum by
restricting the action.
Similarly, using the tangent bundle oQ of Q, we obtain a based space
SoQ equipped with an Q-action. Let
SoN -oQ
be the spectrum whose jth space consists of the stable based maps
from SoQ to the j-fold reduced suspension of SoN . Then SoN -oQis a an
Q-spectrum.
Using the stable bundle isomorphism
Q N ~=i*QoN - oQ
it is elementary to check that the suspension spectrum of S Q N (an
Q-spectrum) has the same weak equivariant homotopy type as SoN -oQ.
Theorem 5.1 (Complement Formula). There is a preferred equivari-
ant weak equivalence of N-spectra
1 SF ' SoN -oQ^h Q ( N)+ ,
where the right side is the homotopy orbits of Q acting diagonally on
the smash product SoN -oQ^ ( N)+ . Alternatively, it is the effect of
inducing SoN -oQ along the homomorphism Q ! N in a homotopy
invariant way.
HOMOTOPICAL INTERSECTION THEORY, I. 15
Remark 5.2. This result recovers the homotopy type of SN (N - Q)
as a space over N in the stable range. Namely, the Borel construction
applied to SoN -oQ^h Q ( N)+ gives a family of spectra over N, and the
result says that this family coincides up to homotopy with the fiberwise
suspension spectrum of SN (N - Q) over N.
Proof of 5.1.Let := Q N denote the normal bundle of Q in N. Us-
ing a choice of tubular neighborhood, we have a homotopy cocartesian
square
S( ) _____//N - Q
| |
| |
fflffl| fflffl|
D( ) _______//N .
There is then a weak equivalence of homotopy colimits
(1) hocolim (D( ) S( ) ! N) ~! hocolim (N N - Q ! N) .
Each space appearing in (1) is a space over N. Take the thick fiber
over N of each of these spaces to get an equivariant weak equivalence
(2) hocolim (D"( ) ! "S( ) *) ~! hocolim (* ! F *) =: SF ,
where F is the thick fiber of N - Q ! N. The proof will be completed
by identifying the domain of (2).
If Q" denotes the thick fiber of i :Q ! N, then the domain of (2)
is, by definition, the Thom space of the pullback of along the map
"Q! Q.
We can also identify "Qwith the homotopy orbits Q acting on N:
Q" ' E Q x Q N .
This space comes equipped with a spherical fibration given by
(3) E Q x Q ( N x S ) ! E Q x Q ( N x *) = "Q
where S denotes the thick fiber of the spherical fibration given by
fiberwise one point compactifying . The spherical fibration (3) comes
equipped with a preferred section (coming from the basepoint of S .
It is straightforward to check that this fibration coincides with the
fiberwise one point compactification of the pullback of to Q".
Consequently, the Thom space of the pullback of to Q" coincides
up to homotopy with the effect collapsing the preferred section of (3)
to a point. But the effect of this collapse this yields
( N)+ ^h Q S .
16 JOHN R. KLEIN AND E. BRUCE WILLIAMS
Hence, what we've exhibited is an N-equivariant weak equivalence of
based spaces
SF ' ( N)+ ^h Q S .
The proof is completed by taking the suspension spectra of both sides
and recalling that 1 S is SoN -oQ.
6. Proof of Theorem A
Returning the the situation of the introduction, suppose
N;-;Q_
____
_____|_
______ |
_____ fflffl|
P ___f___//N
is an itersection problem. As already mentioned, the obstructions to
lifting f up to homotopy coincide with the obstructions to sectioning
the fibration
p :E ! P
where E is the homotopy fiber product of P ! N N - Q.
Choose a basepoint for P . Then N gets a basepoint via f. The
the thick fiber p at the basepoint is identified with the thick fiber of
N - Q ! N. Call the thick fiber of the latter F . Then F is an N-
space; using the homomorphism P ! N, we see that F is also an
P -space.
Lemma 6.1. The map N - Q ! N is (n - q - 1)-connected.
Proof. Using the tubular neighborhood theorem, N - Q ! N is the
cobase change up to homotopy of the spherical fibration S( Q N ) ! Q
of the normal bundle of Q. The fibers of this fibration are spheresd of
dimension n-q -1, so the fibration is that much connected. Then N -
Q ! N is also (n- q- 1)-connnected because cobase change preserves
connectivity.
Corollary 6.2. The map E ! P is also (n - q - 1)-connected.
Proof. E ! P is the base change of the the (n - q - 1)-connected map
N - Q ! N converted into a fibration. The result follows from the
fact that base change preserves connectivity.
We will now apply Corollary 4.4. For this we note that the manifold
P is homotopically a cell complex of dimension p, Consequently, if
p 2(n - q - 2) + 1 = 2n - 2q - 3 ,
HOMOTOPICAL INTERSECTION THEORY, I. 17
4.4 implies that a section exists if and only if the stable homotopy Euler
characteristic
O(p) 2 ss0(S-oP ^h P SF )
is trivial.
To complete the proof, we will need to identify the homotopy type
of the spectrum
(4) S-oP ^h P SF .
By the Complement Formula 5.1, there is a preferred weak equivalence
of N-spectra
1 SF ' SoN -oQ^h Q ( N)+ .
Substituting this identification into (4), we get a weak equivalence of
spectra
(5) S-oP ^h P SF ' S-oP ^h P SoN -oQ^h Q ( N)+ .
To identify the right side of (5)as a Thom spectrum, rewrite it again
as
SoN -oP-oQ^h( Px Q) ( N)+ .
Here, the action of P x Q on
SoN -oP-oQ = SoN ^ S-oP ^ S-oQ
is given by having P act trivially on S-oQ , having Q act trivially
on S-oP , and having P and Q act on SoN by restriction of the N
action.
Clearly, this is the Thom spectrum associated to the (stable) spher-
ical fibration
(SoN -oP-oQx E (P x Q)) x Px Q N ! E (P x Q) x (PxQ) N .
It is straightforward to check that the base space of this fibration is
weak equivalent to the homotopy fiber product E(f, iQ ) described in
the introduction.
Hence, the right side of (5) is just a Thom spectrum of the virtual
bundle , over E(f, iQ ) (where , is defined as in the introduction).
Therefore, we get an equivalence of Thom spectra,
S-oP ^h P SF ' E(f, iQ ), .
Consequently, the stable homotopy Euler characteristic becomes iden-
tified with an element of the group
ss0(E(f, iQ ),)
which, by transversality, coincides with the bordism group
p+q-n (E(f, iQ ); ,) .
18 JOHN R. KLEIN AND E. BRUCE WILLIAMS
Therefore, O(p) can be regarded as an element of this bordism group.
The proof of Theorem A is then completed by applying 4.4.
7. Proof of Theorem B
Recall the weak equivalence
Ff ' sec(E ! P ) ,
where Ff is the homotopy fiber of
map (P, N - Q) ! map (P, N)
at f :P ! N, and
E ! P
is the fibration in which E is the homotopy pullback of
f
P --- ! N - - - N - Q .
Using 6.2, we have that E ! P is (n- q- 1)-connected. So by Corollary
3.7, there is a (2n - 2q - p - 3)-connected map
Ff ' sec(E ! P ) ! secst(SP E ! P ) .
By 4.1 and the Complement Formula 5.1, there is a weak equivalence
secst(SP E ! P ) ' 1 E(f, iQ ),
where the right side is the Thom spectrum of associated with the bundle
, appearing in the introduction. Looping this last map, and using the
previous identifications, we obtain a (2n - 2q - p - 3)-connected map
Ff ! 1+1 E(f, iQ ), .
This completes the proof of Theorem B.
8. A symmetric description
It is clear from its construction that the stable homotopy Euler
characteristic depends only upon the homotopy class of f :P ! N
and the isotopy class of iQ :Q ! N. Using a different description of
the invariant, we will explain why it is an invariant of the homotopy
class of iQ .
The new description is more general in that it is defined not just for
inclusions iQ :Q ! N but for any map, and it is symmetric in P and
Q.
HOMOTOPICAL INTERSECTION THEORY, I. 19
Given maps f :P ! N and g :Q ! N, consider the intersection
problem
N x7N7-___
________|____
________ |
____ fflffl|
P x Q __fxg__//N x N
which asks to find a deformation of f xg to a map missing the diagonal
of N x N. Then we have a stable homotopy Euler characteristic
O(f x g, i ) 2 p+q-n (E(f x g, i ); ,0)
for a suitable virtual bundle ,0 defined as in the introduction. A
straightforward chasing of definitions shows there to be a homeomor-
phism of spaces
E(f x g, i ) ~= E(f, g) .
Furthermore, if , is the virtual bundle on E(f, g) defined as in the
introduction, it is clear that ,0 and , are equated via this homeomor-
phism. The upshot of these remarks is that we can think of O(f xg, i )
as an element of the bordism group
p+q-n (E(f, g); ,) .
When g = iQ is an embedding, this is the same place where the our
originally defined invariant O(f, iQ ) lives.
Theorem 8.1. In fact, the invariants O(f x iQ , i ) and O(f, iQ ) are
equal.
Remark 8.2. Theorem 8.1 immediately shows that O(f, iQ ) depends
only on the homotopy classes of f and iQ . It also gives extends our
intersection invariant to the case when iQ isn't an embedding.
Proof of 8.1.(Sketch). The way to compare the invariants is to con-
sider the commutative diagram
N<-