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<-