Double point manifolds of immersions of spheres in Euclidean space

Double point manifolds of immersions of spheres in Euclidean space Peter J. Eccles

Department of Mathematics, University of Manchester, Manchester, Great Britain. M13 9PL e-mail: [email protected]

To William Browder on his sixtieth birthday

Abstract

Anyone who has been intrigued by the relationship between homotopy theory and dierential topology will have been inspired by the work of Bill Browder. This note contains an example of the power of these interconnections. We prove that, in the metastable range, the double point manifold a selftransverse immersion S n # R n+k is either a boundary or bordant to the real projective space R P n?k . The values of n and k for which non-trivial double point manifolds arise are determined.

1 Introduction

Given a self-transverse immersion f : S n # R n k , the r-fold intersection set Ir (f ) is de ned as follows: Ir (f ) = f f (x ) = f (x ) = : : : = f (xr ) j xi 2 S n; i 6= j ) xi 6= xj g: The self-transversality of f implies that this subset of R n k is itself the image of an immersion (not necessarily self-transverse) r (f ): Ln?k r? # R n k of a manifold L of dimension n ? k(r ? 1) called the r-fold intersection manifold of f . It is natural to ask for a given manifold L whether it can arise as an intersection manifold of an immersed sphere and if so for which dimensions. Alternatively we can consider simply the bordism class of L. In the stable range n < k the map f is necessarily an embedding as Ir (f ) is empty for r > 2. In this note we consider immersions in the metastable range, n < 2k. In these cases the intersection manifolds are empty for r > 3 and so the double point manifold is embedded by  (f ). +

1

2

+

(

1)

+

2

This work began during a visit to the Departments of Analysis and of Geometry of E}otv}os University, Budapest. The author is grateful to these departments and Andras Sz}ucs for their invitation and hospitality, and to the British Council for support with travel costs. 

1

Theorem 1.1 In the metastable range, k 6 n < 2k, (a) if n ? k is odd then the double point manifold of a self-transverse immersion S n # R n+k is a boundary;

(b) if n ? k is even then the double point manifold of a self-transverse immersion S n # R n k is either a boundary or bordant to the real projective space R P n?k . +

It should be recalled that odd dimensional real projective spaces are boundaries.

Theorem 1.2 In the metastable range, 0 6 4p < n, there exists an immersion S n # R n? p with double point manifold bordant to R P p if and only if n  2q ? 1 mod 2q , where q = p if p  0 or 3 mod 4 and q = p + 1 if p  1 or 2 mod 4. 2

2

2

In fact, it is known ([17]) that for p > 5 there is an immersion with double point manifold homeomeorphic to R P p in the dimensions given by this theorem. I am grateful to Andras Sz}ucs for drawing this reference to my attention. In addition, it was my eorts to understand his result that there exists an immersion M n # R n? with double point manifold bordant to the projective plane if and only if n  3 mod 4 ([14]) which led to the present work. Theorem 1.2 shows that in this result the manifold M can always be taken to be a sphere. The proof of these theorems is an application of the general method introduced in [3]. It may be summarized as follows. 2

2

2

Step 1. Write R Pk1 for the truncated real projective space R P 1 =RP k? . Then to an element  2 n R Pk1 we can associate an immersion i: S n # R n k well-de ned up 1

+

to regular homotopy. In the metastable range every regular homotopy class of such immersions arises in this way.

Step 2. Let MO(k) be the Thom space of the universal O(k)-bundle so that each element of n MO(k) represents a bordism class of (n ? k)-dimensional manifolds. Then

there is a map

Mk: R Pk1 ! MO(k) of Thom complexes, induced by the bundle k over R P 1 , the Whitney sum of k copies of the Hopf line bundle. The element (Mk)  2 n MO(k) represents the bordism class of the double point manifold of the immersion i .

Step 3. Since the bordism classes of a manifold is determined by its Stiefel-Whitney numbers we apply the Hurewicz map in Z=2-homology to the class of the double point manifold. The following diagram commutes by naturality. (Mk) - nMO(k) nR Pk1 h

?

Z=2  = Hn(R Pk1 ; Z=2)

(Mk)

h

?

- Hn(MO(k); Z=2)

2

Results about the S -reducability of truncated projective spaces ([1]) imply that, for k 6 n < 2k, the left hand Hurewicz homomorphism is zero if and only if k < n + 1 ? (n + 1). Here (n + 1) is the Hurwitz-Radon-Eckmann number which has the property that (n +1) ? 1 is the maximum number of linearly independent tangent vector elds on S n. Thus, in this case, the double point manifold is a boundary. The complete solution is obtained by evaluating the map (Mk) in homology. This paper is organized as follows. In x2 we discuss the generalized J -homomorphism which leads to the rst step of the above proof. In x3 we consider the relationship between this homomorphism and the Hopf invariant. The relationship between the Hopf invariant and the double point manifold described in x4 then leads to the second step. The calculations required for the third step are described in x5.

2 The generalized J -homomorphism Let G(m) be a closed subgroup of the orthogonal group O(m) with inclusion map i: G(m) ! O(m). Then, if  is an m-dimensional vector bundle on S n represented by  2 n BO(m), a G(m)-structure on  is a choice of element  2 n BG(m) such that i  =  . The normal bundle  of the standard embedding S n ,! R n m is trivial and so each element of n O(m)=G(m) determines a G(m)-structure on  . With this structure, the embedded sphere represents an element of n m MG(m) by the Pontrjagin-Thom construction. This process de nes the generalized J -homomorphism +

+

J : nO(m)=G(m) ! n m MG(m) introduced by Bruno Harris ([4]). The image of this map J consists of those elements which may be represented by the standard embedding of S n in R n m with some G(m)-structure. When G(m) is the trivial group, this reduces to the classical J -homomorphism nO(m) ! n mS m . We now consider the case of G(m) = O(k), for k 6 m, embedded in the standard way. In this case O(m)=O(k) is the real Stiefel manifold Vm?k (R m ) and MG(m) is the suspension m?k MO(k) so that we have the following map: J : nVm?k (R m ) ! n m m?k MO(k): As an aside, we now describe a map of spaces inducing this J -homomorphism which will be needed later. Recall that, as the classifying space BO(k), we can take the in nite Grassmannian Gk (R 1 ) of k-dimensional linear subspaces of R1 (see for example [9]). In this case, the total space EO(k) of the universal bundle is given by EO(k) = f (u; U ) j u 2 U; U 2 Gk (R 1 g: Now, let v = (v ; : : : ; vm?k ) 2 Vm?k (R m ) be an orthogonal (m ? k)-frame. Write U = hv ; : : : ; vm?k i?  R m  R 1 , the subspace of R m orthogonal to each vi. Then a point of R m may be written uniquely as u + t v + : : : + tm?k vm?k where ti 2 R and u 2 U . So we may de ne a continuous map J (v): Rm ! EO(k)  R m?k by J (v)(u + t v + : : : + tm?k vm?k ) = ((u; U ); t ; : : : ; tm?k ): +

+

+

+

1

1

1 1

1 1

1

3

This induces a map J (v): S m ! MO(k) ^ S m?k , i.e. J (v) 2 m m?k MO(k). Checking the de nitions carefully gives the following result.

Proposition 2.1 The continuous map J : Vm?k (R m ) ! m m?k MO(k) induces the gen-

eralized J -homomorphism

J : n Vm?k (R m ) ! n m m?k MO(k)  = n m m?k MO(k): +

Returning to the dierential topology, recall ([16]) that the bordism group of immersed manifolds M n # R n k is isomorphic to the stable homotopy group nS MO(k). An n-manifold embedded in Rn m with an O(k)-structure, as above, has (m ? k) linearly independent normal vector elds and by [5] these lead to a regular homotopy of the embedding to an immersion in R n k  R n m . This process corresponds to the stabilization map n m m?k MO(k) ! nS k MO(k): Composing this with J gives a stable J -homomorphism +

+

+

+

+

+

J S : nVm?k (R m ) ! nS k MO(k) +

whose image consists of classes represented by immersed spheres, and all such classes if m > n since then the stabilization map is an epimorphism. Recall the classical result of Stephen Smale ([10]) that for m > n + 1 the homotopy group nVm?k (R m ) represents regular homotopy classes of immersed spheres S n # R n k ; the stable J -homomorphism maps regular homotopy classes to bordism classes. Finally, recall that hyperplane re ection de nes a (2k ? 1)-equivalence : R Pkm? ! Vm?k (R m ). The composition +

1

J S  : nR Pkm? ! nS k MO(k) 1

+

provides the map  7! [i : S n # R n k ] claimed in Step 1 of the introduction. From the above discussion, all bordism classes represented by immersed spheres lie in the image of this map if m > n and 2k > n. +

3 The Hopf invariant and the J -homomorphism In this section we demonstrate that the multiple suspension Hopf invariant is closely related to the generalized J -homomorphism described in the previous section. To describe this invariant we need a preliminary de nition. The quadratic construction on a pointed space X is de ned to be

D X = X ^ X oZ= S 1 = X ^ X Z= S 1=fg Z= S 1; 2

2

2

2

where the non-trivial element of the group Z=2 acts on X ^ X by permuting the coordinates and on the in nite sphere S 1 by the antipodal action. This space has a natural ltration given by Di X = X ^ X oZ= S i. The following homeomorphisms may be obtained directly from this de nition. 2

2

Proposition 3.1 For a pointed space X , 4

(a) (b) (c) (d)

DX = X ^ X; for i > 1, Di X=Di? X  = i(X ^ X ); for i; j > 1, Di j X  = j (Di j X=Dj? X ); in particular, for i; j > 1, Di S j  = j R Pji j . 0 2

2

2

1

+ 2

2

1

2

+

2

We can now describe the basic properties of the multiple suspension James-Hopf invariants hi2 . Write QX for the direct limit 1 1 X = lim n n X . For a connected space X and i > 0, there is a natural transformation (see [11])

hi = hi (X ): i i X ! QDi X: 2

+1

2

+1

2

In the case i = 0 this is simply the stabilization of the classical James-Hopf invariant:

X ! (X ^ X ) ! Q(X ^ X ): The main property of these maps is a generalization of the classical EHP -sequence.

Theorem 3.2 ([8]) Let X be a (k ? 1)-connected space where k metastable range j < 3k, there is an exact sequence as follows:

> 1.

Then, in the

hi i  i i X ?! j X ?! j QDi X ?! j? X: j +1

2

+1

2

1

Using adjointness and stability we can rewrite this as follows: i+1

j X ?! i

j

+ +1

hi i X ?! j Di X ?! j? X: 2

+1

2

1

We also need to record the relationships between these invariants for dierent values of i. These follow directly from their de nitions.

Proposition 3.3 (a) For each i > 0 the following diagram is commutative. hi2

i i X +1

+1

?

i+2 i+2 X

hi

QDi X 2

? - QDi X

+1 2

+1 2

(b) For each i; j > 0, the following diagram is commutative. hi j (X ) - QDi j X

i j i j X + +1

+ 2

+ +1

1

?

j i+1 i+1j X

+ 2

? - j QDi j X = Q(Di j X=Dj? X )

j hi (j X ) 2

2

5

+ 2

2

1

The rst part of this proposition means that the invariants hi combine to form the stable James-Hopf invariant hS : QX ! QD X . We can now state the main result relating the Hopf invariant and the J -homomorphism. Theorem 3.4 For 1 6 k < m, the following diagram is commutative.  -  V (R m ) J -  m?k MO(k)  R P m? 2

2

2

n

n m?k

1

k

n+m

=

?

n+k m?k m?k MO(k)

k

hm?k?

?

1

= - S m?k? k i - S m?k?? n k D S n k D MO(k) 2

nS+k k R Pkm?1

1

2

+

2

+

1

Here the isomorphism in the bottom row comes from the homeomorphism of Proposition 3.1(d) and the map i is induced by the map S k ! MO(k) arising from the inclusion of a bre in the universal bundle. This result may be proved by making use of the map inducing the J -homomorphism introduced in the previous section beginning with a lemma which corresponds to the k = 0 version of the theorem. Lemma 3.5 For m > 2, the following diagram is homotopy commutative.  - O(m) R P m? 1

i

?

J

?

hm?2 (S 1)

QR P m? 

m S m = QDm? S  Proof. The proof is by induction on m. For m = 2, the result is a formulation of the statement that the Hopf map S ! S has Hopf invariant 1. The inductive step is based on the following commutative diagram. 1

3

-

hm? (S ) 2

2

1

?

QR P m?1

1

?

?

- O(m + 1)

- Sm

J

?

m S m

- Sm

RP m



?

O(m) J

2

1

2

R P m?1



2

2

? - m S m +1

hm? (S ) 2

A

1

1

C

+1

m m Q h (S ) 0 2

QQ B QQs ? m -

- QR P

6

i

?

QS m

In this diagram, A is commutative by Proposition 3.3(a), B is commutative by Proposition 3.3(b) and C is commutative by [6]. Adjointing the vertical maps in the diagram gives the following diagram of stable maps.

-

R P m?1

m?

?

m

1

?

1

-

R P m?1

- Sm

RP m

?

- Sm

RP m

The inductive hypothesis is that m? ' 1. Since the rows are co bre sequences this implies that the stable map m ' 1 as required to complete the inductive step. 2 Proof of Theorem 3.4. Consider the following commutative diagram. 1

- R P m?

R P k?1



1



?



?

?

- O(m)

O(k) J

Vm?k (R m )

J

?

J

?

?

- m S m

k S k 2

QR P k?1

2

- k m?k m?k S k - m m?k MO(k)

hm? (S )

hk? (S )

?

- R Pkm?

1

1

? - QR P m?

2

1

2

k hm?k? (S k )

1

k hm?k? (MO(k))

1

1

? ? - k QDm?k? S k - k QDm?k? MO(k) 2

2

2

1

1

2

In this case, adjointing the vertical maps and using Dm?k? S k  = k R Pkm? we obtain the following commutative diagram of stable maps. 2

k R P k?

1

k ?

?

1

k R P k?1

-k R P m?

1

-k R Pkm? ?? ? ? ? m ? -k R Pk - Dm?k? MO(k) 1

1

m?

? -k R P m?

1

1

1

1

2

1

Here the stable map exists by the basic properties of co bre sequences. By the lemma, k? ' 1 and m? ' 1. It follows that the stable map ' 1. Now applying n to the unstable adjoint of the right hand triangle gives the theorem. 2 1

1

4 The Hopf invariant and the double point manifold The nal ingredient required to enable us to read o the bordism class of the double point manifold of an immersed sphere is the observation that this is determined by the 7

stable Hopf invariant hS : nS k MO(k) ! nS k D MO(k) in the following sense. If the self-transverse immersion f : M n # R n k represents the element  2 nS k MO(k), then the double point manifold  (f ): Ln?k # R n k respresents the element hS (). This has been proved independently by Pierre Vogel ([15]), by Andras Sz}ucs ([12], [13]), and by Ulrich Koschorke and Brian Sanderson ([7]). Notice that  (f ) represents an element in the stable homotopy of D MO(k) because the immersion of the double point manifold automatically acquires some additional structure on its normal bundle, namely that at each point f (x ) = f (x ), the normal 2k-dimensional space may be decomposed as the direct sum of the two (unordered) kdimensional normal spaces of f at the points x and x . The universal bundle for this structure is EO(k)  EO(k) Z= S 1 ! BO(k)  BO(k) Z= S 1 which has the Thom complex D MO(k). In the case of immersed spheres in the metastable range, Theorem 3.4 shows that the double point manifold has a more re ned structure corresponding to the restriction of the above universal bundle to fg  fg Z= S 1: 2

+

2

+

+

+

+

2

2

2

2

1

1

2

2

2

2

2

2

R k  R k Z=2 S 1 ! R P 1 :

As before, the involution on R k  R k is obtained by interchanging coordinates, but in this case by a change of basis it can be written as (x; y) 7! (?x; y). This demonstrates that this bundle is simply the Whitney sum k  k where  is the Hopf bundle. This is another way of proving that the Thom complex D S k is homeomorphic to the suspended truncated real projective space k R Pk1 (using [2]). In fact we are here not interested in this additional structure but merely in the unoriented bordism class of the double point manifold. Forgetting the additional structure induces a map of Thom complexes  : D MO(k) ! MO(2k). The bordism class of the double point manifold is then given by the composition 2

2

 S n k MO(2k) ! n?k MO nS k D MO(k) ! 2

+

+

where the nal stabilization map to the homotopy of the MO-spectrum corresponds to forgetting the immersion. The above identi cation of the restricted universal bundle imples that the composition i : nS k k R Pk1  = nS k D S k ! nS k D MO(k) ! S k n k MO(2k) is given by  (Mk) . We can sum this up in the following result. +

2

+

+

2

+

Theorem 4.1 Let  : nS k MO(k) ! n?k MO be the map de ned geometrically by the 2

+

double point manifold of a self-transverse immersion. Then the following diagram is commutative.

nS+k MO(k) 

?

2

n?k MO 

- nS k D MO(k) nS k D S k = n k k R Pk1

hS 2

+

2

+

2



?

nS+k MO(2k)

 8

nS MO(k)

+

(Mk)



 =

?

nS RPk1

Some geometric comments

It is easy to see directly that the double point manifold of an immersed sphere has the re ned structure discussed above. Let L~ be the double cover of the double point manifold L so that there is the following commutative diagram. - S n  - BO(n + k) L~ n?k

f

?  (f ) -

?

2

Ln?k

R n+k

Then  jL~ n?k is trivial since n ? k < n. A choice of trivialization induces a Z=2-structure on the normal bundle of Ln?k corresponding to the universal bundle discussed above with Thom complex D S k  = k R Pk1 . Notice that this construction applies to any immersion of a sphere and suggests that a stable retraction of Vm?k (R m ) onto R Pkm? can be included in the diagram of Theorem 3.4. This ought to lead to an extension of the results of this paper beyond the metastable range to the general case. 2

1

5 Determining the double point manifold In this section we prove Theorems 1.1 and 1.2. The calculation is based upon the following immediate corollary of Theorems 3.4 and 4.1.

Proposition 5.1 For k 6 n < 2k, the following diagram is commutative. i

- n R Pk1 (Mk-)n MO(k)

nR Pkm?

1



J S  

? ?  - n?k MO nS k MO(k) Thus, given  2 nR Pkm? , if i : S n # R n k is a self-transverse immersion representing J S   () then the bordism class of the double point manifold of i is represented by (Mk) i () 2 nMO(k). 2

+

1

+

For m > n + 1 the above map i is an isomorphism. We may complete the proof by applying the Hurewicz homomorphism in Z=2-homology to Mk because this Hurewicz homomorphism for MO is a monomorphism ([9]). (Mk) - nMO(k) - n?k MO nR Pk1

h

h

h

? (Mk) ? ?  1 Hn?k MO Z=2  HnMO(k) = HnR Pk 9

Here and from now on we write HnX for Hn(X ; Z=2). It is well-known ([1]) that, for k 6 n < 2k, the left hand Hurewicz homomorphism is zero if and only if k < n +1 ? (n +1). So in this case   (Mk) = 0: nR Pk1 ! n?k MO and the double point manifold is a boundary.

Proposition 5.2 For k 6 n < 2k, if k > n + 1 ? (n + 1), let  2 nR Pk1 be such that h() = 6 0. Let Ln?k be the double point manifold of a corresponding immersion i : S n # R n k . Then the (normal) Stiefel-Whitney numbers of L are given by +

wI

[Ln?k ] =

k i

!

!

k ::: k it i

!

2

1

for partitions I = (i1 ; i2 ; : : : ; it ) such that i1 + i2 + : : : + it = n ? k.

Proof. The Thom complex MO(k) is homotopy equivalent to the quotient space BO(k)=BO(k ? 1) so that H MO(k)  = wk Z=2[w ; : : : ; wk ] with the Thom isomorphism i k i  ~ H MO(k) = H BO(k) given by wI wk $ wI . Hence the Stiefel-Whitney number 1

+

wI [L] = hh(Mk) (); wI wk i by the Thom isomorphism = hh(); wI (k)ak i by naturality where a 2 H R P 1 is the generator so that ak 2 H k R Pk1 is the Thom class. k  1   Now the total Stiefel-Whitney class w(k ) = (1 + a) 2 H R P so that wi(k ) = k ai . The result follows. 2 i To complete the calculation we con rm that these are the normal Stiefel-Whitney numbers of R P n?k . 1

Proposition 5.3 If k > n + 1 ? (n + 1), then the (normal) Stiefel-Whitney numbers of R P n?k are given by

wI

[R P n?k ] =

k i

!

1

!

!

k ::: k : it i 2

Proof. Put n + 1 = (2a + 1)2b. It is clear from the de nition of  that (n + 1) 6 2b so that k > n + 1 ? (n + 1) ) n ? k 6 (n + 1) ? 1 < 2b. The total tangent Stiefel-Whitney class of R P n?k is given by (1+ a)n?k since   1  = n?k ) = (1 + a)?n? k 2 (n ? k + 1). Hence the total normal class is given by w ( R P    H R P n?k . Hence, for i 6 n ? k, wi(R P n?k ) = ?n?i k ai = ki ai since, by the above, i 6 n ? k ) i < 2b and n + 1 is a multiple of 2b. The result follows. 2 +1

1+

1+

This completes the proof of Theorem 1.1. To complete the proof of Theorem 1.2 we simply examine the de nition of the function . We have proved that in the case of even n ? k, say 2p, there exists an immersion S n # R n k with double point manifold bordant to R P n?k if and only if k > n+1?(n+1), i.e. (n + 1) > 2p + 1. Recall that, for n + 1 = (2a + 1)2b where b = c + 4d for 0 6 c 6 3 and d > 0, the value of  is given by (n + 1) = 2c + 8d. It follows that the least value of b for which (n + 1) > 2p + 1 is given by +

10

for p  0 mod 4: c = 0 and 8d = 2p, i.e. b = p; for p  1 mod 4: c = 2 and 8d = 2(p ? 1), i.e. b = p + 1; for p  2 mod 4: c = 3 and 8d = 2(p ? 2), i.e. b = p + 1; for p  3 mod 4: c = 3 and 8d = 2(p ? 3), i.e. b = p. This completes the proof of Theorem 1.2. September 1994 Published in `Prospects in Topology,' (ed Frank Quinn), Annals of Math. Studies 138, Princeton University Press (1996).

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[14] [15] [16] [17]

A. Sz}ucs, Double point surfaces of smooth immersions M n ! R n? , Math. Proc. Cambridge Philos. Soc. 113 (1993) 601{613. P. Vogel, Cobordisme d'immersions, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 317{358. R. Wells, Cobordism groups of immersions, Topology 5 (1966) 281{294. R. Wells, Double covers and metastable immersions of spheres, Can. J. Math. 26 (1974) 145{176. 2

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2