Immersions of ManifoldsAuthor(s): Ralph L. CohenSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 79, No. 10, [Part 2: Physical Sciences] (May 15, 1982), pp. 3390-3392Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/12423 .

Accessed: 02/05/2014 09:51

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 62.122.73.92 on Fri, 2 May 2014 09:51:09 AMAll use subject to JSTOR Terms and Conditions

Proc. Nati Acad. Sci. USA Vol. 79, pp. 3390-3392, May 1982 Mathematics

Immersions of manifolds (normal bundle/classifying space/Thom space)

RALPH L. COHEN

Stanford University, Stanford, California 94305

Communicated by George W. Whitehead, February 22, 1982

ABSTRACT This paper outlines a proof of the conjecture that every compact, differentiable, n-dimensional manifold immerses in Euclidean space of dimension 2n - a(n), where a(n) is the num- ber of ones in the dyadic expansion of n.

An old problem in differential topology is that of finding the smallest integer k(n) with the property that every compact, Co, n-dimensional manifold Mn immerses in [n + k(n)]-dimensional Euclidean space, Rn+k(n). A well-know-n conjecture is that k(n) = n - a(n), where a(n) is the number of ones in the dyadic expansion of n. The classical immersion theorem of Whitney (1) states that k(n) c n - 1 and, in particular, implies that this conjecture is true when n is a power of 2. The purpose of this note is to announce and to outline a proof of this conjecture for all integers n. Details will appear elsewhere,

The primary obstructions to finding an immersion of Mn in Rn?k are the Stiefel-Whitney characteristic classes of the stable normal bundle, coi(Mn), for i > k. In 1960, Massey (2) proved that ci(Mn) = 0 for i > n - a(n) and thereby gave the first evi- dence for this conjecture. The fact that Massey's result is best possible follows from the observation that, if we write

n = 2i + ... + 2r

where il < i2 < ... < ir [so that a(n) = r] and let- Mn be the product of projective spaces

Mn = RP2e' X ... X Rp2fr,

then a standard calculation shows that con-a(n)(Mn) #? 0. This in particular implies that k(n) ? n - a(n).

A well-known theorem of Hirsch (3) translates the problem of immersing manifolds into homotopy theoretic problems. This is the approach we shall take. More precisely, let BO(k) be the classifying space of k-dimensional vector bundles, BO = lim

k BO(k), and if Mn is an n-manifold, let vM: M - BO classify the stable normal bundle of M. That is, vm classifies the normal bundle of an embedding of M into a high-dimensional Euclidean space. Our goal is to prove the following.

THEOREM A. If Mn is a compact, C', n-dimensional mani- fold, then there exists a map

vI: M - BO[n - a(n)]

so that the composition Mn -- BO[n - a(n)] C BO is homotopic VM

to the stable normal bundle map vM. Indeed, given a map v14: Mn -* BO[n - a(n)] as in this theo-

rem, Hirsch's result guarantees the existence of an immersion f: Mn - R2n,-(n) having normal bundle classified by vi4. Thus, Theorem A implies the truth of the immersion conjecture.

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. ?1734 solely to indicate this fact.

A scheme for proving this theorem has been developed and partially carried out by Brown and Peterson (4-6). This work can be viewed as a completion of their program.

The outline of the proof of Theorem A will be spread over the following two sections. In Section 1 I recall some of Brown and Peterson's work, describe our main technical lemma, and show how it implies Theorem A. In Section 2 I give an outline of the constructions needed to prove this lemma.

Throughout the rest of this paper all (co)homology will be taken with Z2 coefficients, and by the term "n-manifold" I shall mean a compact, C', n-dimensional manifold. 1. Some preliminaries and a key lemma We begin with some recollections of Brown and Peterson's work toward the solution of the immersion problem.

As above, let VM: M" - BO classify the stable normal bundle of an n-manifold M, and let

Vm: H*(BO) -- H*(Mn)

be the induced homomorphism in (mod 2) cohomology. Let In C H*(BO) be the ideal

In = n ker MM

where the intersection is taken over all n-manifolds. In ref. 4, Brown and Peterson computed In explicitly.

Notice that, by the definition of In, for every n-manifold M' there exists a homomorphism

mi: H*(BO)/In -- H*(M)

making the following diagram of groups and homomorphisms commute:

H*(BO)

p P* 4 M

H*(BO)/In H*(M)

where p* is the projection map. In ref. 6, Brown and Peterson showed that this diagram can be realized by a diagram of spaces and continuous maps. That is, they proved the following.

THEOREM 1.1. For each n 2 0 there exists an n-dimensional C.W. complex BO/In together with a map p: BO/In l BO sat- isfying the following properties:

a. H*(BO/In) = H*(BO)/In and the induced homomorphism p*: H*(BO) -* H*(BO/In) is the natural projection.

b. If Mn is any n-manifold and VM : M -* BO classifies its stable normal bundle, then there exists a map 'M : M -* BO/In making the following diagram homotopy commute:

BO/In

PM / B M BO.

VM

3390

This content downloaded from 62.122.73.92 on Fri, 2 May 2014 09:51:09 AMAll use subject to JSTOR Terms and Conditions

Mathematics: Cohen Proc. Natl. Acad. Sci. USA 79 (1982) 3391

Our goal is to prove the following. THEOREM 1.2. For every n there exists a map pn: BO/In -

BO[n - a(n)] making thefollowing diagram homotopy commute:

BO[n - a(n)]

Pn I BO/In - BO.

p Notice that Theorems 1.1 and 1.2 taken together imply Theo-

remA and therefore the immersion conjecture. Indeed, wemay let vM: Mn -> BO[n - a(n)] be the composition

VM : Mn - BO/In -- BO[n - a(n)]. PM Pn

The main lemma needed to prove Theorem 1.2 is the following.

LEMMA 1.3. There exist spaces Xn together with maps fn:Xn - BO[n - a(n)] and gn: Xn -> BO/In satisfying the following properties:

a. The following diagram homotopy commutes:

Xn BO[n - a(n)]

BO/In - BO. p

b. If MO/In is the Thom spectrum of the stable vector bundle over BO/In classified by p, then 2-locally MO/In splits off of the Thom spectrum TXn. That is, after localizing at the prime 2, there exists a map

Oan: MO/In -> TXn

so that the composition Tgn o 0on: MO/In -> TXn -> MO/In is ho- motopic to the identity.

c. The following diagram homotopy commutes:

TXn - MO[n - a(n)] Tfn K

Tgn j T Tfn

MO/In - TXn. Oan

We now proceed to show why Theorem 1.2 (and therefore the immersion conjecture) follows from Lemma 1.3. We will indicate a proof of this lemma in the next section.

We shall show that Lemma 1.3 implies the existence of a map pn: BO/II -> BO[n - a(n)] so that the composition

Pn ? gn Xn -> BO /In -> BO[n - a(n)]

is homotopic tofn. To do this we consider the Moore-Postnikov tower for the fibration sequence

Vn-a(n) -> BO[n - a(n)] -> BO.

When n - a(n) is odd, it is well known that this fibration se- quence is simple. That is, 1l(BO) = Z2 acts trivially on wT*[Vn-a(n)]. This, however, is not true when n - a(n) is even. For this technical reason we shall restrict our attention in this note to the case n - a(n) is odd. Let

Dnr- - -/-Nl D - D D - Dn~~~I';

be the Postnikov tower for this fibration, where each 4i is a product of Eilenberg-MacLane spaces. Now, since for n > 3, n s 2[n - a(n)], it is well known that lTq[Vna,t(n)] is a finite abelian 2-group for q s n. By modifying this tower if necessary, we may therefore assume that through dimension n each K1 is a product of Eilenberg-MacLane spaces of type K(Z2, q). (See ref. 7 for a discussion of modified Moore-Postnikov towers.)

Suppose inductively that there exists a lifting pi-l: BO/In - Bi-, of p making the following diagram homotopy commute:

BO[n -a(n')]--- ---> BBi_1 -*--)Bo = --BO

Xn BO/II . n ~~gnn To complete the inductive step we need to construct a map

Pi: BO/In -> Bi, making the above diagram homotopy commute. This can be done if and only if the induced map of pairs

(BO/l, Xn) -0(Bi_, 1Bi) (pi 17 fn i) is null homotopic. Since the fibration Bi-, - Bi is principal with fiber 4i a product of K(Z2, q)s, it is sufficient to prove that the above map of pairs induces the zero map in mod 2 cohomology. This is equivalent, by the relative Thom isomorphism theorem, to showing that the induced map of pairs of Thom spectra in- duces the zero map in mod 2 cohomology. To do this we shall prove that, when localized at 2, the map of pairs

(Tpt-1l Tfn,t)

is null homotopic. That is, we shall prove that the map Tpi-l lifts (up to homotopy) to a map Ai: MO/In -> TBi with the prop- erty that the composition Ai o Tgn is homotopic to Tfn,i. For this we define Ai to be the composition

Ai = T(fn,i) ? on: MO/In - TXn - TBi.

One can easily check that properties b and c in the statement of Lemma 1.3 imply that Ai satisfies the required properties.

This completes the inductive.step in our proof that Lemma 13 implies Theorem 1.2 (and therefore the immersion conjecture). 2. Construction of the spaces Xn In this section we indicate how the spaces Xn and mapsfn: Xn 4 BO[n (a(n)] and gn Xn -> BO/In are constructed so as to satisfy Lemma 1.3. First, some preliminaries.

Let MO be the Thom spectrum of the universal stable vector bundle over BO and recall Thom's computation of the unorient- ed cobordism ring (8):

N = *r*(MO) = Z2[bk: k # 2' - 1]

where IbkI = k. If c = (il, , im) is a sequence of integers, none of which is of the form 2' - 1, let bo = ... b-. Let IwI =

q= qi. A strong version of Thom's theorem states that there is a splitting of spectra

MO - V 11'01K(Z)

where K(Z2) is the mod 2 Eilenberg-MacLane spectrum and the wedge is taken over all monomials b. E I*(MO). In cohomology we therefore have

H*(MO) = 3I IA

where A is the mod 2 Steenrod algebra.

This content downloaded from 62.122.73.92 on Fri, 2 May 2014 09:51:09 AMAll use subject to JSTOR Terms and Conditions

3392 Mathematics: Cohen Proc. Natl. Acad. Sci. USA 79 (1982)

In ref. 6, Brown and Peterson proved that

H*(MO/In) = ) ;11oIA/Jn,I1I IlIln

as A-modules, where Jk C A is the left ideal

Jk-= A{y(Sqt): 2i > k}

where X is the canonical antiautomorphism of A. Moreover, Brown and Peterson showed that there is a splitting of 2-local spectra

MO/I N V 1"IwB(n - Ill) n lwln

where B(k) is the spectrum first defined by Brown and Gitler (9) having the property that

H*[B(k)] = A/Jk

Remark: The indexing of the Brown-Gitler spectra here dif- fers from the original indexing used by. Brown and Gitler.

Now, by work of Brown and Peterson (10) and the author (11, 12), the Brown-Gitler spectra themselves can be realized as Thom spectra. For example, let F(R2,k) be the configuration space

F(R2,k) = {(x,,: Xi E=- R2, x j # X if i # j}

and let Fk = F(R2,k)/lk, where the symmetric group lk acts on F(R2,k) by permuting coordinates. Let yk be the vector bundle

F(R2,k) X -k(R I--> F(R2,k)/lk= Fk.

In ref. 10-it was proven that the Thom spectrum T(yk) is, 2-locally equivalent to B(k).

An easy calculation shows that Fk has homological dimension k - a(k), and therefore the classifying map yk: Fk 4 BO of this bundle lifts (up to homotopy) to a map yk: Fk 4 BO[k - a(k)].

With this information we can define Xn to be the disjoint union space

Xn= Mw x Fn-lx l lcl| '< n

where M. is a manifold of dimension lw representing the co- bordism class b< E 7r*(MO) = N *

Now, a theorem of Brown (13) says that every n-manifold is cobordant to one that. immerses in R2n-a(n). Therefore M< can be chosen to be a representative of b. that immerses in R2+l-alGl . Let

: M.-- BO(Iwl - aIwI)

classify the normal bundle of such an immersion. We define fn :Xn +- BO[n - a(n)] to be given by the compositions

MWX Fn,lw, --- BO(IwI .- aIwIl) I B X Yl(-n I)

x BO[n - lwl) - a(n - JJ1i)

-* BO{n - [aIwI + a(n - IwI)I} -- BO[n - a(n)] p

where p is the-Whitney sum pairing. The map gn X 4 BO/II is defined similarly by using the

liftings P.: M., - BO/II1 given in Theorem 1.1 and by studying the Postnikov tower for BO/II constructed by Brown and Pe- terson in ref. 6 to define liftings Yk Fk -> BO/lIk of yk, and pairings

lr,s:B X BOIlI -x BO/Ir+s

that lift the Whitney sum multiplication on BO. Finally, observe that the Thom spectrum

TXn = V T(VM,)A B(n - IwI)

and so we may define a splitting on: MO/In 4 TXn to be the composition

an: MO/ln = V S1lQ) A B(n - IwI)

VT Al 1 V T(JM,) A B(nI-wIWI) = TXn

where Ti: S 4l - T(VM.) is given by the Thom-Pontrjagin construction.

The fact that the quadruple (Xn fn gn, on) satisfies properties (a) and (b) in the statement of Lemma 1.3 is immediate, and we leave its verification to the reader. Proving that we cani make the choices of the manifolds Mo and maps v', y', Vk and A,s so that diagram (c) -in the- statement of Lemma 1.3 homotopy com- mutes involves a quite deep and technical study of the homo- topy theoretic properties of the spaces BO/In and will not be presented in this sketch. Having done this, the proof of Lemma 1.3 would then be complete.

The author is grateful to G. Brumfiel, S. Gitler, M. Mahowald, and R. J. Milgram for helpful conversations concerning this work. He owes a special debt of gratitude to Ed Brown and Frank Peterson for many illuminating conversations and for their diligence in reading through several roughly written manuscripts. This research was partially sup- ported by National Science Foundation Grant MCS 79-06085-AOI.

1. Whitney, H. (1944) Ann. Math. 45, 247-293. 2. Massey, W. S. (1960) Am. J. Math. 82, 92-102. 3. Hirsch, M. W. (1959) Trans. Am. Math. Soc. 93, 242-276. 4. Brown, E. H. & Peterson, F. P. (1964) Topology 3, 39-52. 5. Brown, E. H. & Peterson, F. P. (1977) Adv. Math. 24, 74-77. 6. Brown, E. H. & Peterson, F. P. (1979) Comment. Math. Helv. 54,

405-4306 7. Mahowald, M. (1964) Trans. Am. Math. Soc. 110, 315-349. 8. Thom, R. (1954) Comment. Math. Helv. 28, 17-86. 9. Brown, E. H., Jr., & Gitler, S. (1973) Topology 12, 283-295. 10. Brown, E. H. & Peterson, F. P. (1978) Trans. Am. Math. Soc.

243, 287-298. 11. Cohen, R. L. (1979) Invent. Math. 54, 53-67. 12. Cohen, R. L. (1980) in Proceedings of the Topology Symposium

at Siegen, 1979, Springer Lecture Notes (Springer, Heidelberg), Vol. 788, pp. 399-417.

13. Brown, R. L. (1971) Can. J. Math. 23, 1102-1115.

This content downloaded from 62.122.73.92 on Fri, 2 May 2014 09:51:09 AMAll use subject to JSTOR Terms and Conditions