Proc. NatL 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 proofofthe conjecture that A scheme for proving this theorem has been developed and every compact, differentiable, n-dimensional manifold immerses partially carried out by Brown and Peterson (4-6). This work in Euclidean space ofdimension 2n - a(n), where a(n) is the num- can be viewed as a completion of their program. ber of ones in the dyadic expansion of n. The outline ofthe proofofTheoremA will be spread over the following two sections. In Section I I recall some of Brown and An old problem in differential topology is that of finding the Peterson's work, describe our main technical lemma, and show smallest integer k(n) with the property that every compact, C', how it implies Theorem A. In Section 2 I give an outline of the n-dimensional manifold M" immerses in [n + k(n)]-dimensional constructions needed to prove this lemma. Euclidean space, Rn+k(n). A well-known conjecture is that k(n) Throughout the rest of this paper all (co)homology will be = n - a(n), where a(n) is the number of ones in the dyadic taken with Z2 coefficients, and by the term "n-manifold" I shall expansion of n. The classical immersion theorem of Whitney mean a compact, C , n-dimensional manifold. (1) states that k(n) < n - 1 and, in particular, implies that this and a key lemma conjecture is true when n is a power of 2. The purpose of this 1. Some preliminaries note is to announce and to outline a proofofthis conjecture for We begin with some recollections ofBrown and Peterson's work all integers n. Details will appear elsewhere. toward the solution- of the immersion problem. The primary obstructions to finding an immersion of Mn in As above, let vM: M' + BO classify the stable normal bundle Rn+k are the Stiefel-Whitney characteristic classes ofthe stable of an n-manifold M, and let normal bundle, w1(MW), for i > k. In 1960, Massey (2) proved v~m: H*(BO) H*(Mn) that Wi(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 be the induced homomorphism in (mod 2) cohomology. Let In possible follows from the observation that, ifwe write C H*(BO) be the ideal In = n ker v~m where il < i2< .. < i, [so that a(n) = r] andlet M" be the where the intersection is taken over all n-manifolds. In ref. 4, product of projective spaces Brown and' Peterson computed In explicitly. Notice that, by the definition ofIn, for every n-manifold M" Mn = RP2 X ... XIIP2F, there exists a homomorphism then a standard calculation shows that wn-.(.)(Mn) # 0. This in vm: H*(BO)/In -- H*(M) particular implies that k(n) 2 n - a(n). A well-known theorem of Hirsch (3) translates the problem making the following diagram of groups and homomorphisms ofimmersing manifolds into homotopy theoretic problems. This commute: is the approach we shall take. More precisely, let BO(k) be the H*(BO) classifying space of k-dimensional vector bundles, BO = lim k BO(k), and ifMn is an nmanifold, let vM: M + BO classify the H*(BO)/In H*(M) stable normal bundle of M. That is, vM classifies the normal bundle ofan embeddingofM into a high-dimensional Euclidean where p* is the projection map. In ref. 6, Brown and Peterson space. Our goal is to prove the following. showed that this diagram can be realized by a diagram ofspaces THEOREM A. If Mn is a compact, C", n-dimensional mani- and continuous maps. That is, they proved the following. fold, then there exists a map THEOREM 1. 1. For each nO there exists an n-dimensional together with a map p: BO/In + BO sat- -- C.W. complex BO/In VM Mn BO[n - a(n)] isfying the following properties: so that the composition Mn -r BO[n - a(n)] C BO is homotopic a. H*(BO/In) = H*(BO)/In and the induced homomorphism p : H*(BO) + H*(BO/In) is the natural projection. to the stable normal bundle map vM. b. IfMn is any n-manifold and vM : M M BO classifies its stable Indeed, given a map vi : Mn + BO[n - a(n)] as in this theo- normal bundle, then there exists a map PM : M + BO/In making rem, Hirsch's result guarantees the existence of an immersion thefollowing diagram homotopy commute: f: M" R2,-"(n) having normal bundle classified by v'M. Thus, BO/In Theorem A implies the truth of the immersion conjecture. 0/in PM P The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- M - BO. ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. VM 3390 Downloaded by guest on September 28, 2021 Mathematics: Cohen Proc. Natl. Acad. Sci. USA 79 (1982) 3391

Our goal is to prove the following. be the Postnikov tower for this fibration, where each K1 is a THEOREM 1.2. For every n there exists a map pn : BO/In + product of Eilenberg-MacLane spaces. Now, since for n > 3, BO[n - a(n)] making thefollowing diagram homotopy commute: n < 2[n - a(n)], it is well known that ;T[Vn-,n)] is a finite abelian 2-group for q s n. By modifying this tower ifnecessary, BO[n - a(n)] we may therefore assume that through dimension n each K1 is Pn a product of Eilenberg-MacLane spaces of type K(Z2, q). (See ref. 7 for a discussion of modified Moore-Postnikov towers.) BO/In -. BO. Suppose inductively that there exists a lifting Pi- : BOIL P Bi,1 of p making the following diagram homotopy commute: Notice that Theorems 1.1 and 1.2 taken together imply Theo- BO[n a(n)] * B I B.+B** - = BO remA and therefore the immersion conjecture. Indeed, we may Bo let v$,: Me+ BO[n - a(n)] be the composition tfn M"- BO/I,,-* ,/ t v: BO[n a(n)]. Xn pal p. 9BOln. The main lemma needed to prove Theorem 1.2 is the To complete the inductive step we need to construct a map following. Pi: BO/ln . Bi, making the above diagram homotopy commute. LEMMA 1.3. There exist spaces Xn together with maps fn: Xn This can be done if and only if the induced map of pairs BO[n - a(n)] and gn : Xn + BO/In satisfying the following properties: (BO/In; Xn) X(Bi-,, Bd) a. Thefollowing diagram homotopy commutes: (P1-141.3)

0 BO[n - a(n)] is null homotopic. Since the fibration Bi-I + Bi is principal with fiber 1i a product of K(Z2, q)s, it is sufficient to prove that the above map ofpairs induces the zero map in mod 2 cohomology. gnj This is equivalent, by the relative Thom isomorphism theorem, BO/In BO. to showing that the induced map of pairs of-Thom spectra in- P duces the zero map in mod 2 cohomology. To do this we shall b. IfMO/In is the Thom spectrum ofthe stable vector bundle prove that, when localized at 2, the map of pairs over BO/In classifted by p, then 2-locally MO/In splits off of (MO/lnv TXJ) (T T (TBi-1, TBi) the Thom spectrum TX,,. That is, after localizing at the prime (TA1, Tf,.,) 2, there exists a map is null homotopic. That is, we shall prove that the map TA1-, arn,: MO/In TXn lifts (up to homotopy) to a map Ai: MO/In . TBi with the prop- erty that the composition A, o Tgn is homotopic to Tfn,i. For this so that the composition Tg,, o an: MO/In TX, MO/In is ho- we define A{ to be the composition motopic to the identity. c. The following diagram homotopy commutes: Ai = T(fn,d Cran: MO/In + TXn + TBi TXn ' MO[n - a(n)] One can easily check that properties b and c in the statement ofLemma 1.3 imply that Ai satisfies the required properties. Tgn ITfn This completes the inductive step in our proof that Lemma 13 implies Theorem 1.2 (and therefore the immersion conjecture). MO/In - TXn. an 2. Construction of the spaces X.

We now proceed to show why Theorem 1.2 (and therefore the In this section we indicate how the spaces X. and maps n: Xn immersion conjecture) follows from Lemma 1.3. We will indicate + BO[n - a(n)] and gn, Xn + BO/ln are constructed so as to a proof of this lemma in the next section. satisfy Lemma 1.3. First, some preliminaries. We shall show that Lemma 1.3 implies the existence of a map Let MO be the Thom spectrum ofthe universal stable vector pn : BO/In + BO[n - a(n)] so that the composition bundle over BO and recall Thom's computation ofthe unorient- ed cobordism ring (8): pn o gn: Xn x BO /In + BO[n - a(n)] '7* = 'TT*(MO) = Z2[bk: k # 2' - 1] is homotopic tof,. To do this we consider the Moore-Postnikov tower for the fibration sequence where IbkI = k. Ifw = (il, it") is a sequence ofintegers, none of which is of the form 2' - 1, let bit = bil b'". Let = + BO[n - a(n)] BO. IwI Vn-On) £qi,9i. A strong version ofThom's theorem states that there When n - a(n) is odd, it is well known that this fibration se- is a splitting of spectra quence is simple. That is, 7r1(BO) = Z2 acts trivially on MO = V £"'K(Z2) 1r*[Vn -

T fl14i where A is the mod 2 Steenrod algebra.

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In ref. 6, Brown and Peterson proved that -- BO{n - [ajca + ca(n - IwI)J} -- BO[n - Up ca(n)] H*(MO/I1) = l X10A/1J_1i where p is the Whitney sum pairing. The map g,,: Xn+ BOIl, is defined similarly by using the as A-modules, where Ik C A is the left ideal liftings z',: M. e BOII given in Theorem 1.1 and by studying Ik = A{X(Sq') : 2i > k} the Postnikov tower for BOil, constructed by Brown and Pe- terson in ref. 6 to define liftings ok : Fk e BO/Ik of )kk and where X is the canonical antiautomorphism. of A. Moreover, pairings Brown and Peterson showed that there is a splitting of 2-local spectra B011r x B1I., -+BOXl+* MO/In3 V £10B(n - kol) that lift the Whitney sum multiplication on BO. Finally, observe/l~r,s that the Thom spectrum where B(k) is the spectrum first defined by Brown and Gitler TX, = V T(vm) A B(n - lol) (9) having the property that Wsn H*[B(k)] = Allk and so we may define a splitting orn: MO/I,,+ TXn to be the composition Remark: The indexing ofthe Brown-Gider spectra here dif- fers from the original indexing used by Brown and Gider. a'n: MO/In = V SI" A B(n - 11l) Now, work of Brown and Peterson and the author - by (10) .-T A = (11, 12), the Brown-Gitler spectra themselves can be realized A 1 VT(VM) B(n lol) TXn as Thom spectra. For example, let F(R2,k) be the configuration VrA 1 where 7 : S ek+ T(vM.) is given by the Thom-Pontragin space construction. F(R2,k) = {(xl,.. ,x) : xi E R2, xi #-xj if i #f} The fact that the quadruple (Xn,,f,, gn, a.) satisfies properties (a) and (b) in the statement ofLemma 1.3 is immediate, and we and let Fk = F(R2,k)/lxk, where the symmetric group £k acts leave its verification to the reader. Proving that we can make on F(R2,k) by permuting coordinates. Let be the vector )k the choices ofthe manifolds M. and maps P', iP, k and ut, so bundle that diagram (c) in the statement ofLemna 1.3 homotopy com-

F(R2,k) X 1k(R)_ _ = Fk. mutes involves a quite deep and technical study of the homo- F(R2,k)&k topy theoretic properties of the spaces BOIl, and will not be In ref. 10 itwas proven that the Thom spectrum T(yo) is 2-locally presented in this sketch. Having done this, the proofofLemma equivalent to B(k). 1.3 would then be complete. An easy calculation shows that Fk has homological dimension k - a(k), and therefore the classifying map Vk : Fk+ BO ofthis The author is grateful to C. Brumfiel, S. Gitler, M. Mahowald, and bundle lifts (up to homotopy) to a map Ak: Fk4 BO[k - a(k)]. R. J. Milgram for helpful conversations concerning this work. He owes With this information we can define Xn to be the disjoint a special debt of gratitude to Ed Brown and Frank Peterson for many union space illuminating conversations and for their diligence in reading through several roughly written manuscripts. This research was partially sup- Xn = HLL M, x F" ported by National Science Foundation Grant MCS 79-06085-AO1. J4 -< n 1. Whitney, H. (1944) Ann. Math. 45, 247-293. where M, is a manifold of dimension 1l representing the co- 2. Massey, W. S. (1960) Am. J. Math. 82, 92-102. bordism class b, E 7r*(MO) = 11*. 3. Hirsch, M. W. (1959) Trans. Am. Math. Soa 93, 242-276. Now, a theorem of Brown (13) says that every n-manifold 4. Brown, E. H. & Peterson, F. P. (1964) Topology 3, 39-52. is cobordant to one that immerses in R2n-,n). Therefore M< 5. Brown, E. H. & Peterson, F. P. (1977) Adv. Math. 24, 74-77. can be chosen to be a representative of b, that immerses in 6. Brown, E. H. & Peterson, F. P. (1979) Comment. Math. Helv. 54, Rti-&W. Let 405-430. 7. Mahowald, M. (1964) Trans. Am. Math. Soc. 110, 315-349. P,: -, BO(IwI - 8. Thom, R. (1954) Comment. Math. Helv. 28, 17-86. M., alwI) 9. Brown, E. H., Jr., & Gider, S. (1973) Topology 12, 283-295. classify the normal bundle of such an immersion. We define 10. Brown, E. H. & Peterson, F. P. (1978) Trans. Am. Math. Soc. * - to be the 243, 287-298. f: Xn BO[n a(n)] given by compositions 11. Cohen, R. L. (1979) Invent. Math. 54, 53-67. BO(cI - alol) 12. Cohen, R. L. (1980) in Proceedings ofthe Topology Symposium MOXF3.1*A P. x , Bn-1- - at Siegen, 1979, Springer Lecture Notes (Springer, Heidelberg), Vol. 788, pp. 399-417. x BO[n - Jw - a(n - Jw)] 13. Brown, R. L. (1971) Can. J. Math. 23, 1102-1115. Downloaded by guest on September 28, 2021