The relationship between framed bordism and sew-framed bordism Pyotr M. Ahmet ev and Peter J. Eccles Abstract A sew-framing of an immersion is an isomorphism between the normal bundle of the immersion and the Whitney sum of copies of some line bundle. This reduces to a framing if the line bundle is oriented. In this note we investigate the relationship between the bordism groups of framed immersions in Euclidean space and the bordism groups of sew-framed immersions. We show that in certain codimensions (i) framed immersions in Euclidean space are sew-framed boundaries and (ii) framed immersions in Euclidean space are framed bordant to the double cover of a sew-framed immersions. These results are simple consequences of the Kahn Priddy Theorem and James Periodicity. 1 Framed immersions and sew-framed immersions Recall that a framing of an immersion M n R n+ of a smooth n-dimensional manifold M n into Euclidean (n+)-space is a trivialization of its normal bundle. A manifold which has a framed immersion into some Euclidean space is stably parallelizable. A framing may be thought of as an isomorphism between the normal bundle of the immersion and the Whitney sum of copies of the (unique) oriented line bundle over M. So, by analogy, we may define a sew-framing (or projective framing) of an immersion to be an isomorphism between the normal bundle of the immersion and the Whitney sum of copies of some (not necessarily oriented) line bundle over M. It should be noted that given a self-transverse framed immersion f : M n R n+ then the double point manifold, a smooth manifold of dimension n, has an immersion 2 (f) n R n and the framing of f induces a natural sew-framing on this immersion (see Proposition 5). Hence sew-framed immersions arise naturally in the study of framed immersions. Clearly, a framing is a particular sort of sew-framing in which the line bundle used is oriented (and so trivial). The main purpose of this note is to observe that given n then, for infinitely many values of, every framed immersion M n R n+ = R n+ {0} R n+ [0, 1) of a manifold without boundary is the boundary of a sew-framed immersion W n+1 R n+ [0, 1). More generally, we can use a sew-framed immersion F : W n+1 R n+ [0, 1] of a manifold with boundary W n+1 = M1 n M2 n such that F 1 (R n+ {i}) = Mi n to define The first author has been supported in part by RFBR grant 05-01-00993. Both authors have been supported in part through a grant from the Royal Society of London. 1
a sew-framed bordism between the two sew-framed immersions f i = F M n i : M n i R n+ {i} in the usual way. The sew-framed bordism group of sew-framed immersions of n-manifolds in codimension will denoted by Imm sfr (n, ) and the framed bordism group of framed immersions of n-manifold in codimension will denoted by Imm fr (n, ). Treating a framing as a sew-framing leads to the forgetful homomorphism σ : Imm fr (n, ) Imm sfr (n, ). To state the theorem precisely it is necessary to recall the definition of some wellnown numbers. For a positive integer a, let φ(a) be the number of positive integers r such that r < a and r 0, 1, 2 or 4 modulo 8. Put b(a) = 2 φ(a). Theorem 1. For n 1, let M n be a stably parallelizable compact manifold of dimension n without boundary. Then, for 2m n, each framed immersion M n R n+b(2m+1) 1 = R n+b(2m+1) 1 {0} is the boundary of a non-oriented framed immersion W n+1 R n+b(2m+1) 1 [0, 1) or, equivalently, the forgetful homomorphism is trivial. σ : Imm fr (n, b(2m + 1) 1) Imm sfr (n, b(2m + 1) 1) On the other hand, it is also possible to associate a framed immersion to each sewframed immersion in the following manner. Suppose that (f : N n R n+, φ) is a sewframed immersion with φ: ν f = κ, where νf is the normal bundle of the immersion and κ is a line bundle over N n. Let p: Ñ n N be the double covering of N n given by the sphere bundle of κ. Then the sew-framing φ induces a framing φ: ν f p = p (κ) = ε of the immersion f p: Ñ n N n R n+. It is routine to chec that this construction respects bordisms so that a transfer homomorphism may be defined by τ : Imm sfr (n, ) Imm fr (n, ) τ[f : N n R n+, φ] = [f p: Ñ n R n+, φ]. Theorem 2. For n 1, let M n be a stably parallelizable compact manifold of dimension n without boundary. Then, for 2m n, each framed immersion M n R n+b(2m+1) is framed bordant to the double cover of a sew-framed immersion or, equivalently, the transfer homomorphism τ : Imm sfr( n, b(2m + 1) ) Imm fr( n, b(2m + 1) ) is an epimorphism. 2 Bordism classes of immersions and the stable homotopy groups of Thom spaces Theorems 1 and 2 are most readily obtained by interpreting the homomorphisms between the bordism groups as homomorphism between the stable homotopy groups of the appropriate Thom complexes as described below. 2
The basis of the application of stable homotopy theory to the study of immersions is the observation that the bordism group of immersions of (smooth compact closed) n-manifolds in R n+ with a G()-structure (where G() is a subgroup of the orthogonal group O()) is isomorphic to the stable homotopy group πn+ S MG(). Here MG() is the Thom complex of the universal G()-bundle i γ where i: BG() BO() is the inclusion map and γ is the universal -plane bundle over BO(). This was first observed, in the case G() = O() by Wells ([7]) using the Smale-Hirsch Theorem, which relates immersions of manifolds to monomorphisms of their tangent bundles, and the Pontrjagin- Thom construction, which gives an isomorphism between bordism groups of embeddings and homotopy groups of Thom complexes. The result for a general subgroup of O() follows by the same method. The simplest case is that of framed immersions for which G() is the trivial group and so the universal G()-bundle can be taen to be the trivial -dimensional bundle over a point with Thom complex S. This gives the well-nown result that Imm fr (n, ) is isomorphic to the stable stem πn+ S S = π S n. For sew-framings the group G() is O(1) embedded diagonally in O(), for a sewframing of an immersion M n R n+ corresponds to a factorization of the classifying map of the normal bundle M n BO() through the diagonal map d: BO(1) BO(1) BO(). In this case the universal bundle d γ is the Whitney sum γ 1 of copies of the universal line bundle. If γ 1 is taen to be the canonical line bundle over RP then the Thom complex of γ 1 is homeomorphic to the truncated real projective space P = RP /RP 1 ([3]). Thus, by the Pontrjagin-Thom-Wells Theorem, the bordism group Imm sfr (n, ) is isomorphic to the stable homotopy group πn+ S P. Lemma 3. The forgetful homomorphism σ : Imm fr (n, ) Imm sfr (n, ) corresponds under the Pontrjagin-Thom-Wells isomorphism to the map induced by the inclusion of the bottom cell. π S n+s π S n+p Proof. The map of universal bundles corresponding to viewing a framing as a sewframing can be described as γ0 1 γ 1 where γ0 1 is the canonical line bundle over RP 0 RP. This induces a map of Thom complexes S = P P which is the inclusion of the bottom cell, as required. Lemma 4. The transfer homomorphism τ : Imm sfr (n, ) Imm fr (n, ) corresponds under the Pontrjagin-Thom-Wells ismomophism to the map π S n+p π S n+s induced by the co-attaching map of the bottom cell of P 1, i.e. the map q in the cofibre sequence S 1 i P 1 j P arising from the inclusion of the bottom cell in P 1. 3 q S. (1)
Proof. Given a sew-framed immersion (f : N n R n+, φ) with φ: ν f = κ, where νf is the normal bundle of the immersion and κ is a line bundle over N n, we can define an embedding, g : Ñ n E(κ), of the double cover of N n into the total space of the bundle κ by g(a, x) = (a, 0,..., 0, x), for a N n and x a unit vector in the fibre of κ over a. Composing this with the bundle map E(κ) N n gives the double cover map p: Ñ n N n. Let F : E(ν f ) R n+ be a tubular neighbourhood of the immersion f. Then it is clear that the composition f 1 : Ñ n g E(κ) φ 1 E(ν f ) F R n+ is regularly homotopic to the immersion f : Ñ n R n+. The embedding g has a tubular neighbourhood G: Ñ n R E(κ) given by G ( (a, x), (t 1, t 2,..., t ) ) = (a, t 1 x, t 2 x,..., t 1 x, exp(t )x). Then Ñ n R G E(κ) φ 1 E(ν f ) F R n+ gives a tubular neighbourhood of f 1. The universal case of the embedding G is the map S R E(γ 1 ) given by (x, t 1, t 2,..., t ) ([x], t 1 x, t 2 x,..., t 1 x, exp(t )x) where γ 1 is the canonical line bundle over the Grassmanian G 1 (R ) = RP which can be taen as the universal line bundle over BO(1). It follows that the transfer homomorphism corresponds to the homomorphism in stable homotopy induced by the map T (γ 1 ) = P S obtained by applying the Pontrjagin-Thom construction to this universal map S R E(γ 1 ). It is sufficient to consider the embedding S N R E(γN 1 ) where γ1 N is the canonical line bundle over RP N. The total space E(γN 1 ) can be identified with the space RP N+ RP 1 via ([x], t 1 x, t 2 x,..., t x) [t 1, t 2,..., t, x] for x S N. Under this identification the embedding S N R E(γN 1 ) = RPN+ RP 1 is given by (x, t 1, t 2,... t ) [t 1, t 2,..., t 1, exp(t ), x]. Applying the Pontrjagin-Thom construction to this embedding gives a pointed map T (γn 1 ) = (RP+N RP 1 ) = P +N (R ) = S given by [s 1, s 2,..., s 1, s, x] { (s1, s 2,..., s 1, ln(s )) if s > 0 if s = 0, where x S N and s 0. It is now sufficient to recognize this map as the map q in the cofibre sequence S 1 i P +N 1 j P +N q S. To see this observe that the map i is the one point compactification of the map R 1 P +N P 2 given by (t 1, t 2,..., t 1 ) [t 1, t 2,..., t 1, 1, 0]. Hence the map j is the pointed map (P +N P 2 ) (P +N P 1 ) given by [x] [x] if [x] P +N P 1 and [x] otherwise. It follows that the above formula gives the map q. Finally in the section we give the result about the double manifolds of framed immersions referred to earlier. Proposition 5. Given a self-transverse framed immersion f : M n R n+ then the natural immersion of the ((n )-dimensional) double point manifold 2 (f) R n+ gives rise to an immersion 2 (f) R n and the framing of f induces a natural sewframing on this immersion. 4
Proof. Let 2 (f) = { (x 1, x 2 ) M M f(x 1 ) = f(x 2 ), x 1 x 2 }. By the self-transversality of f this is a submanifold of M M of dimension n. The group of order 2 acts freely on 2 (f) by interchanging coordinates and the double point manifold 2 (f) is the quotient space. Clearly f induces an immersion δ 2 (f): (f) R n+. The framing on the immersion f gives a basis for each fibre of the normal bundle of f. Each fibre of the normal space of δ 2 (f) has a natural decomposition as the direct sum of two unordered normal fibres of f each of which has a basis. Thus in this case the group of the bundle is the wreath product 1 Z 2 O(2) where 1 is the trivial subgroup of O() (see for example [2], 2.1). This subgroup is conjugate to the subgroup O(1) embedded through the diagonal map as O(1) d O() O(2). Thus the normal bundle of δ 2 (f) has a normal -frame field so that by the Smale-Hirsch Theorem there is an immersion of 2 (f) in R n. The structure group for this immersion is O(1) embedded diagonally in O() and so the immersion has a sew-framing as required. 3 James periodicity and the Kahn-Priddy Theorem For 0 a b, the truncated real projective space RP b /RP a 1 will be denoted by P b a. In the case of a = 0, RP 1 is interpreted as the empty set so that P b 0 is RP b with a disjoint base point. The theorem we need is the following. Theorem 6 (James Periodicity Theorem). For each positive integer a and each non-negative integer there is a homeomorphism Σ b(a) P +a 1 = P +a 1 S b(a) P +b(a)+a 1 +b(a). Proof. This result is implicit in [6]. It may be obtained quite explicitly as follows. The well-nown Hurwitz-Radon-Ecmann theorem in linear algebra (see [5]) tells us that there is a non-singular bilinear map This induces a homeomorphism f : R a R b(a) R b(a). (RP +a 1 RP 1 ) R b(a) RP +b(a)+a 1 RP +b(a) 1. by ([x, y], z) [x, y, f(x, z)] for x R a {0}, y R, z R b(a). On taing one point compactifications this gives the homeomorphism in the theorem. Adams version of the Kahn-Priddy Theorem ([1], Formulation 2.3) gives the following result. Theorem 7 (Kahn-Priddy Theorem). A stable map RP 2m S 0 which induces an isomorphism π S 1 RP 2m π S 1 ( = Z/2) induces an epimorphism of 2-primary stable homotopy groups 2 π S nrp 2m 2 π S n for 1 n 2m. 5
These two results now enable us to prove the main results of this paper. Proof of Theorem 2. For m 1, let q : P b(2m+1) S b(2m+1) be the co-attaching map of the bottom cell of Pb(2m+1) 1, i.e. the map q in the cofibre sequence (1) of Lemma 4 in the case = b(2m + 1). By Lemma 4, the theorem is equivalent to the statement that q induces an epimorphism of stable homotopy groups ( ) for 1 n 2m. π S n+b(2m+1) i Firstly, observe that S b(2m+1) Pb(2m+1) S b(2m+1) is a map of degree 2 since b(2m + 1) is even. This means that q i induces isomorphisms of p-primary stable homotopy groups p πb(2m+1)+n S ( ) for p 2 and n 1 and so q induces epimorphisms of such groups. Now, by the James Periodicity Theorem, P b(2m+1)+2m b(2m+1) is homeomorphic to Σ b(2m+1) P0 2m and so the restriction of q determines a stable map q : P b(2m+1)+2m b(2m+1) q 0 : P 2m 0 S 0. q S b(2m+1) Composing this with the inclusion map RP 2m P 2m 0 gives a stable map q 1 : RP 2m S 0. The cofibre of q is ΣP b(2m+1)+2m b(2m+1) 1. Since b(2m + 1) is a multiple of 4, π S b(2m+1)p b(2m+1)+2m b(2m+1) 1 From the cofibre sequence = π S b(2m+1) P b(2m+1)+1 b(2m+1) 1 = π S 4 P 5 3 (by James Periodicity). S 3 2ι S 3 P3 4 S 4 2ι S 4, π4 S P3 4 = Z/2 generated by the image of the suspension of the Hopf map η 3 π4 S S 3. This maps to zero under the epimorphism π4 S P3 4 π4 S P3 5 (since Sq 2 : H 3 (P3 5 ; Z/2) H 5 (P3 5 ; Z/2) is non-zero implying that the 5-cell in P3 5 is attached by the generator of π4 S P3 4 ) and so πb(2m+1)+1 S b(2m+1)+2m ΣP b(2m+1) 1 = π S b(2m+1) P b(2m+1)+2m b(2m+1) 1 = π S 4 P3 5 = 0. This shows that q induces an epimorphism of πb(2m+1)+1 S ( ) or equivalently that q 0 induces an epimorphism of π1 S ( ). It follows that q 1 induces an epimorphism and so an isomorphism of π1 S, since, as above, the stable map S 0 i P0 2m q 0 S 0 is a map of degree 2 and so induces the zero homomorphism on π1 S = Z/2. It now follows from the Kahn-Priddy Theorem that the stable map q 1 induces an epimorphism of 2 πn( S ) for 1 n 2m. The same must therefore be true of the stable map q 0 and so the map q induces an epimorphism of 2 πn+b(2m+1) S ( ) for 1 n 2m. Combining this with the earlier observation about mod p stable homotopy we obtain that the map q induces an epimorphism of πn+b(2m+1) S ( ) for 1 n 2m as required. Proof of Theorem 1. For 1 n 2m, continuing the long exact sequence in stable homotopy coming from the cofibre sequence (1) in Lemma 4 with = b(2m + 1), the 6
above result that q induces an epimorphism of stable homotopy groups is equivalent to the vanishing of the homomorphisms (Σi) : π S n+b(2m+1)s b(2m+1) π S n+b(2m+1)σp b(2m+1)+2m b(2m+1) 1. By Lemma 3, this observation is equivalent to Theorem 1. Notice that the above arguments show that for an odd prime p there are p-local versions of Theorems 1 and 2 in which b(2m + 1) is replaced by any even positive integer. 4 Examples Proposition 8. The bordism group Imm sfr (2, 3) is cyclic of order 2. Proof. This is clear since, by the Pontrjagin-Thom-Wells Theorem, this bordism group is isomorphic to the stable homotopy group π5 S P3. An easy calculation shows that this is cyclic of order 2 generated by an element η4 which maps to the element η 4 π5 S P4 given by composing the double suspension of the Hopf map η π 3 S 2 = Z with the inclusion of the bottom cell. A generator for Imm sfr (2, 3) = Z/2 may be described as follows. Let f : K 2 R 3 be the immersion of the Klein bottle obtained by constructing a cylinder on a figure eight and identifying the ends after rotating through an angle π. This represents an element of order 4 in the group Imm sfr (2, 1) = Z/8 ([4]). Let κ be the normal line bundle of this immersion of the Klein bottle. This bundle is the pull bac of the canonical line bundle over the circle under the standard bundle map K 2 S 1 and so the bundle 2κ over K 2 is trivial. We fix an arbitrary bundle isomorphism 2κ = 2ɛ where ɛ is the trivial line bundle over K 2. Let g : K 2 R 5 be the composition of f with the standard inclusion R 3 R 5 (this is in fact regularly homotopic to an embedding as we are in the stable range). Then there is an isomorphism φ: ν g = κ 2ɛ = 3κ giving a sew-framing on g. Proposition 9. This sew-framed immersion (g : K 2 R 5, φ) represents a generator of Imm sfr (2, 3) = Z/2. In order to prove this result we describe a geometrical construction corresponding to the homomorphism of stable homotopy induced by the middle map of the cofibre sequence S P P +1 S+1. Given an immersion f : M n R n+ with a sew-framing ν f = κ we can construct a submanifold L n 1 of M n representing the Poincaré dual of w 1 (κ) H 1 (M; Z/2) by taing the transverse self-intersection of the zero section of the bundle κ. The normal bundle of the inclusion i: L M is the pull-bac of the bundle κ and so the immersion f i: L M R n+ has a sew-framing ν f i = ( + 1)i κ. This construction is clearly bordism invariant and gives a homomorphism ω : Imm sfr (n, ) Imm sfr (n 1, + 1). Lemma 10. This homomorphism ω : Imm sfr (n, ) Imm sfr (n 1, + 1) corresponds under the Ponrtjagin-Thom-Wells isomorphism to the map induced by the collapse of the bottom cell. π S n+p π S n+p +1 7
Proof. Let κ: M RP N be the map representing the line bundle κ where N is large. Then, if choose κ transverse to RP N 1 R N, the submanifold L is given by κ 1 (RP N 1 ). Since γn 1 may be taen to be the normal bundle of RPN in RP N+ the sew-structure on M gives the following diagram. L i M E(ν f ) RP N 1 RP N RP N+ This induces the following diagram of Thom complexes. T (ν f ) T (ν f i ) P N+ P N+ +1 This gives the result by applying the Pontrjagin-Thom construction. Proof of Proposition 9. In the example, since κ is the normal bundle of the immersion f : K 2 R 3, the submanifold L is a circle S 1 embedded in K as one figure eight section. The restriction of κ to this circle is trivial and so the induced sew-framing of the immersion S 1 K R 3 is a framing, clearly regularly homotopic to the standard Hopf framing representing a generator of π 3 S 2. It follows that the element of π S 5 P 3 represented by g : K R 5 maps to η 4 π S 5 P 4 as required.. It should be observed from Theorem 2 that the transfer homomorphism Imm sfr (2, 3) Imm fr (2, 3) is an isomorphism (since Imm fr (2, 3) = π 5 S 3 = Z/2). Applying the double cover construction to the above immersion g : K 2 R 5 of the Klein bottle gives a framed immersion of the torus which represents the generator of Imm fr (2, 3). An exact sequence Finally, we observe that the following result. Theorem 11. For all n and 0 the following sequence is exact: Imm fr (n, ) σ Imm sfr (n, ) ω Imm sfr (n 1, + 1) τ Imm fr (n 1, + 1) σ Imm sfr (n 1, + 1). Proof. This follows from the long exact sequence in stable homotopy arising from the cofibre sequence S P P +1 S+1 using Lemma 3, Lemma 4 and Lemma 10. References [1] J.F. Adams, The Kahn Priddy theorem, Proc. Cambridge Phil. Soc. 73 (1973) 43 55. [2] M.A. Asadi-Gomanhhaneh and P.J. Eccles, Determining the characateristic numbers of self-intersection manifolds, J. London Math. Soc. (2) 62 (2000), 278 290. 8
[3] M.F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961) 291 310. [4] P.J. Eccles, Multiple points of codimension one immersions, Lecture Notes in Mathematics, 788 (Springer, 1980), 23 38. [5] B. Ecmann, Gruppentheoretischer Beweis des Satzes von Hurwitz Radon über die Komposition quadratischer Formen, Comment. Math. Helv. 15 (1942) 358 366. [6] I.M. James, Spaces associated with Stiefel manifolds, Proc. London Math. Soc. (3) 9 (1959) 115 140. [7] R. Wells, Cobordism groups of immersions, Topology 5 (1966) 281 294. 4 April 2005 P.M. Ahmet ev Stelov Mathematical Institute RAS Moscow Russia pmahmet@mi.ras.ru P.J. Eccles School of Mathematics University of Manchester Manchester Great Britain pjeccles@manchester.ac.u 9