Cohomotopy sets of -manifolds
Elementary geometric arguments are used to compute the group of homotopy classes of maps from a -manifold to the -sphere, and to enumerate the homotopy classes of maps from to the -sphere. The former completes a project initiated by Steenrod in the 1940’s, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented -manifolds and provide a framework for the study of Morse -functions on -manifolds, a subject that has garnered considerable recent attention.
Fix a smooth, closed, connected, oriented -manifold . For each positive integer , consider the cohomotopy set of free homotopy classes of maps . The Pontryagin-Thom construction gives a refinement of Poincaré duality
where is the set of closed -dimensional submanifolds of with a framing on their normal bundle, up to normally framed bordism in . The maps pull back the cohomology fundamental class of whereas the “forgetful” maps use the normal framing to orient the submanifold and then push forward its homology fundamental class.
The purpose of this note is to compute for all . In fact the only “interesting” cases are (framed links in ) and (framed surfaces in ), computed in Theorems 1 and 2 below; in all other cases is an isomorphism, by classical arguments recalled in Section 0. The set also has a group structure, inherited from the target group , and has an action of this group, induced by the action of on . We will compute both structures geometrically, the group structure (corresponding to disjoint union in ) and the action (corresponding to framed link translation in , defined in Section 2).
Recall that the -manifold is odd if it contains at least one closed oriented surface of odd self-intersection, and otherwise it is even. It is spin if every surface in , orientable or not, has even self-intersection, or equivalently the second Stiefel-Whitney class .
If is odd, then the forgetful map is an isomorphism. If is even, then there is an extension of abelian groups
classified by the unique element of that maps to in the universal coefficient sequence In particular (e) splits, or equivalently if and only if is spin.
Remark. As stated, Theorem 1 is also true for non-compact -manifolds . The proof given in Section 1 easily adapts to this case. In fact, as noted by the referee, the smoothness assumption can be dropped as well since topological -manifolds can be smoothed away from a point (in the closed case).
The set was first investigated by Steenrod , building on work of Pontryagin  and Eilenberg , in the more general context of the study of maps from an arbitrary finite complex to a sphere of codimension one. Steenrod succeeded in enumerating the elements of , but did not address the question of its group structure. Recently, this group structure was analyzed by Larry Taylor  from a homotopy theoretic point of view, building on work of Larmore and Thomas . We discuss in Remark 1.4 below how Taylor’s Theorem 6.2 can be interpreted in a way that it implies our Theorem 1.
The set is a “sheaf of torsors” over the discrete space of classes in of self-intersection zero. More precisely, the forgetful map has image equal to . The fiber over any has a transitive -action by framed link translation (defined in Section 2 below), with stabilizer equal to the image of the map that records twice the intersection with any framed surface representing an element in . Thus is an -torsor, where .
This computation of provides a framework for the study of Morse -functions on -manifolds, a subject that has garnered considerable recent attention (see for example ) and that was the original motivation for our work.††† A Morse -function on is a “generic” map . Every smooth, closed, connected, oriented -manifold admits such a fibration, in fact, one with no “definite folds” and with connected fibers. Furthermore, the moves generating homotopic fibrations have been determined. This sets the stage for a new approach to the study of smooth -manifolds.
It should be noted that Theorem 2 can be deduced from Taylor’s Theorem 6.6 in , which applies to a general -complex, although the two proofs are completely different. Our differential topological approach gives a more geometric perspective, and also provides an answer to a question that was left open in [22, Remark 6.8] regarding the existence of 4-manifolds of type III; see the end of Section ‣ 0. Preliminaries and Example 2.5.
In fact, there is a homotopy theoretic version of Theorem 2 that applies to a general CW-complex. It is the special case of the following theorem, which in turn follows from the existence of the fiber bundle constructed at the beginning of Section 3 (where we also explain the necessary translations).
Let be a topological group and be a closed abelian subgroup of . For any CW-complex there is an “exact sequence”
in the following sense: A map lifts to a map if and only if it becomes null homotopic when composed with the map induced by the inclusion . Moreover, the natural action of the group on the set is transitive on the fibers of and the stabilizer of equals the image of the homomorphism
which is induced by the continuous map defined by .
If is a compact connected Lie group with maximal torus , we invite the reader to check that the degree of equals the order of the corresponding Weyl group. This gives a satisfying explanation for the factor of in Theorem 2, as the order of the Weyl group of the Lie group . We suspect that for non-maximal the map is null homotopic; it certainly does factor through a manifold of lower dimension.
The proofs of Theorems 1–3 are given in the correspondingly numbered sections below. In this preliminary section, we first recall the classical arguments that
and that all homology classes in below the top dimension are represented by embedded submanifolds (recall that is assumed smooth) – a fact that we will often appeal to. We then set the context for our proofs of Theorems 1 and 2, in which and are computed, introducing a notion of “twisted” homology classes, and discussing a partition of -manifolds into types I, II, III and III from properties of their intersection forms.
The generator is a homotopy equivalence for , which implies that for any CW-complex . If is an -complex then because is -connected. Thus for we have when or , as stated above. (Note that the case was first proved by Hopf, and the cases are immediate from general position.)
A similar discussion applies to the Thom class , where is the Thom-space of the universal bundle over the oriented Grassmannian . The map is a homotopy equivalence for or , and in general is -connected . Hence induces an isomorphism for any -complex , and so for we have for all . Now the oriented Pontryagin-Thom construction gives a diagram like the first of this paper, with replaced by and normal framings replaced by normal orientations. It follows that for , the group is isomorphic to the set of closed oriented -dimensional submanifolds of , up to oriented bordism in . Note that the orientation of allows us to translate normal orientations on submanifolds into tangential orientations.
Similarly, using the Thom class one can conclude that for but , is isomorphic to the set of closed -dimensional (unoriented) submanifolds of , up to bordism in . For , there is an exact sequence
(see  and also ) so classes in are still represented by submanifolds. In the following, we will frequently use these interpretations of homology classes in terms of submanifolds in . In particular, the forgetful maps are then just given by only remembering the orientation from a framing. The reduction mod 2 map forgets the orientation on the submanifold.
Our study of the framed bordism sets and will feature two special subsets and , whose elements we call twisted classes. Here denotes the subgroup of of all elements of order at most , and denotes the subset of of all classes of self-intersection zero.
To define these subsets, first observe that is an abelian group with respect to the operation of disjoint union, and that the forgetful map is an epimorphism. Each is represented by a knot whose normal bundle has exactly two trivializations (up to homotopy) since . If the resulting pair of framed knots are framed bordant, then they represent the unique element in the preimage , and if not, then they represent the two distinct elements in . Thus is either an isomorphism or a two-to-one epimorphism (Theorem 1 refines this statement).
Now consider the image subgroup under of the -torsion subgroup of , and define to be its complement in :
Evidently splits over (that is, there is a homomorphism with ) and so we refer to the elements in as split classes. The elements in , which we call twisted classes, are exactly those -torsion classes that generate subgroups over which does not split. In geometric terms, if a twisted class is represented by a knot , then , equipped with either framing, represents an element of order in . Clearly is nonempty if and only if does not split over , and in this case it is the nontrivial coset of the index two subgroup .
A class in will be called twisted if its intersection with at least one -dimensional homology class in is a twisted -dimensional class:
Evidently is empty when is empty, but as will be seen below, the converse may fail.
4-manifold types I, II and III.
The parity of the -manifold is determined by the self-intersections of the orientable surfaces in . If they are all even, then is said to be even, and otherwise it is odd. The even ones includes all the spin -manifolds – those whose tangential structure groups lift to Spin(4) – which are characterized by the condition that all surfaces in , orientable or not, have even self-intersections (only defined modulo for non-orientable surfaces); by the Wu formula, this is equivalent to the condition . As is customary, we say is of type I, II or III, according to whether it is odd, spin, or even but not spin.
All -manifolds of type III must have -torsion in their first homology. In fact by Theorem 1 (proved in the next section) is of type III if and only if contains twisted classes, i.e. is nonempty. The simplest such manifold is the unique nonspin, oriented -manifold that fibers over with fiber . This manifold is even since the fiber, which generates , has zero self-intersection, whereas any section has odd self-intersection. A handlebody picture of , minus the and -handle, is shown in Figure 1(a) using the conventions of [11, Ch.1]. The generator of , represented by the core of the obvious Mobius band bounded by the (attaching circle of the) 1-framed 2-handle, is a twisted -dimensional class.
Note that is one of a doubly indexed family of -manifolds shown in Figure 1(b), with . Here is the linking number between the dotted circle (the -handle) and the -framed 2-handle. Alternatively, these manifolds can described as the boundaries of with a single -handle added. An easy exercise in link calculus (or even easier -dimensional argument) shows that for fixed , the diffeomorphism type of depends only on the parity of when is even, and is independent of when is odd. Thus there are really only two such manifolds and when is even, and only one when is odd. The ’s are all of type II, while the ’s (for even ) are all of type III. These manifolds will feature as building blocks for our examples below.
Types III and III.
When computing their second framed bordism sets, the 4-manifolds of type III will be seen to display the most interesting behavior, suggesting a partition into two subclasses: is of type III if it contains some twisted 2-dimensional classes, i.e. is nonempty, and is otherwise of type III. Thus the 4-manifolds of type III are exactly those for which some of their twisted 1-dimensional classes (which they have because they are of type III) arise as intersections of homology classes of dimensions 2 and 3.
There exist -manifolds of both subtypes. In particular, the ’s for even are all of type III since they have vanishing third homology. In Example 2.5, we will construct a -manifold of type III, which answers a question raised in [22, Remark 6.8]. It will be shown in Remark 2.2 that with regard to their cohomotopy theory, the manifolds of type III behave more like odd manifolds, while those of the type III behave more like spin manifolds, which explains the choice of subscripts.
1. Computation of the group
As noted in the introduction, the set of framed bordism classes of framed links in is an abelian group under the operation of disjoint union. This operation corresponds to the product in inherited from the group structure on the target -sphere. Indeed, given two maps , one can pull back regular values to get disjoint framed links in . Choose disjoint tubular neighborhoods of . Up to homotopy, is given by mapping to , wrapping the disk fibers in around the -sphere using the framing, and mapping to . It follows that the product has as a regular value, with pre-image equal to the (disjoint) union . This yields an easy indirect proof that is abelian; there is also a direct proof, well known to homotopy theorists, following from the observation that can be replaced with the fiber of the map , which is a homotopy-abelian -space (cf. [22, §6.1]).
It is evident that the forgetful map
sending the bordism class of a framed link to the homology class of the underlying link, is a surjective homomorphism.
The kernel of is either trivial or cyclic of order two, according to whether is an odd or even -manifold.
This fact was known to Steenrod. In his 1947 paper  (where he introduced his squaring operations) he identified the kernel of the map dual to with the cokernel of the squaring map . Since is dual to mod 2 self-intersection of integral classes, this cokernel is or according to whether is odd or even.
Thus for odd we have , or dually . For even there is an abelian extension
(cf. [22, §6.1]), or dually
as asserted in Theorem 1.
For completeness, we provide a geometric proof of Claim 1.1, in terms of framed links, which also yields a simple description for the map in : Any element in is represented by an unknot with one of its two possible framings. Exactly one of these framings extends over any given proper -disk in bounded by . Let denote the resulting -framed unknot (this may depend on the choice of ) and denote with the other framing. Then clearly in , while generates and is of order at most two. The claim is that if and only if is odd. But if , then capping off the boundary of a framed bordism in from to and projecting to produces a (singular) surface of odd self-intersection, so is odd. Conversely, if is odd, then removing two disks from an embedded oriented surface in of odd self-intersection and then “tilting” this surface in gives a framed bordism between and . This proves the claim, and shows that in the even case, the map in sends the generator of to .
To complete the proof of Theorem 1, it remains to show that the element in defined by the extension (e) maps to the Stiefel-Whitney class under the monomorphism
in the universal coefficient sequence for .
To see this, first recall the definition of . View as , where is the inclusion of singular -boundaries to -cycles in . Then maps the equivalence class of a functional to the cohomology class of the cocylce , where is the boundary map on chains in , that is, .
Alternatively, can be viewed as the dual of the Bockstein homomorphism
for the coefficient sequence . To explain this, we appeal to:
There is a classical method for describing extensions by 2-torsion groups due to Eilenberg and MacLane [5, Theorem 26.5]. Let and be abelian groups fitting into an extension
Given any , choose with . Then there exists such that , and it is easy to check that is uniquely determined by because consists of 2-torsion only. In fact, this leads to a homomorphism of groups , inducing an isomorphism .
For the case at hand we have , mapping to the functional , where is any integral cycle representing a 2-torsion element in . From this one can easily check that the diagram
commutes, which is the sense in which and are dual.
Now is represented by any functional for which the diagram
commutes; see for example Spanier [19, §5.5.2]. This requires an initial choice of a map making the right square commute, and then is forced. A different choice of will only change by the restriction of a functional on , and so its equivalence class in is well-defined.
Under the identification , the element corresponds to the “characteristic functional”
whose kernel is subgroup of split classes, introduced in Section ‣ 0. Preliminaries. Thus if is a knot of order 2 in , then or according to whether (endowed with either framing) has order or in , or equivalently, whether (meaning two copies of with the same framing on each) is framed bordant to or to . Note that up to framed bordism, the framing on does not depend on the choice of framing on , since this choice is being doubled.
Thus to prove , we must show that the functional corresponding to is equal to the composition . But sends the class of a surface to the class of any curve that is characteristic in , meaning Poincaré dual to ,††† This is well known, and easily verified. Note that does in fact represent an element in : It is null-homologous when is orientable, and of order in otherwise, since can then be viewed as an integral -chain with boundary . and reports self-intersections, by the Wu formula, so it remains to show
for any surface and characteristic curve in .
If is orientable, both sides of equation (w) vanish since is empty and is even. If is non-orientable, we can choose to be connected (it is then characterized up to homology by the orientability of its complement ) with closed tubular neighborhood in . Then is either an annulus or a Möbius band, according to whether is orientation preserving or orientation reversing in . In either case, is bordant in to ; the bordism is trivial when is an annulus, and a pair of pants when is a Möbius band, as indicated in Figure 2 with one suppressed dimension.
Starting with any framing on in , the induced framing on extends across (tilted in ) to give a normal framing on . We can assume that is normal to along , pointing in the suppressed direction in the figure, is shown there by the thin push-off, and is determined by , and the orientation on . Then extends to a normal vector field across all of , and since is orientable, across the rest of off of a disk ; we use this vector field to push off itself in order to compute . Similarly, there is no obstruction to extending the framing on across (tilted in ) to give a framing on the unknot , which we also call . By construction is framed bordant to . Now the mod self-intersection is or according to whether the framing on extends over or not, or equivalently, whether is framed bordant to or . Thus by definition , proving equation (w) and completing the proof of Theorem 1. ∎
If has no -torsion, then it is immediate from the theorem that is isomorphic to or to , depending upon whether is odd or even. In particular for simply connected, is trivial when is odd, and isomorphic to when it is even.
If has -torsion, then for some sequence of integers, where has no -torsion. For each let be a knot in representing times the generator of the summand above. Then generate . Now recall the functional from the proof of the theorem, given by , where is any surface in in which is characteristic. If , or equivalently is spin, then of course . Otherwise choose the smallest for which , and rewrite , where incorporates the remaining summands. Then
This is a consequence of the identity established in the proof of the theorem.
and the Steenrod square induces a homomorphism
With a little bit of work, the statement of Theorem 6.2 in  can be interpreted as saying that . Since , this statement implies our Theorem 1.
More directly, the dual of Taylor’s Theorem 6.2 for the case and is exactly the assertion (w), which is the key step in our proof of Theorem 1.
2. Computation of the set
The forgetful map
sends the bordism class of a framed surface in to the homology class of the underlying oriented surface. The image of is the set of all classes in of self-intersection zero, that is, those represented by surfaces in with trivial normal bundle. Thus to compute , it suffices for each to classify up to framed bordism the framings on surfaces representing , that is, to enumerate the framed bordism classes in the fiber
To accomplish this, fix a surface representing and choose a framing of the normal bundle of . Then represents one element in , and all others arise as “framed link translates” of , in the following sense:
Let be an oriented link in with normal framing . We can assume that is disjoint from , after an isotopy if necessary. Now form a new framed surface , called the -translate of , by adjoining one new framed torus for each component of , where is the boundary of a small -dimensional solid torus thickening of in in the direction of and . This is illustrated schematically in Figure 3.
To describe the framing on precisely, we set up coordinates as follows. Identify a small tubular neighborhood of in with via the framing , with . Then set
where is the equatorial disk , as shown in Figure 4(a) at a crossection . Now we require to spin once relative to the “constant” framing , using cylindrical coordinates in , so that in particular it does not extend across the disk fibers of pushed into . Explicitly and at any , as illustrated in Figure 4(b) by showing (up to homotopy) the tip of in each frame.
The -framed unknot acts trivially on any framed surface , that is is framed bordant to . We leave this as an instructive exercise for the reader. Of course in general, need not be framed bordant to . For example, if is a framed -sphere in (note that all framings on embedded -spheres in -manifolds are isotopic since ) then is not framed bordant to . Indeed generates , while .
There is an another more direct way to modify the framing using a link lying on . Simply add a full right handed twist in the normal fibers along any transverse arc to in . This results in a new framing on the same surface, which we call the -twist of and, by abuse of notation, denote by .
In fact any framing on can be obtained in this way. Indeed where is any embedded representative of the Poincaré dual of the cohomology class in (with integer coefficients) that measures the difference (meaning that this difference on any given -cycle in is full right handed twists, where is the intersection number of the cycle with ). Thus any two framings on are related by “link twists”.
It will be seen in the proof of the “action lemma” below that is framed bordant to , where is the framing on given by together with the normal to in , and so link twisting can be thought of as a special case of framed link translation.
Framed link translation defines an action of the group on the set , given by . The orbit of under this action is the set of all classes of framed surfaces that, forgetting the framing, represent .
Since addition in is disjoint union, it is clear that the operation in the lemma defines an action provided it is well-defined. To see that it is in fact well-defined, assume that and are bordant via a framed surface , and that and are bordant via a framed -manifold . We must show that and are framed bordant.
To do so, first adjust and to intersect transversally in finitely many points . Then “pipe” these intersections to the upper boundary of in the usual way: Choose disjoint arcs in joining the points to points in , and thicken these into disjoint solid cylinders in that meet and in disks and ; the are meridional disks for in . Now construct a new surface , removing the disks from and replacing them with the cylinders , and then rounding corners. The framing on clearly extends across , restricting to the -framing on each unknot . Thus provides a framed bordism between and , where is a -framed unlink in . But as noted in Remark 2.1 above, acts trivially, and so and are framed bordant.
To complete the proof of the action lemma, we must show that any framed surface belonging to is framed bordant to acted on by some framed link . We can assume that and are connected, and by hypothesis, they are bordant. Hence we can find a bordism from to with only and -handles. The framing extends (not uniquely) over the -handles to give a normal framing on the middle level of . Similarly, on extends to a normal framing on the middle level. Abusing notation, call the middle level and the two normal framings and .
Now let be a link in representing the Poincaré dual of the cohomology class in that measures the difference , and so in the notation above. Then acquires a framing by appending its normal vector field in onto , and we propose to show that is framed bordant to .
To that end, we first “blister” . That is, we create a -dimensional bordism obtained from (using the first vector of ) by removing an open disk bundle neighborhood of so that where is the torus boundary of the disk bundle. Then “tilt” into so that and .
The framing on induces a framing on by the link translation construction above. Together these give a framing on that extends over by adding an extra full twist when going past the arc , the sign of the twist depending on the orientations of and . By construction, this framing restricts to on . This is shown in Figure 5 with two of the five dimensions suppressed, one tangent to and the other normal to in