Stiefel - Whitney Currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, Vl, . . . , Vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called "atomicity." Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n m + 1 and hence determines a Z2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle. More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the rood 2 linear dependency current. The cohomology class o f the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents rood p computes cohomology rood p. An alternate approach to Federer's theorem is offered in an appendix. This approach is simpler and is via sheaf theory. 1. Introduct ion It is well known [3, 17, 18, 22, 24] that the linear dependency set of a collection of sections of a vector bundleis related to the characteristic classes of the bundle. In particular, the zero set of a regular section defines a cohomology class which is the Chem-Euler class of the bundle. In [12] canonically defined current representatives of the Chern classes of a complex vector bundle were associated to collections of smooth sections of the bundle. These currents are called linear dependency currents since they are supported on the linear dependency set of the collection of sections. The main aim of this paper is to study the linear dependency currents of a collection of sections of a real vector bundle. These are either mod 2 or bundle-twisted currents which represent either the mod 2 or twisted-integer Stiefel-Whitney classes of the bundle. Since they are either mod 2 or 2-torsion, these currents were overlooked in [12]. The linear dependency current associated with an ordered collection v of sections of a real vector bundle is defined in Section 3 paralleling a standard construction in enumerative geometry (see, for example, [19, 7]). In general, the most one can say is that this linear dependency current, 9 1998 The Journal of Geometric Analysis ISSN 1050-6926 810 Reese Harvey and John Zweck LDm~ is a rood 2 current and that it determines a Z2-cohomology class which is well defined independent of the particular collection of sections of the bundle (Theorem 3.14). However, if the degree of LI) m~ (v) is odd or equal to the rank of the bundle it is also possible to define a (bundle-twisted) linear dependency current, LD(v), which encodes certain (twisted) integer multiplicities of dependency among the sections (Proposition 3.16). The mod 2 reduction of this current LD(v) is the rood 2 current LDm~ The current LD(v) determines a (twisted) integer cohomology class well defined independent of the choice of collection v (Theorem 3.6). If the degree of LD(v) is less than the rank of the bundle (which occurs when the collection consists of more than one section), this cohomology class is a torsion class of order 2 (Corollary 3.8). A major advantage of the approach taken here is that the lincar dependency current is defined whenever the collection v satisfies a weak measure theoretic condition called "atomicity", which is vastly more general than the usual general position hypothesis. For example, a real analytic collection of m sections of a rank n bundle is atomic provided that, for all j c {0, 1 . . . . . m 1}, the codimension of the set of points over which exactly j of the sections are linear independent is at least the expected codimension n j (see [12], Proposition 2.14). Another important property of the (mod 2) linear dependency current is that it is a (rood 2) locally integrally flat current. Recall that the integrally flat currents are those of the form R + dS, where R and S are rectifiable. Federer [5] proved that the complex of locally integrally flat currents (or such currents rood p) can be used to compute integer (or mod p) cohomology. In the Appendix we offer an alternate approach to the theory of (mod p) integrally flat currents and their cohomological properties. This simple approach is via sheaf theory and is distinct from the form of the theory given in the geometric measure theory literature. The theory of dependency currents relies heavily on the theory of zero divisor currents, which was originally developed in [14] for "atomic" sections of an oriented vector bundle over an oriented manifold. The notion of an atomic section provides a generalization of the notion of a section being transverse to zero, one that is both useful and vastly more general. The zero divisor is a d--closed locally integrally flat current which determines a unique integer cohomology class, the Euler class. In this paper it is crucial that the notion of a zero current be understood in the non-orientable case. This is done in Section 2 where the zero divisor is defined as a bundle twisted current. This current determines a cohomology class (Theorem 2.5), which is the twisted Euler class, 7 E H n (X, ~), of the vector bundle. The reduction mod 2 of the zero divisor eliminates the twisting, yielding a rood 2 current which represents the top Stiefel-Whitney class, wn ~ Hn(X, Z2). In Section 4 we identify the Z2-cohomology class of the degree q rood 2 current LD m~ 2(v) as Wq, the qth Stiefel-Whitney class of the bundle (Theorem 4.1). Moreover, if q is odd, the Z class of the twisted current LD(v) is identified as the qth twisted integral Stiefel-Whitney class Wq E Hq(x , ~) (Theorem 4.10). This result is a corollary of the fact that the Bockstein of the mod 2 dependency current LDm~ of degree q 1 is the degree q twisted dependency current associated with a subcotlection of the collection v (c.f. [24, 21]). The Stiefel-Whitney classes were originally defined [22, 24, 21] as the primary obstruction to the existence of certain collections of linearly independent sections of a bundle F --+ X. In Section 5 we examine the relationship between linear dependency currents and obstruction cocycles. Given a triangulation of X it is possible to choose a particular atomic collection of sections of F so that the Steem'od obstruction cocycle of the collection is defined. The Poincar6 dual of such a cocycle is a cycle which defines a current on X by integration. We then show that this obstruction current is equal to the linear dependency current of the particular collection of sections. Among other things, this provides an alternate proof of the results of Section 4. Stiefel-Whitney Currents 811 In Section 6 higher dependency currents and general degeneracy currents of vector bundle maps are discussed, further expanding the results of [12]. Some of the degeneracy currents studied in Section 6 were not included in [12] since they are either not defined as twisted currents or their real cohomology class is zero. In these cases we can define mod 2 and/or twisted degeneracy currents. The integer cohomology classes of the twisted degereracy currents were first studied by Ronga [20] who proved that they are uniquely determined by their torsion-free part and rood 2 reduction. We expand upon Ronga's result by explicitly identifying the integer cohomology classes of the higher dependency currents as certain polynomials in the integer Pontrjagin and Stiefel-Whitney classes (Theorem 6.15). In Section 7 applications of the theory to singularities of projections and maps are given. In particular we recover the well-known fact that the Steifel-Whitney classes of the tangent bundle TX and normal bundle NX of a submanifold X C R N can be defined in terms of singularities of projections. The original version of this result is due to [17, 18], [23] (see also [1]). Note, however, that they only consider generic projections whose critical sets are non-degenerate, with multiplicity • The atomic theory enables us to consider degenerate critical sets of arbitrary integer multiplicity (Proposition 3.16). In particular, if X is a real analytic submanifold, the tangent and normal StiefelWhitney classes can be defined in terms of the singularities of any projection whose degeneracy subvarieties have at least the expected codimension. Integer and mod 2 cohomological obstructions to the existence of smooth immersions and surjections between manifolds are also given, c.f. [20]. Two further applications are worth noting. Following [12] we can define mod 2 and twisted integer degeneracy currents associated with higher self-intersections of plane fields and invariants of pairs of foliations. Mod 2 and integer umbilic currents of hypersurfaces can also be studied using these ideas. Details of these two applications are left to the reader. Secondary (Cheeger-Chern-Simons) Stiefel-Whitney classes will be introduced in a later paper. 1 Canonical Llo c representatives of these classes will be associated to each atomic collection of sections of a bundle with Riemannian connection. In the case of a single section o~, the secondary Euler class is represented by the Chern-Euler potential cr(~). As is discussed in [10] this potential satisfies the important equation dcr(ot) = X Div(ot), where X is the Euler form and Div(~) the divisor of the section. If the collection v consists of more than one section, then there is a canonical L~o c current T(v) satisfying the current equation dr(v) = LD(v), which represents the appropriate secondary Stiefel-Whitney class. This current equation is related to a formula of Eells [4]. Finally, we would like to draw the attention of the reader to recent work of Fu and McCrory [6] who, in the spirit of [15], have constructed current representatives for the tangential Stiefel-Whitney homology classes of a singular variety. 2. Divisors and atomicity Harvey and Semmes defined the zero divisor current of an atomic section of an oriented real rank n vector bundle over an oriented manifold. The divisor is a codimension n current which is supported on the zero set of the section and which encodes the integer multiplicity of vanishing of the section. Furthermore, it is a d-closed locally integrally flat current whose cohomology class in H n (X, Z) is well defined independent of the choice of section. This class is the Euler class of the bundle. The aim of this section is to define and study the zero divisor current in the case in which neither the vector bundle nor the base manifold are assumed to be orientable. In this case, the zero divisor is defined to be a current that is twisted by the orientation bundle of the vector bundle. It is also useful to define the rood 2 divisor to be the mod 2 reduction of the divisor. Both of these notions of divisor 812 Reese Harvey and John Zweck will be important in the study of linear dependency currents in Section 3. We begin by recalling some definitions. Let V --+ X be a real rank n vector bundle over an Ndimensional manifold. No orientation assumptions will be made on V or X. Let Ox and Ov denote the principal g2-bundles of orientations of TX and V over X. An O-twisted k-form is a section of (,9 @~2 AleT*X --+ X. (Often the subscript •2 will be dropped when tensoring with (.9.) A density is a top degree smooth Ox-twisted form on X. Note that densities can be integrated over X. A generalized function is a continuous linear functional on the space of compactly supported smooth densities on X. A current of degree p is a differential p fo rm on X whose coefficients (with respect to each coordinate system) are generalized functions. Equivalently, a degree p current is a continuous linear functional on the space of compactly supported Ox-twisted (N p)-forms. Similarly, an Or-twisted current is an Or-twisted form whose coefficients are generalized functions, i.e., it acts on Ov | Ox-twisted forms. An L~o c form is a form whose coefficients are L~o c functions. Therefore, L~o c forms are currents that are not twisted. On the other hand, an oriented compact submanifold of X defines an OK-twisted current by integrating (untwisted) forms over it. Note that exterior differentiation is a well defined operation on (twisted) currents. On a contractable open subset U of X, each Ortwis ted current T can be written in the form = [e] | r where [e] 6 Ov is the orientation class of a local frame e for V over U and where T is a current on U. If V is orientable, each choice of orientation defines an isomorphism between Ortwisted currents and currents. These two isomorphisms differ by a minus sign. Note that the definition of a current on a non-orientable manifold agrees with that given in [13] but disagrees with that in [25, 26]. A. Divisors in the nonorientable case In this subsection we define and study the divisor of a section of V --> X. The divisor is defined to be an Ov-twisted current. Note that if V and X are oriented, the definition of divisor given below agrees with that of [14]. The solid angle kernel, 0, is the L~o c form on R n obtained by pulling back the normalized volume form on the unit sphere to R '~ ~ {0} by the radial projection map. The current equation d 0 = [0] on R n, where [0] denotes the point mass at the origin, motivates the definition of divisor. Definition 2.1. Let X be a smooth manifold and let y = (yl . . . . . yn) denote coordinates on R n. dy 1 ~n In the case n > 1 a smooth function u : X -+ R n is called atomic if, for each form i7[5 on with d 1 P = I I ] < n 1, the pullback u*(~@lp) to X has an L~oc(X) extension across the zero set Z of u. y Also assume that u does not vanish identically in any connected component of its domain X. In the case n = 1, it is convenient to define a smooth function u : X ---> R to be atomic if log l ul 6 L]o c (X), (c.f. [14]). If u is atomic, then the zero set Z has measure zero in X (see [14]) so that the L]o c (X) extensions are unique. In particular, the smooth form u*(O) on X ~ Z has a unique L]oc(X) extension across Z, and therefore defines a current on X. Def in i t ion 2.2. Let u : X --> R n be an atomic function. The divisor of u is the degree n current Div(u) on X defined by Div(u) := d (u*O) . Atomicity is a weak condition which ensures the existence of a zero divisor. Harvey and Semmes proved that a large class of smooth functions are atomic. More specifically those functions which Stiefel-Whitney Currents 813 vanish algebraically and whose zero sets are not too big in thesense of Minkowski content are atomic. In particular, real analytic functions whose zero sets have codimension n are atomic. L e m m a 2.3. Let g be a smooth GL(n , R)-valued function on an oriented manifold X and let u : X --+ R ~ be atomic. Then v := ug is atomic and Div(v) = -4-1 Div(u) , where -t-1 := sgn det(g) is constant on connected components o f X. This result of [14] allows one to extend the notion of divisor to sections of vector bundles. First, a section v of a smooth vector bundle V --+ X is called atomic if for each choice of local frame e for V the function v, defined by v = re, is atomic. Def in i t ion 2.4. Let v be an atomic section of a rank n bundle V --+ X. The divisor, Div(v), of v is the Or twis ted current on X defined locally on an open subset U of X as follows. Choose a local frame e for V over U and let v : U --+ ~n be the coordinate expression for v determined by e. Then Div(v) := [e] | Div(v) on U . In particular, if V is oriented, then Div(v) is a current on X. As described in the Appendix, the locally integrally flat currents are those currents that can be expressed as R + dS where R and S are locally rectifiable. Furthermore, the complex 5~/*oc(X) of Ortwis ted currents on X which are locally integrally flat may be used to compute the cohomology, H*(X, Zv ) , of X with integer coefficients twisted by (_gv, i.e., Zv :----(gv | Z. T h e o r e m 2 . 5 . Le tv beanatomicsec t ionofarealrankn vectorbundle V --~ X. Thezerodivisor, Div(v) c 5~oc(X), o f v is an Or tw i s t ed d-closed locally integrally fiat current o f degree n on X, whose support is contained in the zero set o f the section v. Furthermore, i f # is another atomic section o f V, then there is an Or tw i s t ed locally rectifiable current R so that Div(v) Div(/~) ---d R . (2.5.1) That is, the cohomology class o f Div(v) in H n ( X, Z v ) is well defined independent o f the choice o f section v. This class is the twisted Euler class "Y o f V. In particular, i f V is oriented, the Euler class e ~ H n ( x , Z) o f V is the cohomologyclass o f Div(v). C o r o l l a r y 2.6. Suppose that v is an atomic section o f an odd rank bundle V --~ X. Then there is an Or tw i s t ed locally rectifiable current R on X so that