Stiefel-Whitney Classes Research Articles

Axioms for Stiefel-Whitney Homology Classes of Some Singular Spaces

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds $\text {mod} 2$ and homology manifolds $\text {mod} 2$. As a consequence, it is shown that on homology manifolds $\text {mod} 2$ the generalized Stiefel conjecture holds.

Axioms for Stiefel-Whitney homology classes of some singular spaces

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds mod 2 \text {mod}\, 2 and homology manifolds mod 2 \text {mod}\, 2 . As a consequence, it is shown that on homology manifolds mod 2 \text {mod}\, 2 the generalized Stiefel conjecture holds.

Real Algebraic Realization of Characteristic Classes

Throughout this paper X denotes a real non-singular affine algebraic variety of dimension n. We will give a realization of the characteristic classes (the Stiefel-Whitney classes, the Pontrjagin classes and the Euler classes) of real affine algebraic vector bundles over X by algebraic subvarieties (Theorems 1, 2). For the complex field, Grothendieck [4] showed that the Chern classes of an algebraic vector bundle over a complex non-singular quasi-projective variety are realized by algebraic cycles. Morimoto [6] considered the complex analytic case. We prove Theorems 1, 2 by the method used there. If we work over a real analytic vector bundle, Thorn's transversality theorem shows easily a realization of the characteristic classes by analytic subsets (see Suzuki [10]). Theorem 1 was partially proved in [2], [8], and two different applications of them were given in [2], [9]. In Section 4, Theorems 3, 4 will show that the smoothing of algebraic subvarieties of X of codimension 1 for homological equivalence is always possible. The proof uses an idea in [8]. Given two cohomology classes of X which are realized by algebraic subvarieties, it seems likely that their cup product is realized by an algebraic subvariety. We prove this under some assumptions, applying Theorems 1, 2 (Theorem 5). We must remark that a realization of the cup product by an analytic subset is always possible according to the transversality theorem. Section 6 considers an affine algebraic structure of a topological vector bundle over X. If the rank is 1, and if the Stiefel-Whitney class is realized by an algebraic subvariety, then the bundle has an affine algebraic structure.

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C ∗-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.

A spinor field representation of the Stiefel-Whitney class

Manifolds with a non-vanishing second Stiefel-Whitney class do not admit globally defined spinors. Generalized spinors transforming under the Spin c group do however exist and we show how they may be used to represent the U(1) connection in the corresponding bundle and hence the Stiefel-Whitney class.

Cup products in Grassmannians

This note calculates the height of the first Stiefel-Whitney class in the cohomology of the real Grassmannians and determines the length of the longest nontrivial cup-product in H ∗(G k(R n+k);Z 2) (k⩽n) with k⩽4.

The imbedding of bounding manifolds in Euclidean space

Theorem 1. I / n ~_ 5, then any n-dimensional mani/old which is the boundary o/a~ orientable mani/old will be also the boundary o / a simple connected manifold which imbeds in R 2n-i. In [4] M. Hirsch proved the Theorem above for many values of n, depending upon the dyadic expansion of n. By "manifold" we mean a compact C ~~ manifold. Let y be a cohomology class of the classifying space BSO, with any group of coefficients and let W be an oriented manifold, (with boundary or not). Then y(W) will denote the corresponding normal characteristic class of W. By Wi we denote the/-dimensional Stiefel-Whitney class of the universal oriented bundle. I f i equals 1 or an even number, then Wi is an element of H i (BSO; Z2). On the other hand ff i is an odd number greater than 1, then Wi is an element of H i (BSO; Z). Let wi be the rood 2 reduction of Wi. The main result of this paper, which is exactly what is needed to complete Hirseh's proof, is the following:

Characteristic classes for permutation representations

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney classin the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and(b) stable, i.e. w(V) = 1 when F is a trivial representation.

Stiefel-Whitney classes in $H\sp{\ast} B{\rm O}\langle \varphi (r)\rangle $

We determine the Stiefel-Whitney classes in ${H^*}(BO;{{\mathbf {Z}}_2})$ which are mapped nontrivally by the homomorphism induced by the covering projection $p:BO\left \langle {\phi (r)} \right \rangle \to BO$.

New topological indices in SO(3) Einstein-Yang-Mills theory

The topological classification of solutions to the Euclideanised SO(3) Einstein-Yang-Mills equations is discussed. As well as the usual Pontriagin index, two new two-valued indices, associated with the second Stiefel-Whitney class, are required to specify the bundles. The Charap-Duff solution is classified in this framework.

Involutions with fixed point set of constant codimension

The cobordism classes of manifolds admitting involutions with fixed point set of codimension 5 are determined by means of Stiefel-Whitney classes.

A continuous version of the Borsuk-Ulam theorem

Let p : E → B p:E \to B be an n n -sphere bundle, q : V → B q:V \to B be an R n {{\mathbf {R}}^n} -bundle and f : E → V f:E \to V be a fibre preserving map over a paracompact space B B . Let p ¯ : E ¯ → B \overline p :\overline E \to B be the projectivized bundle obtained from p p by the antipodal identification and let A ¯ f {\overline A _f} be the subset of E ¯ \overline E consisting of pairs { e , − e } \{ e, - e\} such that f e = f ( − e ) fe = f( - e) . If the cohomology dimension d d of B B is finite then the map ( p ¯ | A ¯ f ) ∗ (\bar {p} | \overline {A}_f)^* is injective for a continuous cohomology theory H ∗ {H^*} . Moreover, if the j j th Stiefel-Whitney class of q q is zero for 1 ⩽ j ⩽ r 1 \leqslant j \leqslant r then ( p ¯ | A ¯ f ) ∗ (\bar {p} | \overline {A}_f)^* is injective in degrees i ⩾ d − r i \geqslant d - r . If all the Stiefel-Whitney classes of q q are zero then ( p ¯ | A ¯ f ) ∗ (\bar {p} | \overline {A}_f)^* is injective in every degree.

Relations Among Characteristic Classes of n-Manifolds Imbedded in R n+k

ABsTRAcr. Let In C H*(BO; Z2) be the (graded) set of those normal characteristic classes which are zero on all compact, closed C' manifolds. Let Ink C H*(BO; Z2) be the set of those characteristic classes which are zero on all n-manifolds which imbed in R n+k. Let K be the (graded) ideal in H*(BO; Z2) generated by the Stiefel-Whitney classes Wk, Wk+ 1, Wk+2, Wk+3..... We will prove the following result: If 1 < i < min{(2k 2), (n + k 1)/2}, then Inik = Ini + K'. Also, we will prove an analogous result for manifolds with an extra structure.

On the Height of the First Stiefel-Whitney Class

Let ${G_k}({{\mathbf {R}}^{n + k}})$ denote the grassmann manifold of k-planes in real $(n + k)$-space and ${w_1} \in {H^1}({G_k}({{\mathbf {R}}^{n + k}});{{\mathbf {Z}}_2})$ the first Stiefel-Whitney class of the universal bundle. We determine, for many (k, n), the exact height of ${w_1}$ in the cohomology ring. We also indicate the combinatorial significance of the complex analogue of these computations.

On the cohomology of real Grassmanians

LetGk(Rn+k){G_k}({\textbf {R}^{n + k}})denote the grassman manifold ofk-planes in real(n+k)(n\, +\, k)-space andw1∈H1(Gk(Rn+k);Z2){w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height ofw1{w_1}in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.

Universal metaplectic structures and geometric quantization

The recently developed concepts of generalized and universal spin structures are carried over from the orthogonal to the symplectic and unitary cases. It turns out that the analogues ofSpin c-structures, namely theMp c-structures andMU c-structures, are sufficient to avoid topological obstructions to their existence. It is indicated how this fact can be used in the geometric quantization of certain suitably polarized symplectic manifolds with arbitrary second Stiefel-Whitney class, where the usual Kostant-Souriau quantization scheme breaks down.

On the Vanishing of the Dual Stiefel-Whitney Classes of Orientable Manifolds

We will determine for which values of n and k the dual $(n - k)$-Stiefel-Whitney class of all orientable n-manifolds vanishes.

Wall Manifolds

In the calculation of the oriented cobordism ring, it is standard to consider so-called Wall manifolds, for which the first Stiefel-Whitney class is the reduction of an integral class. This paper studies the Wall-type structures in the equivariant case.

Fibrations and a transfer map in algebraic K-Theory

In [l] D. R. Anderson showed that a fibration F -+ E -% B, with fiber F and base B both finite CW complexes, defines a homomorphism p* : Wh(rriB) -+ Wh(rrE) which relates the Whitehead torsions of a fiber homotopy equivalence. In this paper we define a similar homomorphism p* : I?0(7riB) + kO(rIE), which relates the Wall finiteness obstruction of the total space, O(E), with that of the base O(B). Here &,(r) denotes the reduced Grothendieck group of finitely generated projective modules over the group ring 2~. We will call this homomorphism a transfer, because its behavior resembles the homology transfer of Becker-Gottlieb and Dold and also because it generalizes the classical transfer in K-theory. The properties of this transfer will be used in a subsequent paper to compute the Wall finiteness obstruction (and Whitehead torsion) of many fiber spaces. Before stating the Theorem let us set some notation. Given a space F let F -+ EF --j BF denote the classifying (universal) fibration; i.e. any fibration F -+ E -+ B is determined by a map B -+ BF. Let w denote the induced map rIB + 7rIBF. We shall call w (first Stiefel-Whitney class) the orientation of the fibration. In particular, a fibration will be said orientable if w is the trivial homomorphism. Given a map f:A -+ B, we will denote by f* the induced map 7rIA -+ 7rIB and also (by abuse of notation) the corresponding map &(rIA) + gO(rrIB), which is determined by VIA -+ rlB. x(A) denotes the ordinary Euler characteristic of the space A.

An axiomatic proof of Stiefel’s conjecture

Stiefel’s combinatorial formula for the Stiefel-Whitney homology classes of a smooth manifold is proved, by verifying that the classes defined by his formula satisfy axioms which characterize the Stiefel-Whitney classes.