Local formulae for Stiefel-Whitney classes

We construct an explicit Čech cocycle representing the k-th Stiefel-Whitney class of a vector bundle. This construction involves only the transition functions of the bundle. We also give local formulae for the secondary Stiefel-Whitney classes. These may be useful in determining whether the Stiefel-Whitney numbers of a flat bundle are zero.     

Stiefel-Whitney classes and toral actions

Certain Stiefel-Whitney classes of manifolds with smooth, effective toral actions are shown to be computable in terms of Poincare duals of fixed point sets of isotropy subgroups. As an application the toral degrees of symmetry of certain Dold manifolds are determined.     

On the dual Stiefel-Whitney classes of some Grassmann manifolds

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 ^s + 2 and n = 2 ^s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned. Part of this research was carried out while the first author was a member of three research teams supported in part by the grant agencies VEGA and APVV, and the second author was a member of a research team supported in part by VEGA.     

Stiefel-Whitney classes and the conormal cycle of a singular variety

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety is given by means of the conormal cycle of an embedding of in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.

     

On trivialities of Stiefel-Whitney classes of vector bundles over highly connected complexes

It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel-Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel-Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces.     

Divisibility classes are seldom closed under flat covers

We show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring $R/sR$ satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.     

The curve complex and covers via hyperbolic 3-manifolds

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.     

The number of genus 2 covers of an elliptic curve

The main aim of this paper is to determine the number c _N,D of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈Div (E). In the case that D is reduced, this formula is due to Dijkgraaf.The basic technique here for determining c _N,D is to exploit the geometry of a certain compactification C =C _E,N of the universal genus 2 curve over the Hurwitz space H _E,N which classifies (normalized) genus 2 covers of degree N of E. Thus, a secondary aim of this paper is to study the geometry of C. For example, the structure of its degenerate fibres is determined, and this yields formulae for the numerical invariants of C which are also of independent interest.     

The Second Stiefel-Whitney Class of Small Covers

Let π : Mⁿ→Pⁿ be an n-dimensional small cover over Pⁿ and λ : F(Pⁿ)→Z₂ⁿ be its characteristic function. The author uses the symbol c(λ) to denote the cardinal number of the image Im(λ). If c(λ) = n + 1 or n + 2, then a necessary and sufficient condition on the existence of spin structure on Mnis given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when Pⁿ is n-...     

On certain classes of curve singulariries with reduced tangent cone

We study a class of rational curves with an ordinary singular point, which was introduced in [GO]. We find some conditions under which the tangent cone is reduced and we show that the tangent cone is not always reduced. We construct another class of rational curves with an ordinary singular point satisfying the condition required in [GO] and whose tangent cone is always reduced.     

The chern classes of the eigenbundles of an automorphism of a curve

LetX be smooth complex projective curve. Leth be an automorphism ofX of orderp. We improve a formula to compute the characteristic classes of the normal bundles of certain components of the fixed point set ofh acting on the symmetric products ofX. The author was supported by the Emmy Noether Research Institute for Mathematics.     

Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies

Every closed nanorientable 3-manifold M can be obtained as the union of three orientable handlebodies V₁, V₂, V₃ whose interiors are pairwise disjoint. If g_i denotes the genus of V_i, g₁g₂g₃, we say that M has tri-genus (g₁, g₂, g₃), if in terms of lexicographical ordering, the triple (g₁, g₂, g₃) is minimal among all such decompositions of M into orientable handlebodies. We relate the tri-genus of M to the genus of a surface that represents the dual of the first Stiefel-Whitney class of M. This is used to determine g₁ and g₂.     

Cycles of covers

We initially consider an example of Flynn and Redmond, which gives an infinite family of curves to which Chabauty’s Theorem is not applicable, and which even resist solution by one application of a certain bielliptic covering technique. In this article, we shall consider a general context, of which this family is a special case, and in this general situation we shall prove that repeated application of bielliptic covers always results in a sequence of genus 2 curves which cycle after a finite number of repetitions. We shall also give an example which is resistant to repeated applications of the technique.     

Flows, flow-pair covers and cycle double covers

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, Discrete Math. 244 (2002) 77-82] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f₁ is an h-flow and f₂ is a k-flow, and . Then G admits a nowhere-zero 6-flow if and only if G admits a (4,3)-flow parity-pair-cover; and G admits a nowhere-zero 5-flow if G admits a (3,3)-flow parity-pair-cover. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f₁ is an h-flow and f₂ is a k-flow, and for each {i,j}={1,2}. Then G has a 5-cycle double cover if G admits a (4,4)-flow even-disjoint-pair-cover; and G admits a (3,3)-flow parity-pair-cover if G has an orientable 5-cycle double cover.     

Acyclic covers

The following question, which is directly related to the Whitehead problem of subcomplexes of acyclic 2-complexes, is studied: If is a class of groups, X is a 2-dimensional CW-complex and X' is an acyclic, infinite cyclic cover of X with in , must X' be contractible? A positive answer is given if X is finite and is the class of amenable groups. Received: April 24, 1995.