Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading

TL;DR

This paper constructs explicit examples of orientable manifolds with maximal nonzero dual Stiefel-Whitney classes, improving upon nonconstructive existence proofs by providing concrete instances for certain dimensions.

Contribution

The authors explicitly construct real Bott manifolds with maximal dual Stiefel-Whitney classes for all dimensions not divisible by 4, advancing the understanding of characteristic classes in manifold topology.

Findings

Explicit real Bott manifolds with desired properties for n ≠ 0 mod 4

Confirmation of maximal dual Stiefel-Whitney classes in constructed examples

Extension of known existence results to concrete constructions

Abstract

It is known that, for all n, there exist compact differentiable orientable n-manifolds with dual Stiefel-Whitney class wbar_{n-ahat(n)} nonzero, and this is best possible, but the proof is nonconstructive. Here ahat(n) equals the number of 1's in the binary expansion of n if n equiv 1 mod 4 and exceeds this by 1 otherwise. We find, for all n nonzero mod 4, examples of real Bott manifolds with this property.