Explicit examples for n ≡ 0 (mod 4), n not a power of two, with maximal nonzero dual Stiefel-Whitney class

Construct explicit orientable n-dimensional manifolds M for every integer n ≡ 0 (mod 4) that is not a power of two such that the dual Stiefel-Whitney class of grading n − (n), denoted \bar{w}_{n−(n)}(M), is nonzero. Here (n) denotes the function defined for orientable manifolds by: (n) equals the number of ones in the binary expansion of n when n ≡ 1 (mod 4), and equals that number plus 1 otherwise.

Background

Massey’s and subsequent results establish upper bounds on dual Stiefel-Whitney classes and, in the non-orientable generality, provide explicit realizations with nonzero top possible class. Davis and Wilson proved an analogue for orientable manifolds showing that for an orientable n-manifold M, \bar{w}i(M)=0 for i>n−(n), and they proved nonconstructively that there exists an orientable n-manifold with \bar{w}{n−(n)}(M)≠0.

The present paper constructs explicit examples realizing \bar{w}{n−(n)}≠0 using real Bott manifolds for all n not congruent to 0 modulo 4, and also notes that complex projective space covers the case when n is a power of two. However, for n ≡ 0 (mod 4) that are not powers of two, explicit orientable manifolds realizing \bar{w}{n−(n)}≠0 are not known.

Section 4 documents attempts in two prominent families: specific real Bott manifolds in dimension 12 (where computations yield \bar{w}_9=0) and generalized Dold manifolds (where only limited n are covered). These attempts underscore the gap for n ≡ 0 (mod 4), n not a power of two, motivating the open problem of producing explicit constructions in these remaining cases.