Span of Dold Manifolds

On parallelizability and span of the Dold manifolds

“…Writing r = c +4d, where 0 ≤ c ≤ 3 and d ≥ 1, one knows that 1 + span(S m ) = 2 c + 8d, and it is readily seen that 2 c + 8d < 2 c+4d = 2 r for all d ≥ 1. In other words, the lower bound 2 c + 8d for span(P (m, 1)) given in [15] is, for m ≡ 15 (mod 16), always smaller than the smallest upper bound 2 r implied by the highest non-vanishing Stiefel-Whitney class of P (m, 1). For instance, if m = 2 k+4 + 15 for some k ≥ 1, then r = 4, c = 0, d = 1; and by Theorem 4.1 we know that span(P (m, 1)) = 9, while the best "Stiefel-Whitney" upper estimate is span(P (m, 1)) ≤ 16.…”

“…Cases m = 16t + i, i = 1, 3, 5, 7, 9, 11, 13. In all these cases (which were already covered, in a different way, in [15]) we have d = 0. Clearly, if i = 1, 5, 9, 13, then c = 1; if i = 3, 11, then c = 2; and if i = 7, then c = 3.…”

“…Finally, in Section 4, we apply (8) to determine the span of the manifolds P (m, 1), including those having m ≡ 15 (mod 16). The latter manifolds were not covered by Novotný's results in [15,Section 4].…”

“…In particular, P (m, 0) is the m-dimensional real projective space RP m and P (0, n) is CP n . For further information and details about this family of manifolds, in addition to [4], we recommend [14], [15], [16], [18].…”

“…In his Ph.D. thesis (partially published in [15]), Novotný determined the number span(P (m, 1)) for all m ≡ 15 (mod 16). We complete his results by proving the following.…”