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Span of Dold manifolds. (English) Zbl 1161.57016

For each pair \((m,n)\) of nonnegative integers, let \(S^m \times \mathbb{C}^n/\sim\), with \((x,z) \sim (-x, \overline{z})\), be the Dold manifold \(P(m,n).\) Its span is the maximum number of everywhere linearly independent vector fields on \(P(m,n)\), denoted by \(\operatorname{span} P(m,n)\) and its stable span is the number \(\operatorname{span} (P(m,n) \times S^1)-1\), denoted by \(\operatorname{span} P(m,n).\)
Denoting by \(2^{\nu(t)}\) the highest power of \(2\) dividing the positive integer \(t\), it is proved that
\[ \text{(stab)span\,} P(m,n) \leq 2^{\nu(n+1)}(2^{\nu(k+1)} + 1 ) -2, \]
where \( m= 2^{\nu(n+1)}\). \(k + l\), with \( 0 \leq l < 2^{\nu(n+1)}.\)
Interesting conclusions follow from these upper bounds and from further results on \(P(m,n).\) They include exact values for \(\text{(stab)span\,} P(m,n)\) in some particular cases, as well as examples showing the improvement of the results for \(P(1,7)\) and \(P(9,7)\).
In the case \(P(m,1), \, m \not\equiv 15\pmod {16}\), \( m\) odd, it is presented how the \(\operatorname{span} P(m,1)\) can be constructed.

MSC:

57R25 Vector fields, frame fields in differential topology
55S40 Sectioning fiber spaces and bundles in algebraic topology
57R20 Characteristic classes and numbers in differential topology
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Full Text: Euclid