## 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

### Keywords:

Dold manifold; span; stable span; vector field; Stiefel-Whitney class
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