On a Problem of Mazurkiewicz concerning the Boundary of a Covering Surface.

Proof: Jf there exists a closed integral curve, then both r* and C* are in the kernel of F*, so F*0. If there is no closed integral curve, let r' be a cycle without contact.3 Any integral curve of F meets r' infinitely often; let P be a point of accumulation of these intersections. We may then construct a C' closed curve C' which differs from the integral curve only in an arbitrarily small neighborhood of P. C' and r' always cross in the same direction; hence their-intersection number is non-zero, so that they neither bound nor are homologous. The winding number of r' is zero because it is without contact, while that of C' is zero because it differs from an integral curve by a small amount. Hence F*-0 as before. COROLLARY 1. If there exists on T2 any curve whose winding number with respect to F is non-zero, then there exists a closed integral curve of F. 3. Making use of the cycle r, we may generalize the rotation number X(r), which is classically defined for the case that there exists a cycle without contact.2 3 Let r and r, form a basis for H,(T2), and let F*(r) = it, F*(r,) = j{. PROPOSITION. A (r) = i and x(r) = -j/i, the latter holding only in case ,A(r) $ 0. By considering the number of points of tangency on the cycle r, we may classify qualitatively the various kinds of integral curve families of non-oriented lineelement fields.4 From this classification, the following corollaries are immediate: COROLLARY 2. The non-oriented line element field F is orientable if and only if i andj are both even. COROLLARY 3. The number of closed integral curves is at least the greatest common divisor of i and j.