In any fixed closed oriented 4-manifold, same-sign mod-2-null families of disjoint nonorientable surfaces have total normal-Euler excess bounded by a constant depending only on the manifold.
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Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold
In any fixed closed oriented 4-manifold, same-sign mod-2-null families of disjoint nonorientable surfaces have total normal-Euler excess bounded by a constant depending only on the manifold.