On the toric lifting properties for simplicial $3$-spheres

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X({\mathbb{Z}}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.