Finite-to-one mappings on infinite-dimensional compacta

Weakly infinite-dimensional compacta can be classified by means of essential maps onto ''transfinite cubes'' (Smirnov's compacta). We investigate the behavior of this classification under finite-to-one mappings. In particular, we show that this topic is closely related to an open problem about the invariance of strong infinite-dimensionality under light mappings on compacta. We provide also an analogue for the transfinite dimension ind of Hurewicz's theorem on dimension-rising mappings.