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Stratified operations on maniplexes

There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reﬂexible. Many such polytopes and maniplexes arise as the result of constructions such as truncations and products. Here we show that for a wide variety of these constructions, the connection group of the output can be described in a nice way in terms of the connection group of the input. We call such operations stratiﬁed . Moreover, we show that, if F is a maniplex operation in one of two broad subclasses of stratiﬁed operations, and if R is the smallest reﬂexible cover of some maniplex M , then the connection group of F ( R ) is equal to the connection group of F ( M ). In particular, we show that this is true for truncations and medials of maps, for products of polytopes (including pyramids and prisms over polytopes), and for the mix of maniplexes. As an application, we determine the smallest reﬂexible covers of the pyramids over the equivelar toroidal maps.