  Consensus is one of the most fundamental problems in distributed computing.
This paper studies the consensus problem in a synchronous dynamic directed
network, in which communication is controlled by an oblivious message
adversary. The question when consensus is possible in this model has already
been studied thoroughly in the literature from a combinatorial perspective, and
is known to be challenging. This paper presents a topological perspective on
consensus solvability under oblivious message adversaries, which provides
interesting new insights. Our main contribution is a topological
characterization of consensus solvability, which also leads to explicit
decision procedures. Our approach is based on the novel notion of a
communication pseudosphere, which can be seen as the message-passing analog of
the well-known standard chromatic subdivision for wait-free shared memory
systems. We further push the elegance and expressiveness of the "geometric"
reasoning enabled by the topological approach by dealing with uninterpreted
complexes, which considerably reduce the size of the protocol complex, and by
labeling facets with information flow arrows, which give an intuitive meaning
to the implicit epistemic status of the faces in a protocol complex.
