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.
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