Acknowledgments: We Would like to Thank 7.1 Renaming

Captured in our combinatorial topology language, the k-set consensus task is the triple hD n ; D n ; i. D n is P(D), for some domain D, and maps each n 2 D n to the subset of n-simplexes in D n that contain at most k diierent values from the values in n. In the full version of the paper, we use Theorem 12 and Sperner's Lemma to prove that any wait-free protocol for this problem must have at least one execution in which k +1 diierent values are output. This implies: Theorem22. If k n then there does not exists a wait-free protocol that solves the k-set consensus task. This bound is tight, by the protocol of 7]. 8 Discussion This paper presents a study of wait-free solvability based on combinatorial topol-ogy. Informally, we have deened the notion of a chromatic divided image, and proved that a necessary condition for wait-free solvability is the existence of a simplicial chromatic mapping from a divided image of the inputs to the outputs that agrees with the problem speciication. We were able to use theorems about combinatorial properties of divided images to derive tight lower bounds for renaming and k-set consensus. Our results do not use homology groups, whose computation may be complicated. We also derive a new necessary and suucient condition, based on a speciic, well structured chromatic divided image. Many questions remain open. First, it is of interest to nd other applications to the necessary and suucient condition presented here; in particular, can we derive interesting protocols from the suucient condition? Second, there are several directions to extend our framework, e.g., to allow fewer than n failures (as was done for one failure in 3]), to handle other primitive objects besides read/write registers (cf. 14, 6]), and to incorporate ongoing tasks. In the renaming task ((2]), processes start with an input value (original name) from a large domain and are required to decide on distinct output values (new names) from a domain which should be as small as possible. Clearly, the task is trivial if processes can access their id; in this case, process p i decides on i, which yields the smallest possible domain. To avoid trivial solutions, it is required that the processes and the protocol are anonymous 2]. That is, process p i with original name x executes the same protocol as process p j …