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Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots drawn on a sheet of paper, and lasts at most 3p-1 moves: the player who makes the last move wins. Sprouts is a very intricate game and the best known manual analysis only achieved to find a winning strategy up to p=7 spots. Recent computer analysis reached up to p=32. The standard game is played on a plane, or equivalently on a sphere. In this article, we generalize and study the game on any compact surface. First, we describe the possible moves on a compact surface, and the way to implement them in a program. Then, we show that we only need to consider a finite number of surfaces to analyze the game with p spots on any compact surface: if we take a surface with a genus greater than some limit genus, then the game on this surface is equivalent to the game on some smaller surface. Finally, with computer calculation, we observe that the winning player on orientable surfaces seems to be always the same one as on a plane, whereas there are significant differences on non-orientable surfaces.
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Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-1 moves. The first player who cannot play loses. The complexity of the p-spot game is very high, so that the best hand-checked proof only shows who the winner is for the 7-spot game, and the best previous computer analysis reached p=11. We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague-Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we reached up to p=32. The outcome of the 33-spot game is still unknown, but the biggest computed value is the 47-spot game ! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if p is 3, 4 or 5 modulo 6. We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until p=11 in a reasonable amount of time.
Consider a group word w in n letters. For a compact group G, w induces a map G^n rightarrow G$ and thus a pushforward measure {mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsens classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of ErdH{o}s and Turan on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schurs lemma, basic counting, and a divisibility argument.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
We prove that if there are $mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $kappa$ such that $kappa^omega=kappa$, there exists a group topology on the free Abelian group of cardinality $kappa$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
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