We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some families of generalized Petersen graphs. We also extend these ideas to prove that the cop number of any connected I-graph is at most 5. 1 arXiv:1509.04696v1 [math.CO]

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The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad O in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total Oalgebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower is a pro-π * isomorphism. More generally, we also show that similar results remain true if we replace "homotopy fibration sequence" with "homotopy pullback square."

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