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Acyclins might refer to |
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In mathematics, an Acyclic space is a topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point. * In other words, using the idea of reduced homology,* * * * * * * * H * ~ * * * * * i * * * ( * X * ) * = * 0 * , * * ∀ * i * ≥ * 0. * * * {\displaystyle {\tilde {H}}_{i}(X)=0,\quad \forall i\geq 0.} * It is common to consider such a space as a space without "holes," for example, a circle or a sphere is not acyclic but a disc * or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any close loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." * The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic. * If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem. |