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It is common to consider such a space as a nonempty 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 closed
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loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." The condition of acyclicity on a space
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Berrick, A. Jon; Hillman, Jonathan A. (2003), "Perfect and acyclic subgroups of finitely presentable groups",
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one can associate a (canonical, terminal) acyclic space, whose fundamental group is a
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The homotopy groups of these associated acyclic spaces are closely related to
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This gives a repertoire of examples, since the first homology group is the
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are isomorphic to the corresponding homology groups of a point.
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594:{\displaystyle H_{1}(G;\mathbf {Z} )=H_{2}(G;\mathbf {Z} )=0}
167:{\displaystyle {\tilde {H}}_{i}(X)=0,\quad \forall i\geq -1.}
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to the circle or to the higher spheres is null-homotopic.
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do not vanish in general, because the fundamental group
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in which cycles are always boundaries, in the sense of
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may be too technical for most readers to understand
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430:{\displaystyle {\tilde {H}}_{i}(G;\mathbf {Z} )=0}
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605:is superperfect (hence perfect) but not acyclic.
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657:Dror, Emmanuel (1973), "Homology spheres",
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59:Learn how and when to remove this message
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510:{\displaystyle H_{1}(G;\mathbf {Z} )=0}
318:of the fundamental group. With every
41:make it understandable to non-experts
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311:which is not contractible.
296:{\displaystyle \pi _{1}(X)}
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363:is a group
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754:Categories
621:References
437:, for all
207:CW complex
743:EMS Press
448:≥
393:~
276:π
159:−
156:≥
150:∀
119:~
49:June 2012
719:30232002
630:Topology
609:See also
376:homology
250:manifold
243:topology
237:Examples
229:and the
745:, 2001
711:2009444
681:0328926
651:0315713
344:on the
338:Quillen
35:Please
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