I'm studying singular homology, and have come across the term acyclic. I understood that a contractible space is acyclic, but can't find a counterexample showing the inverse is false.

Could anyone suggest me (hopefully the easiest) example?


2 Answers 2


You can consider the exterior of the Alexander Horned Sphere. (see Hatcher-Algebraic Topology, p.169)

  • $\begingroup$ ah, that is a much better example :) $\endgroup$ Nov 14, 2014 at 10:52

A counterexample is the punctured Poincare Homology Sphere.

  • 2
    $\begingroup$ Maybe the punctured Poincare homology sphere $\endgroup$
    – Dario
    Nov 14, 2014 at 10:46
  • $\begingroup$ @Dario fixed, thanks :) $\endgroup$ Nov 14, 2014 at 10:58

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