finding an acyclic but not contractible space

Question

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?

Answer 1

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

Answer 2

A counterexample is the punctured Poincare Homology Sphere.