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?

score 6 · Accepted Answer · answered Nov 14 '14 at 10:50

6

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

answered Nov 14 '14 at 10:50

Dario

5,749
2
24
36

ah, that is a much better example :) – Loreno Heer Nov 14 '14 at 10:52

Loreno Heer · Answer 2 · 2014-11-14T10:57:00.030

4

A counterexample is the punctured Poincare Homology Sphere.

edited Nov 14 '14 at 10:57

answered Nov 14 '14 at 10:43

Loreno Heer

4,460

2

Maybe the punctured Poincare homology sphere – Dario Nov 14 '14 at 10:46
@Dario fixed, thanks :) – Loreno Heer Nov 14 '14 at 10:58

finding an acyclic but not contractible space

2 Answers2