On topological space whose homology groups/modules are trivial

Question

Let $X$ be a simply connected topological space. Consider the following three statements :

(1) $X$ is contractible.

(2) For every commutative ring $R$, $H_0(X,R)=R$ and $H_n(X,R)=0, \forall n \ge 1$

(3) $H_0(X,\mathbb Z)=\mathbb Z$ and $H_n(X,\mathbb Z)=0,\forall n \ge 1$

Now we know that $(1) \implies (2) \implies (3)$.

My question is: Is any of these implications reversible ?

Answer 1

By the Hurewicz theorem, (3) (together with simple connectedness) implies all homotopy groups of $X$ are trivial (if some homotopy group is nontrivial, the least nontrivial homotopy group would give a nontrivial homology group). If $X$ has the homotopy type of a CW-complex, then $X$ is contractible by Whitehead's theorem. So both implications are reversible if $X$ has the homotopy type of a CW-complex.

For arbitrary simply connected spaces, the second implication is reversible but not the first. Indeed, (3) implies (2) immediately by the universal coefficient theorem. Some examples of simply connected spaces that satisfy (3) (and thus (2)) but are not contractible are the (open) long line and the Warsaw circle.

Answer 2

(3) implies (2) by the Universal Coefficient Theorem.

I'm pretty sure that (2) does not imply (1); that would make topology too easy....