What if I want to look for a space with vanishing first homology but nonzero fundamental group? Dear all, it seems any domain in Euclidean space, that is, connected open set will not make it. Am I right or could anyone give me a reference?? 
What should I do if I want to construct a space with fundamental group A5?
 A: For any group $G$ there is a simple explicit construction of a 2-dimensional (connected)  CW-complex $X$ such that $\Pi_1(X)=G$. This can be found in [Hatcher: Algebraic Topology, Corollary 1.28]. 
As an example take the group $G = A_5$ mentioned in the question. Then proceed the following steps: 


*

*Choose a presentation, e.g. $A_5 = \langle a,b\mid a^2=b^3=(ab)^5=1\rangle$

*Form the bouquet $S^1 \vee S^1$ and consider $a$ resp. $b$ as loop around the corresponding  $S^1$.  

*For each relation attach a 2-cell with attaching map just the relation-loop, i.e. we attach 


*

*a 2-cell $D_1$ with attaching map $a^2=a \ast a$ (where $\ast$ is composition) 

*a 2-cell $D_2$ with attaching map $b^3$

*a 2-cell $D_3$ with attaching map $(ab)^5$



In summary we have $X=S^1_a \vee S^1_b \coprod_{a^2} D_1 \coprod_{b^3} D_2 \coprod_{(ab)^5} D_3$. It has one 0-cell, two 1-cells and three 2-cells. 
In general each generator of $G$ corresponds to one copy of $S^1$ and each relation corresponds to one attached 2-cell. 
A: If you want to look for such a space, look at the Wikipedia entry for Homology Sphere. The conjecture in the first sentence is false (take any homology sphere, embed it into some $\mathbb{R}^n,$ then thicken it to be an open set).
A: Your question has been answered, but I thought I'd add something. For any space $X$, $H_1(X,\mathbb{Z})$ is just the abelianization of $\pi_1(X,*)$. So, to find an example of a space with zero first homology but non-trivial fundamental group, you just need to find a space $X$ with $\pi_1$ equal to its commutator subgroup $[\pi_1,\pi_1]$ (such groups are called perfect groups). Then $H_1(X,\mathbb{Z})\simeq \pi_1^{ab} = \pi_1/[\pi_1,\pi_1] = 0$. These spaces are guaranteed to exist - see, for example, Gabriel's comment.
An example of such a space $X$, as has already been mentioned, is the homology 3-sphere. This is the quotient of $S^3$ (viewed as the group of unit quaternions) by the binary icosahedral group $2I$, a perfect (sub)group. In fact, this quotient map is a covering and exhibits $S^3$ as the universal cover of $X$. Hence $\pi_1\simeq 2I$ and $H_1 = 0$.
