Why is second homology/fundamental group of $RP^3$ trivial? I'm looking at a space that I believe is the real projective space $RP^3$. It seems that cutting the space in half with a surface should furnish a non-trivial second homology element and a non-trivial second fundamental group element. By "cutting in half" I am thinking of $RP^3$ as a three ball with opposite points identified, and then slicing this guy down the middle. So I don't see how such a surface can be the boundary of a volume, nor do I see how to contract it. All the references seem to agree that both these groups are trivial so I'm obviously missing something. Perhaps I am being derailed by the intuition from $RP^2$ where it seems clear that $H_1(RP^2)=\pi_1(RP^2)=Z_2$. 

Edit: just to clarify, I'm looking for a geometric intuition here, not a calculation, unless a calculation can show me for example an explicit homotopy to the trivial element or an explicit volume for which this surface is the boundary of.

Another Edit: I've just realized that this "cutting the space in half" produces a surface that is not a 2-sphere (which is what lives in the second fundamental group) but actually it is a $RP^2.$ So even if this "cut" is not contractible (as i believe it is not) it says nothing about $\pi_2(RP^3).$
Write the Serre exact sequence attached to $Z/2\rightarrow S^3\rightarrow RP^3$, you have $\pi_2(S^3)\rightarrow \pi_2(RP^3)\rightarrow \pi_1(Z/2)$, since $\pi_2(S^3)=\pi_1(Z/2)=1$, you have $\pi_2(RP^3)=1$.

For the homology see the link, the standard way to compute the homology of a projective space is to use a cellular decomposition.

http://topospaces.subwiki.org/wiki/Homology_of_real_projective_space
Write the Serre exact sequence attached to $Z/2\rightarrow S^3\rightarrow RP^3$, you have $\pi_2(S^3)\rightarrow \pi_2(RP^3)\rightarrow \pi_1(Z/2)$, since $\pi_2(S^3)=\pi_1(Z/2)=1$, you have $\pi_2(RP^3)=1$.

For the homology see the link, the standard way to compute the homology of a projective space is to use a cellular decomposition.

http://topospaces.subwiki.org/wiki/Homology_of_real_projective_space