Topological Intuition behind the 2-Homology of $RP^2$

Question

I believe that, intuitively, one can think of the 2-homology of a space as how many ways one can wrap a surface around it and close it up such that it's injective. For example, one can take the entire torus or sphere as the surface that wraps around them to give a generator of the 2-homology.

On the other hand, one cannot do this for the Klein bottle, a cylinder/Möbius strip, a circle, and these can therefore be seen to have a trivial 2-homology.

However, this intuition is struggling when considering the real projective plane. The 2-homology of the real projective plane is trivial, however I'm struggling to see why wrapping the entire surface around it would not give such a generator. Perhaps such a wrapping does not separate two spaces, as with the Klein bottle? Is this because the real projective plane cannot be embedded into $\mathbb{R}^3$?

Answer 1

Honestly, your definition of "wrapping around a surface" is not clear to me. I will list here what geometric meaning you can give to the homology groups of a manifold.

The main "geometric" interpretation is the following theorem you can find on any book of Algebraic Top:

Thm: Let $M$ be a connected $n$-manifold without boundary. Then $H_n(M;\Bbb Z)$ is $\Bbb Z$ if and only if $M$ is closed and orientable, $0$ otherwise.

A geometric intuition (which holds in the smooth case) of this theorem is that in the orientable case, starting from an oriented triangulation of the manifold, you can build a non-exact closed $n$-simplex (could this count as a kind of wrapping?) which generates the top homology group.

you can weak the orientability requirement if you work in $\Bbb Z_2$ coefficient, and you obtain that, with the same hypothesis:

Thm: Let $M$ be a connected $n$-manifold without boundary. Then $H_n(M;\Bbb Z_2)$ is $\Bbb Z_2$ if and only if $M$ is closed. $0$ otherwise.

So you see that in your case $H_2(\Bbb RP^2; \Bbb Z)=0$ detects the non-orientability of the projective space.

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Cellular homology should prove another visual interpretation close to your concept of "wrapping", because it's the homology of the chain complex generated by the cells, whose differentials are the degree of certain maps, so in fact in you case you are studying the $2$-cells (the usual disks) and how they attach to the $1$-skeleton of $\Bbb RP^2$. In our case, you see that the unique $2$-cell in the standard CW-structure of $\Bbb RP^2$ is attached to the unique $1$-cell via a degree $2$ map, therefore you get the $0$ in the second homology group. Cellular homology makes cristal clear why the circle has trivial second (and higher) homology groups: it admits a CW-structure with only 0 and 1 cells

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In the simply connected case, $H_2(M;\Bbb Z)$ is isomorphic to the second homotopy group $\pi_2(M)$, which is the group of pointed homotopy classes of maps $S^2\to M$, via the Hurewicz map