= Euler characteristic with different coefficient fields
{c}
{title2=$\chi(\mathbb{RP}^2)=1$}
For a concrete example of <cellular homology of real projective space>, the <real projective plane> $\mathbb{RP}^2$ has one <topological cell> in each of <dimensions> zero, one and two. Its one-dimensional <topological cell> is a loop, and the face attaches around that loop twice. Choose orientations so that the <cellular chain complex> over the <integers> is
$$
0\longrightarrow\mathbb Z\xrightarrow{\;2\;}\mathbb Z\xrightarrow{\;0\;}\mathbb Z\longrightarrow0,
$$
with nonzero <chain groups> in degrees two, one and zero. The twice-around attachment is why the first <boundary operator> shown is multiplication by two.<Algebraic Topology by Allen Hatcher>[Algebraic Topology by Allen Hatcher, §2.2, Example 2.42, printed p. 144: real projective cellular boundary maps]{ref}
Over the <rational numbers>, multiplication by two is invertible, so $H_2$ and $H_1$ vanish, while $H_0\cong\mathbb Q$. Over the <finite field> $\mathbb F_2$, two equals zero. Both <boundary operators> then vanish, giving one-dimensional <homology groups> in all three degrees. Their respective <Betti numbers> are
$$
(b_0,b_1,b_2)_{\mathbb Q}=(1,0,0),\qquad
(b_0,b_1,b_2)_{\mathbb F_2}=(1,1,1).
$$
Nevertheless,
$$
\boxed{\chi(\mathbb{RP}^2)=1-0+0=1-1+1=1.}
$$
The direct cell count is also $1-1+1=1$.
\Image[/euler-characteristic-projective-plane-coefficients.png]
{title=Changing coefficients changes the projective plane's homology groups, but their alternating dimension sum remains 1}
This calculation makes the strength and loss of information visible. The <Euler characteristic> survives a change of the <field> used for coefficients, while individual <homology groups> can change. Over the <integers>, the same <chain complex> has $H_1\cong\mathbb Z/2\mathbb Z$ and $H_2=0$: a <torsion subgroup> that rational <Betti numbers> do not record. The <rank cancellation in the Euler-Poincare formula> explains why the alternating total still agrees with the cell count.
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