I'm very new at algebraic topology. I'm trying to compute the Euler characteristic of $\mathbb{CP}^2 \# \mathbb{CP}^2$ by using the following known facts. ($\mathbb{CP}^2 \# \mathbb{CP}^2$ represents the connected sum of two complex projective planes.)
Fact 1
Let $M_1$, $M_2$ be connected smooth manifolds of dimension $n\geq3$, and let $M_1\# M_2$ denote their smooth connected
sum. Then $H_{\rm dR}^p(M_1 \# M_2)\cong H_{\rm
dR}^p(M_1)\oplus H_{\rm dR}^p(M_2)$ for $0<p<n-1$. And the same
is true for $p=n-1$ if $M_1$ and $M_2$ are both compact and
orientable.
Fact 2
$$H_{\rm dR}^p(\mathbb{CP}^2)\cong\left\{ \begin{array}{ll} \mathbb R & p=0,2,4, \\ 0& p=1,3. \end{array} \right.$$
Fact 3
$$\chi(M)=\sum_{p=0}^n (-1)^p \dim H_{\rm dR}^p(M).$$
According to fact 1 and fact 2,
$$\dim H_{\rm dR}^p(\mathbb{CP}^2)\cong\left\{ \begin{array}{ll} 1 & p=0,2,4, \\ 0& p=1,3, \end{array} \right.$$
and since $\mathbb{CP}^2$ is compact and orientable,
$$\dim H_{\rm dR}^p(\mathbb{CP}^2 \# \mathbb{CP}^2)\overset{p=1,2,3}{=}\dim H_{\rm
dR}^p(\mathbb{CP}^2)\oplus H_{\rm dR}^p(\mathbb{CP}^2)=2\dim H_{\rm
dR}^p(\mathbb{CP}^2)=\left\{ \begin{array}{ll} 2 & p=2, \\ 0& p=1,3. \end{array} \right.$$
Moreover, $\dim H_{\rm dR}^0(\mathbb{CP}^2 \# \mathbb{CP}^2)=1$ since $\mathbb{CP}^2 \# \mathbb{CP}^2$ is connected, and $\dim H_{\rm dR}^4(\mathbb{CP}^2 \# \mathbb{CP}^2)=1$ since $\mathbb{CP}^2 \# \mathbb{CP}^2$ is compact, connected and orientable.
Therefore,
\begin{equation*}
\begin{aligned}
\chi(\mathbb{CP}^2 \# \mathbb{CP}^2)&=\sum_{p=0}^4 (-1)^p \dim H_{\rm dR}^p(\mathbb{CP}^2 \# \mathbb{CP}^2)\\
&=1-0+2-0+1\\
&=4.
\end{aligned}
\end{equation*}
Is this computation correct? Thank you for your criticism and correction.
Best Answer
Just to get this off of the unanswered list: your computation is correct.