I'm trying to compute the Borel-Moore homology (which, abusing notation, we write $H_i$ for the $i$-th Borel-Moore homology group with coefficients in $\mathbb{C}$ - which is usually denoted by $H_i^{BM}(-;\mathbb{C})$) of the nilcone $\mathcal{N}$ of $\mathfrak{sl}_2(\mathbb{C})$, and I'm wondering if my argument works. My idea is to use the Springer resolution $\pi: \underset{= T^* \mathbb{P}^1}{\tilde{\mathcal{N}}} \rightarrow \mathcal{N}$, and then proceed roughly as follows.
There exists a long exact sequence
$$ \ldots \rightarrow H_k(\mathbb{P}^1) \overset{\phi}{\rightarrow} H_k(T^*\mathbb{P}^1) \oplus H_k(0)\rightarrow H_k(\mathcal{N}) \rightarrow \ldots $$
which can be obtained resulting from a corresponding short exact sequence on Borel-Moore chain level.
Using $H_i(\mathbb{P}^1) = \mathbb{C}$ if $i=0$ or $i=2$, and $0$ otherwise, and $H_i(T^*\mathbb{P}^1) = \mathbb{C}$ if $i=2$ or $i=4$, and $0$ otherwise (which we obtain via the isomorphism $H_i(T^* \mathbb{P}^1) \cong H^{ordinary}_{i-2}(\mathbb{P}^1)$ since $T^* \mathbb{P}^1$ is the total space of the complex line bundle over the compact base $\mathbb{P}^1$) we obtain that $$ H_i(\mathcal{N}) = \begin{cases} \mathbb{C} &\text{if $i=4$} \\ 0 &\text{otherwise} \end{cases}. $$
Here, we use that we can show that the map $\phi$ (induced by the inclusion $\mathbb{P}^1 \hookrightarrow T^*\mathbb{P}^1$) is non-zero and hence and isomorphism. Thanks in advance!
