Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are smooth locally closed subvarieties and $$\overline{X_{\beta}}\subseteq\bigcup_{\gamma\leq\beta}X_{\gamma}.$$ For fixed $\beta\in S$, we have the equivariant Thom-Gysin sequences $$\ldots\rightarrow H_T^{i-2d(\beta)}(X_{\beta})\rightarrow H_T^i(\bigcup_{\gamma\geq\beta}X_{\gamma})\rightarrow H_T^i(\bigcup_{\gamma>\beta}X_{\gamma})\rightarrow\ldots,$$ where $d(\beta)$ is the complex codimension of $X_{\beta}$ in $X$. The idea is to inductively compute $H_T^*(X)$ from a knowledge of $H_T^*(X_{\beta})$ for each $\beta\in S$. This seems eminently possible if the partial order on $S$ is a total order. In this case, $X=\bigcup_{\gamma\leq\beta}X_{\gamma}$, where $\beta\in S$ is the maximal element.

However, there are some interesting examples in which $S$ is not totally ordered. Consider the nilpotent cone $\mathcal{N}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$. The nilpotent cone has a stratification into the nilpotent $G$-orbits, where $G$ is the simply-connected group with Lie algebra $\mathfrak{g}$.

$\textbf{Question}:$ Are there some general ways which to inductively compute $H_T^*(X)$ from the $H_T^*(X_{\beta})$ assuming only that $S$ is partially ordered? Are there some nice examples of this in the literature?