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?



Equivariant Stratifications of a Variety
This will be a partial answer because in the spirit of Christmas I'm not going to go hunt for references. :) 


1. It usually makes no difference whether the set of strata is totally or partially ordered. In the partially ordered case, put $X\_d = $ (disjoint) union of all $d$-dimensional strata.
2. You are probably better off working with compactly supported equivariant cohomology. In fact I don't think the short exact sequence you wrote down exists in the generality you claim, but with compact support it's fine. If the strata are smooth and $X$ is compact then you get a sequence also for ordinary cohomology using Poincare duality (whence the funny degree shift in your short exact sequence) and $H^\bullet(X)=H^\bullet\_c(X)$.
3. There is a spectral sequence associated to the stratification whose first page is given by the various cohomologies $H^\bullet\_{c,T}(X\_\beta)$ computing $H^\bullet\_{c,T}(X)$ (compactly supported equivariant cohomology). I don't have a reference off hand, but I know that the non-equivariant version is given a complete proof in Arapura's paper on the Leray spectral sequence; maybe it can be adapted. Other places I would start looking: the book of Chriss and Ginzburg and Kirwan's thesis.


