# Can one use the equivariant Thom-Gysin sequence on a singular affine variety?

Let $X$ be a smooth complex affine variety. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite stratification of $X$ into smooth locally closed subvarieties. Let $T$ be a complex algebraic torus acting on $X$, and suppose that our stratification is $T$-equivariant (ie. that the strata are $T$-invariant). For fixed $\beta\in B$, there is a $T$-equivariant Thom-Gysin sequence of the form $$\ldots\rightarrow H^{i-d(\beta)}_T(X_{\beta};\mathbb{Q})\rightarrow H^i_T(\bigcup_{\gamma\leq\beta}X_{\gamma};\mathbb{Q})\rightarrow H^i_T(\bigcup_{\gamma<\beta}X_{\gamma};\mathbb{Q})\rightarrow\ldots,$$ where $d(\beta)$ is the real codimension of $X_{\beta}$ in $X$. Furthermore, if the $T$-equivariant Euler class of the normal bundle of $X_{\beta}$ in $X$ is not a zero-divisor in $H_T^*(X_{\beta};\mathbb{Q})$, then the Thom-Gysin sequence splits into short-exact sequences.

$\textbf{Question}$: To what extent can we make sense of the above if $X$ is allowed to be singular?

It occurs to me that there are two natural impediments to our proceeding when $X$ is singular. Firstly, is it clear that $d(\beta)$ will be well-defined? Secondly, will the normal bundle of $X_{\beta}$ in $X$ be well-defined so as to give a $T$-equivariant Euler class?

I would appreciate any and all references/suggestions.

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Probably you want to assume each $X_\beta$ is smooth, and has a well-defined normal cone rather than normal bundle. I don't think the $T$-equivariance plays much role in the question (other than helping to make things not be zero divisors). Worry first about the case of dilation acting on the affine cone over a projective variety. –  Allen Knutson Oct 2 '13 at 0:12