Abstract
Torsion-sensitive intersection homology was introduced to unify several versions of Poincaré duality for stratified spaces into a single theorem. This unified duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. In this paper we consider for torsion-sensitive intersection homology analogues of another important property of classical intersection homology: topological invariance. In other words, we consider to what extent the defining sheaf complexes of the theory are independent (up to quasi-isomorphism) of choice of stratification. In addition to providing torsion sensitive versions of the existing invariance theorems for classical intersection homology, our techniques provide some new results even in the classical setting.
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Notes
Even though \(\mathcal {E}\) is a complex of sheaves, we do not write \(\mathcal {E}^*\) in order to emphasize the role of \(\mathcal {E}\) as coefficients.
This is no longer true if the local system is only defined on a dense open set whose complement has codimension 1. For example let \(X=S^1\) with stratification \(S^1\supset \{pt\}\). Suppose E is the constant sheaf with stalk \(\mathbb {Z}\) on \(S^1-\{pt\}\). Then there are two non-isomorphic extensions of E to \(S^1\), namely the constant sheaf with stalk \(\mathbb {Z}\) and the twisted sheaf with stalks \(\mathbb {Z}\) such that a generator of \(\pi _1(S^1)\) acts by multiplication by \(-1\).
We see in this argument why \(\vec {p}_1(S)\ge -1\) is required, as well as our torsion assumptions; see also Sect. 4.2, below.
We see in this argument why \(\vec {p}_1(S)\le \text {codim}(S)-1\) is required, as well as our torsion-free assumptions; see also Sect. 4.2, below.
These \(\tilde{S}\) are called source strata in [5]. Note that the precise choice of \(\tilde{S}\) does not matter in the conditions that follow as the assumption that \(\vec {p} \) can be pushed forward assures us that any two choices would give the same perversity conditions.
In fact, since \(H^1(\mathcal {E}_x)\) is torsion, we can treat it as a module over \(R/\text {Ann}(H^1(\mathcal {E}_x))\). If \(H^1(\mathcal {E}_x)\ne 0\) then Ann\((H^1(\mathcal {E}_x))\ne 0\), and this is an Artinian ring since R is a PID. If \(H^1(\mathcal {E}_x)=0\), it is clearly Artinian.
We remark that if \(\vec {p}_2(k)=\emptyset \) for all k, which corresponds to \(\mathscr {S}^*\) satisfying the original Goresky–MacPherson axioms, then \(\vec {q}_2(k)=P(R)\) for all k. In this case \(q^{-1}(n-j+1,\mathfrak p)=q^{-1}(n-j,0)\) by Remark 5.3, which is again consistent with the expectation from the classical case.
A version of such a construction for fairly general sheaf complexes can be found in Habegger–Saper [16, Section 3].
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This work was partially supported by a Grant from the Simons Foundation (#839707 to Greg Friedman).
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Friedman, G. Topological invariance of torsion-sensitive intersection homology. Geom Dedicata 217, 105 (2023). https://doi.org/10.1007/s10711-023-00843-6
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DOI: https://doi.org/10.1007/s10711-023-00843-6