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Hodge theory for intersection space cohomology. (English) Zbl 1441.55004

This paper concerns the intersection spaces introduced by the paper’s first-named author in [M. Banagl, Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (2010; Zbl 1219.55001)]. Beginning with an appropriate stratified space \(X\) and a perversity parameter \(\bar p\), one replaces \(X\) with its intersection space \(I^{\bar p}X\), when defined, and considers the intersection space cohomology \(HI_{\bar p}^*(X):=H^*(I^{\bar p}X)\). One thus obtains an invariant of the original space \(X\) that is an alternative to the Goresky-MacPherson intersection homology theory, or the Cheeger \(L^2\) cohomology theory, for studying Poincaré duality phenomena for stratified spaces. In particular, when intersection spaces can be constructed they satisfy duality isomorphisms of the form \(\tilde H^i(I^{\bar p}X;\mathbb{R})\cong \tilde H^{n-i}(I^{\bar q}X;\mathbb{R})\), where \(X\) is \(n\)-dimensional and \(\bar p\) and \(\bar q\) are complementary perversities in the sense of Goresky and MacPherson. We note that, in general, \(H^i(I^{\bar p}X;\mathbb{R})\) is not isomorphic to the perversity-\(\bar p\) intersection cohomology of \(X\).
The authors here consider spaces with two strata \(\Sigma\subset X\) such that the lower stratum has a product bundle neighborhood. For such spaces, they demonstrate that the intersection space cohomology can be obtained as the \(L^2\) cohomology of \(M:=X-\Sigma\) when equipped with an appropriate metric, in particular a metric that is a fibered scattering metric near \(\Sigma\).
Prior work on the computation of intersection space cohomology via complexes of differential forms, in this case written \(HI^*_{dR,\bar p}(X)\), is due to M. Banagl [J. Differ. Geom. 104, No. 1, 1–58 (2016; Zbl 1359.57016)] and T. Essig [About a de Rham complex describing intersection space cohomology in a non-isolated singularity case, Diplomarbeit, Ruprecht-Karls-Universität Heidelberg (2012)]. The new feature here is the introduction of a Hodge-theoretic approach. In other words, the authors show that if \(M=X-\Sigma\) is equipped with a suitable metric then \(HI^*_{dR,\bar p}(X)\) is isomorphic to an associated space of \(L^2\)-harmonic forms on \(M\).
To describe the metric more explicitly, suppose that \(\Sigma\) has a neigborhood \(\Sigma \times cL\) in \(X\), and let \(\bar M\) be the compactification of \(M\) with boundary \(\Sigma \times L\). Then the desired metrics, called product-type fibred scattering metrics, have the following form near \(\partial M\): \[ g_{fs}=\frac{dx^2}{x^4}+g_{\sigma}+\frac{g_L}{x^2}. \] Here \(g_L\) and \(g_\Sigma\) are fixed metrics on \(L\) and \(\Sigma\) and the \(x\)-coordinate is orthogonal to the boundary. For such metrics, the authors define weighted spaces of \(L^2\) differential forms \(x^cL^2_{g_{fs}}\Omega^*_{g_{fs}}(M)\), with \(c\) denoting a weight parameter, and the associated spaces of harmonic forms \(\mathcal H^*_{ext}(M,g_{fs},c)\). The main theorem of the paper is then the following:
Theorem. Let \(X\) be a (Thom-Mather) stratified pseudomanifold with smooth, connected singular stratum \(\Sigma\subset X\). Assume that the link bundle \(Y\to \Sigma\) is a product \(L\times \Sigma\to\Sigma\), where \(L\) is a smooth manifold of dimension \(l\). Let \(g_{fs}\) be an associated product-type fibred scattering metric on \(M=X-\Sigma\). Then \[ HI^*_{dR,\bar p}(X)\cong \mathcal H^*_{ext}\left(M,g_{fs},\frac{1}{2}(l-1)-\bar p(l+1)\right). \]
These spaces of harmonic forms were previously studied by the second-named author in [E. Hunsicker, J. Topol. Anal. 10, No. 3, 531–562 (2018; Zbl 1414.58003)], so one feature of this theorem is to give a topological interpretation to those analytic results.
The proof of the theorem proceeds by showing that the homology and cohomology of the intersection space \(I^{\bar p}X\) can be identified with certain groups, denoted \(IG\), of the conifold transition of \(X\) that are defined in terms of the intersection homology and intersection cohomology of this conifold transition. This intermediary then allows the authors to invoke the previous results of the second author [loc. cit.].
Finally, the authors consider an application to signatures. They show that if \(X\) is a Witt space with dimension a multiple of 4 then the signature induced by the middle-dimensional, middle-perversity intersection pairing of intersection space cohomology is equal both to the Goresky-MacPherson intersection homology signature of \(X\) and to the lower-middle perversity to upper-middle perversity perverse signature of the conifold transition of \(X\). Furthermore, these all agree with the signature of the manifold-with-boundary \(\bar M\).

MSC:

55N33 Intersection homology and cohomology in algebraic topology
58A14 Hodge theory in global analysis
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References:

[1] 10.1007/3-540-38587-8
[2] 10.1007/978-3-642-12589-8 · Zbl 1219.55001
[3] 10.4171/GGD/183 · Zbl 1275.55008
[4] 10.4310/jdg/1473186538 · Zbl 1359.57016
[5] 10.4310/ATMP.2014.v18.n2.a3 · Zbl 1405.14045
[6] 10.1007/s00208-003-0439-4 · Zbl 1034.32021
[7] 10.5427/jsing.2017.16g · Zbl 1394.55006
[8] 10.1142/S1793525312500185 · Zbl 1269.32017
[9] 10.5427/jsing.2012.5d · Zbl 1292.32017
[10] 10.1007/978-1-4757-3951-0
[11] 10.1007/BF00136813 · Zbl 0733.57010
[12] 10.1090/memo/1214 · Zbl 1418.55003
[13] 10.1073/pnas.76.5.2103 · Zbl 0411.58003
[14] 10.1090/pspum/036/573430
[15] 10.4310/jdg/1214438175 · Zbl 0529.58034
[16] 10.1515/crelle.2012.005 · Zbl 1266.55003
[17] 10.1016/j.aim.2013.02.017 · Zbl 1280.55003
[18] 10.1016/0040-9383(80)90003-8 · Zbl 0448.55004
[19] 10.1007/BF01389130 · Zbl 0529.55007
[20] 10.1215/S0012-7094-04-12233-X · Zbl 1062.58002
[21] 10.2140/gt.2007.11.1581 · Zbl 1132.58013
[22] 10.1142/S1793525318500188 · Zbl 1414.58003
[23] 10.1016/0166-8641(85)90075-6 · Zbl 0568.55003
[24] 10.1007/s00229-016-0884-5 · Zbl 1368.55005
[25] 10.1007/978-0-387-21752-9
[26] 10.1016/0377-0257(93)80040-i
[27] 10.1215/ijm/1258138217
[28] 10.2748/tmj/1178244668 · Zbl 0086.15003
[29] 10.2307/2374334 · Zbl 0547.57019
[30] 10.1007/978-1-4684-9322-1
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