Abstract
Circle actions on pseudomanifolds have been studied in Padilla and Saralegi-Aranguren (Topol Appl 154:2764–2770, 2007) by using intersection cohomology (see also Hector and Saralegi in Trans Am Math Soc 338:263–288, 1993). In this paper, we continue that study using a more powerful tool, the equivariant intersection cohomology (Brylinski in Equivariant intersection cohomology, American Mathematical Society, Providence, 1992; Joshua in Math Z 195:239–253, 1987). In this paper, we prove that the orbit space \(B\) and the Euler class of the action \(\Phi :{\mathbb{S }}^{1} \times X \rightarrow X\) determine both the equivariant intersection cohomology of the pseudomanifold \(X\) and its localization. We also construct a spectral sequence converging to the equivariant intersection cohomology of \(X\) whose third term is described in terms of the intersection cohomology of \(B\).
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Notes
\(\Lambda \text{ e}={H}^{{*}}{\left( \mathbb C \mathbb P ^\infty \right)}\).
As Allday pointed out to us, this spectral sequence degenerates into the Skjelbred exact sequence of [13] when \(\overline{p}=\overline{0}\) (cf. Proposition 5.5).
See [10, sec. 1.2] for some examples. Notice that, as Friedman pointed out in [3], there is a misprint in [10, sec. 1.1] in the definition of perverse stratum: it should be \({H}^{{*}}{\left( L_S \backslash \Sigma _{L_S} \right)} \not = {H}^{{*}}{\left( (L_S \backslash \Sigma _{L_S})/{\mathbb{S }}^{1} \right)} \otimes {H}^{{*}}{\left( {\mathbb{S }}^{1} \right)}\), where \(\Sigma _{L_S}\) is the singular part of the link \(L_{S}\). That definition is equivalent to the one we give above.
The Gysin sequence for intersection cohomology has been constructed in [9, sec. 6]. In this article, we use the notations of [10, sec. 1.3]. Notice that, as Friedman pointed out in [3], in the definition of \(\mathcal{G }^*_{\bar{p}}(B)\) given in [10, sec. 1.3], the degree should be shifted by 1, so that \(\mathcal{G }^{*}_{\bar{p}}(B) \subset {\Omega _{{\bar{p}}-{\bar{x}}}(B)}\).
Of the orbit space \(B\).
See (cf. [10, sec. 3.1]) for details.
An element of \(\mathcal{K }^{{*}}_{{\overline{p}}}{\left( B \right)}\) is written \(\overline{\alpha }\) where \(\alpha \in {\Omega }^{{*}}_{{\overline{p}}}{\left( B \right)}.\)
When \(X\) is a manifold, the family of strata \(\mathbb{S }_X\) is reduced to the regular stratum.
In this range the intersection cohomology of \(X\) coincides with its cohomology (see for example [12]).
References
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Acknowledgments
This work has been partially supported by the UPV/EHU Grant EHU09/04 and by the Spanish MICINN Grant MTM2010-15471. The authors wish to thank the referee for the indications given in order to improve this paper.
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Royo Prieto, J.I., Saralegi-Aranguren, M.E. Equivariant intersection cohomology of the circle actions. RACSAM 108, 49–62 (2014). https://doi.org/10.1007/s13398-012-0097-z
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DOI: https://doi.org/10.1007/s13398-012-0097-z