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2 results

  • 8 - Poincaré Duality

  • Greg Friedman, Texas Christian University
  • Book:
    Singular Intersection Homology
    Published online:
    18 September 2020
    Print publication:
    24 September 2020, pp 498-612

  • Summary

    We show that oriented pseudomanifolds possess fundamental intersection homology classes, and we prove that intersection homology possesses a Poincaré duality given by the cap product with the fundamental class. We also prove Lefschetz duality for pseudomanifolds with boundary. We derive from both of these dualities nonsingular cup product and torsion pairings. We include an expositional survey of intersection pairings and the original approach of Goresky and MacPherson to intersection homology duality using such pairings.

  • Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

  • Hector Cordova Bulens, Pascal Lambrechts, Don Stanley
  • Journal:
    Canadian Journal of Mathematics / Volume 70 / Issue 2 / 01 April 2018
    Published online by Cambridge University Press:
    20 November 2018, pp. 265-293
    Print publication:
    01 April 2018
    • Article
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  • Let $W$ be a compact simply connected triangulated manifold with boundary and let $K\,\subset \,W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of $W\text{ }\!\!\backslash\!\!\text{ K}$ out of a model of the map of pairs $\left( K,\,K\cap \partial W \right)\,\to \,\left( W,\,\partial W \right)$ under some high codimension hypothesis.

    We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.