Intersection cohomology assigns groups which satisfy a generalized form of Poincare duality over the rationals to a stratified singular space. This monograph introduces a method that assigns to certain classes of stratified spaces cell complexes, called intersection spaces, whose ordinary rational homology satisfies generalized Poincare duality. The cornerstone of the method is a process of spatial homology truncation, whose functoriality properties are analyzed in detail. The material on truncation is autonomous and may be of independent interest tohomotopy theorists. The cohomology of intersection spaces is not isomorphic to intersection cohomology and possesses algebraic features such as perversity-internal cup-products and cohomology operations that are not generally available for intersection cohomology. A mirror-symmetric interpretation, as well as applications to string theory concerning massless D-branes arising in type IIB theory during a Calabi-Yau conifold transition, are discussed.

Iterated Truncation ; 1.7 Localization at Odd Primes; 1.8 Summary; 1.9 The Interleaf Category; 1.10 Continuity; Properties of Homology Truncation; 1.11 Fiberwise Homology Truncation; 1.12 Remarks on Perverse Links and Basic Sets Spaces; 2.1 Reflective Algebra; 2.2 The Intersection Space in the Isolated Singularities Case; 2.3 Independence of Choices of the Intersection Space Homology; 2.4 The Homotopy Type of Intersection Spaces for Interleaf Links ; 2.5 The Middle Dimension; 2.6 Cap products for Middle Perversities; 2.7 L-Theory; 2.8 Intersection Vector Bundles and K-Theory; 2.9 Beyond Isolated Singularities; 3 String Theory; 3.1 Introduction3.2 The Topology of 3-Cycles in 6-Manifolds; 3.3 The Conifold Transition; 3.4 Breakdown of the Low Energy Effective Field Theory Near a Singularity; 3.5 Massless D-Branes; 3.6 Cohomology and Massless States; 3.7 The Homology of Intersection Spaces and Massless D-Branes; 3.8 Mirror Symmetry; 3.9 An Example; References; Index

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