Duality, Cobordism & Topological Fields

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Abstractions about Poincaré and Lefschetz duality, singularity invariants, cobordism, generalized Poincaré spaces, and topological quantum field theories.

5 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Lefschetz duality — A manifold-with-boundary extension of Poincaré duality pairing absolute cohomology with relative homology, and relative cohomology with absolute homology, through the relative fundamental class.
• Milnor number — Measure an isolated hypersurface singularity by the finite dimension of its local Jacobian algebra, equivalently the number of middle-dimensional spheres in its Milnor fiber.
• Poincaré space — A finite-type space equipped with a fundamental homology class whose cap product realizes Poincaré duality in every degree.
• Semi-s-cobordism — A cobordism whose inclusion of one designated boundary component is a simple homotopy equivalence, with no corresponding requirement on the other boundary.
• Topological quantum field theory — A quantum field theory whose observables depend only on topological structure, mathematically formalized as a symmetric monoidal functor from a cobordism category to vector spaces or related algebraic categories.