Manifold Topology & Homology Theories

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Abstractions about the structure of manifolds and topological spaces, including CW complexes, homology and cohomology theories, spectral sequences, decompositions of 3-manifolds and fibration-related constructions used to classify spaces up to homotopy.

28 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.

• Branched manifold — A manifold-like space locally formed from finitely many smooth sheets sharing regions under compatible charts, with a well-defined tangent structure despite controlled branching.
• Cellular homology — A homology theory for CW complexes computed from a chain complex with one generator per cell and boundary maps determined by attaching maps.
• Change of fiber — The homotopy-equivalence map between fibers of a fibration induced by transporting along a path in the base space.
• Collar neighbourhood — A neighborhood of a manifold's boundary that is identified with the product of that boundary and a half-open interval while fixing the boundary at interval coordinate zero.
• CW complex — A topological space constructed inductively by attaching open cells of increasing dimension under closure-finiteness and weak-topology conditions.
• Cyclic surgery theorem — A three-manifold theorem bounding the distance between two Dehn fillings of an irreducible boundary-torus manifold when both filled manifolds have cyclic fundamental group.
• EHP spectral sequence — A spectral sequence derived from EHP fibrations that inductively relates unstable homotopy groups of spheres to stable homotopy after localization at a prime.
• Excisive triad — A topological triad (X;A,B) in which X is covered by the interiors of subspaces A and B, supplying the cover condition used by excision and Mayer–Vietoris arguments.
• Gelfand–Fuks cohomology — Continuous Lie-algebra cohomology for topological Lie algebras of smooth vector fields.
• Handlebody — A manifold piece obtained from a ball or collar by attaching standard handles, used to decompose manifolds according to topology and critical-point index.
• Homeotopy — A homotopy group of the topological group of self-homeomorphisms of a space.
• Incompressible surface — A properly embedded surface in a three-manifold with no essential loop that bounds a compressing disk in the ambient manifold.
• Induced homomorphism — A homomorphism obtained canonically by applying a functorial algebraic construction to an underlying map.
• JSJ decomposition — A canonical decomposition of an irreducible orientable compact 3-manifold along incompressible tori into atoroidal and Seifert-fibered pieces.
• Kline sphere characterization — A theorem characterizing the topological two-sphere by how simple closed curves and pairs of points separate a compact connected space.
• May spectral sequence — A spectral sequence used to compute the cohomology of the Steenrod algebra and the input to the Adams spectral sequence.
• Mayer–Vietoris sequence — A natural long exact sequence relating the homology or cohomology of a space to those of two covering subspaces and their intersection.
• Morse homology — A homology theory whose chain groups are generated by critical points of a Morse function and whose boundary counts gradient-flow trajectories between adjacent indices.
• Obstruction theory — A family of topological methods that assigns cohomological classes whose vanishing determines whether a partial construction extends to the next dimension.
• Path space (algebraic topology) — The mapping space of continuous interval paths in a topological space, either with a fixed starting point or with both endpoints free.
• Reeb graph — A quotient graph summarizing how connected components of a real-valued function’s level sets merge and split across values.
• Relative contact homology — A contact-topological invariant associated with a contact manifold together with a Legendrian submanifold, constructed from Reeb chords and pseudoholomorphic curves within symplectic field theory.
• 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.
• Simply connected at infinity — A noncompact space property requiring sufficiently remote loops to contract outside any prescribed compact core.
• Smooth functor — A functor on finite-dimensional real vector spaces whose action varies smoothly in smoothly parameterized families.
• Toric manifold — A smooth compact even-dimensional manifold with an effective locally standard action of a half-dimensional torus and a simple convex polytope as orbit space.
• Train track (mathematics) — A smoothly branched graph embedded in a surface that carries measured laminations through compatible nonnegative branch weights.
• Triangulation (topology) — A homeomorphic representation of a topological space by the geometric realization of a simplicial complex.