Algebraic Topology & Homology

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Abstractions about topological spaces analyzed through homology, cohomology, spectra, decompositions, surgery, and induced maps. They connect complexes and manifolds with obstruction theory, duality, path and fiber structure, triangulation, and generalized cohomology theories.

37 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.
• Burau representation — A parameter-dependent linear representation of the braid group obtained from its action on the homology of an infinite cyclic cover of a punctured disk.
• Category of compactly generated weak Hausdorff spaces — A convenient category of spaces whose topology is detected by compact Hausdorff probes and whose compact images are closed.
• 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.
• Dogbone space — Bing’s quotient of three-dimensional Euclidean space that is not R3 even though point preimages are points or tame arcs, while its product with a line is R4.
• Dunce hat (topology) — A compact two-dimensional cell complex obtained by identifying all three edges of a triangle with one orientation reversed.
• 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.
• Eilenberg–Mazur swindle — A proof method using infinite self-similar sums or decompositions to cancel or absorb an object in a seemingly paradoxical way.
• 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.
• KR-theory — A real-equivariant form of topological K-theory classifying complex vector bundles equipped with compatible conjugate-linear involution over an involutive space.
• L-theory — An algebraic theory of quadratic and symmetric forms whose L-groups classify surgery obstructions and stable equivalence over rings with involution.
• 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.
• Rational homotopy theory — The study of topological spaces after replacing homotopy invariants by rational versions that discard torsion and admit algebraic models.
• 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.
• Simple space — A connected topological space whose fundamental group is abelian and acts trivially on every higher homotopy group, usually with a CW-type assumption.
• Simplicial set — A contravariant functor from the simplex category to sets, equivalently graded simplices equipped with compatible face and degeneracy maps.
• Smooth functor — A functor on finite-dimensional real vector spaces whose action varies smoothly in smoothly parameterized families.
• 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.
• Twisted K-theory — A generalized cohomology theory in which K-theory classes are modified by a background twist, often a degree-three integral cohomology class or bundle of operator algebras.
• Verdier duality — A derived-sheaf duality that exchanges proper direct image with exceptional inverse image and extends Poincaré duality to singular spaces and maps.