Algebraic Topology

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32 domain-specific abstractions whose origin domain is Algebraic Topology.

• Bott cannibalistic class — A K-theory characteristic class measuring how an Adams operation acts on the Thom class of a complex vector bundle or representation.
• 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.
• Classifying space — A space BG representing principal G-bundles up to homotopy, obtained from a contractible free G-space EG and characterized by pullback classification.
• CW complex — A topological space constructed inductively by attaching open cells of increasing dimension under closure-finiteness and weak-topology conditions.
• Delta set — A semi-simplicial object consisting of sets of n-simplices with face maps satisfying simplicial identities but no required degeneracy maps, providing flexible combinatorial models for gluing and homology.
• 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–MacLane space — A connected space K(G,n) with exactly one nontrivial homotopy group, G in degree n, serving as a representing space for cohomology.
• 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.
• Extension (simplicial set) — The Ex endofunctor on simplicial sets, right adjoint to subdivision, that replaces a simplicial set by maps from subdivided simplices and iteratively improves horn-filling behavior.
• Homeotopy — A homotopy group of the topological group of self-homeomorphisms of a space.
• Homological Stability — Eventual degreewise invariance in an indexed family: stabilization maps induce homology isomorphisms once the size parameter enters a degree-dependent stable range.
• Homotopy Category — A category that retains objects while replacing maps by homotopy classes or, more generally, formally inverting a designated class of weak equivalences.
• Induced homomorphism — A homomorphism obtained canonically by applying a functorial algebraic construction to an underlying map.
• 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.
• 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.
• Mapping Space — A topological or enriched space whose points are maps between fixed spaces, with topology chosen so families, homotopies, and evaluation become structural.
• 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.
• Moduli Stack of Formal Group Laws — A coordinate-independent moduli stack obtained from formal group laws by quotienting coordinate changes, retaining isomorphisms and organizing formal groups by height for chromatic homotopy theory.
• 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.
• Poincaré space — A finite-type space equipped with a fundamental homology class whose cap product realizes Poincaré duality in every degree.
• Rational homotopy theory — The study of topological spaces after replacing homotopy invariants by rational versions that discard torsion and admit algebraic models.
• 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.
• Sullivan Conjecture — The proved Miller theorem that, for a finite group and finite-dimensional CW complex, the based mapping space from the group's classifying space is weakly contractible, equivalently constant maps give a weak equivalence from the target to the unbased mapping space.
• 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.