Algebraic Topology & K-Theory

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Abstractions about topological invariants and duality built from homotopy and cohomology, spanning generalized K-theory (KR-theory, twisted K-theory, Milnor K-theory), duality theorems (Poincaré, Lefschetz, Verdier, Pontryagin duality), and homotopy-theoretic constructions (Eilenberg-MacLane spaces, rational homotopy theory, virtual fundamental classes).

26 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.

• 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.
• 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.
• Constructible topology — The compact Hausdorff totally disconnected refinement of the Zariski topology on a scheme or spectrum, generated by making quasi-compact open sets and their complements open.
• Double affine braid group — A braid-like group associated with an affine root system that adds a second affine translation structure and whose group algebra leads to double affine Hecke algebras.
• 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.
• Hausdorff completion — The inverse-limit completion of a filtered group formed from its discrete quotients, with the filtration intersection removed by the canonical map.
• Homology Manifold — A finite-dimensional ANR whose local relative homology at every point matches Euclidean space in a stated dimension and coefficient system, whether or not Euclidean charts exist.
• 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.
• Milnor K-theory — The graded ring generated by nonzero field elements modulo Steinberg relations, linking symbols in algebraic K-theory with Galois cohomology.
• Mnëv's universality theorem — A theorem showing that realization spaces of oriented matroids can reproduce, up to stable equivalence, arbitrary integer-defined primary semialgebraic sets and therefore arbitrarily complicated topology.
• Novikov conjecture — The conjecture that higher signatures of closed oriented manifolds are invariant under oriented homotopy equivalence.
• Poincaré space — A finite-type space equipped with a fundamental homology class whose cap product realizes Poincaré duality in every degree.
• Polynomial differential form — An element of the commutative differential graded algebra of polynomial coordinate functions and their differentials on a standard simplex, varying simplicially with face and degeneracy maps.
• Pontryagin duality — A duality for locally compact abelian groups that assigns each group its continuous characters into the circle group and naturally identifies the original group with its double dual.
• 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.
• Splitting principle — A technique that pulls a vector bundle to a space where it decomposes into line bundles, performs calculations there, and transfers valid identities back through an injective cohomology map.
• 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.
• V-topology — A very fine Grothendieck topology in algebraic geometry whose covers are universally subtrusive and can be tested by lifting valuation-ring maps.
• Verdier duality — A derived-sheaf duality that exchanges proper direct image with exceptional inverse image and extends Poincaré duality to singular spaces and maps.
• Virtual fundamental class — Replace the missing ordinary fundamental class of an obstructed moduli space with a cycle of expected dimension derived from a perfect obstruction theory, enabling deformation-invariant enumerative integrals.
• Waraszkiewicz spiral — A member of an uncountable family of planar continua constructed so distinct members are incomparable under continuous surjections.