Geometric Topology

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13 domain-specific abstractions whose origin domain is Geometric Topology.

• Cantor tree surface — Form the unique orientable boundaryless surface of genus zero whose end space is a Cantor set and whose every end is planar, equivalently a sphere with a Cantor set removed.
• Dehn surgery — A 3-manifold construction that removes tubular neighborhoods of link components and glues solid tori back along specified boundary slopes.
• 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.
• 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.
• 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.
• Lens space — A closed manifold obtained by a cyclic quotient of an odd-dimensional sphere, or in dimension three by gluing two solid tori with a slope specified by coprime integers p and q.
• Novikov conjecture — The conjecture that higher signatures of closed oriented manifolds are invariant under oriented homotopy equivalence.
• Orientation character — Classify loops in a manifold by whether their transport preserves or reverses local orientation, encoding the result as a homomorphism from the fundamental group to the two-element sign group.
• 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.
• Triangulation (topology) — A homeomorphic representation of a topological space by the geometric realization of a simplicial complex.