Search results
Results From The WOW.Com Content Network
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
William Schumacher Massey (August 23, 1920 [1] – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology (ISBN 0-387 ...
Absolutely closed See H-closed Accessible See . Accumulation point See limit point. Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.
In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y , intersect transversely in a manifold M , the homology class of the intersection is the Poincaré dual of the cup product D M X ⌣ D M Y {\displaystyle D_{M}X\smile D_{M}Y} of the Poincaré duals of X and Y .
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace X n that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n.
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf.
However, in 2012 Bourbaki resumed the publication of the Éléments with a revised chapter 8 of algebra, the first 4 chapters of a new book on algebraic topology, and two volumes on spectral theory (the first of which is an expanded and revised version of the edition of 1967 while the latter consist of three new chapters).