Search results
Results from the WOW.Com Content Network
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 ...
In other cases, such as for group homology, there are multiple common methods to compute the same homology groups. In the language of category theory, a homology theory is a type of functor from the category of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category ...
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ) and abstract algebra (theory of modules and syzygies ) at the end of the 19th century, chiefly by ...
In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.
A homotopy between two embeddings of the torus into : as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such that ...
In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the excision theorem for singular homology and cohomology , there is no simple known way to calculate the homotopy groups of a space by ...
This means homology groups are homotopy invariants, and therefore topological invariants. In particular, if X is a connected contractible space, then all its homology groups are 0, except (). A proof for the homotopy invariance of singular homology groups can be sketched as follows.