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A torus, one of the most frequently studied objects in algebraic topology. 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.
As such, the existence of an H-space structure on a given space is only dependent on pointed homotopy type. The multiplicative structure of an H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. [9]
Given two directed paths γ and δ, a directed homotopy from γ to δ is a morphism of directed spaces h whose underlying map U(h) is a homotopy –in the usual sense– between the underlying paths U(γ) and U(δ). In algebraic topology, there is a homotopy from α to β if and only if there is a homotopy from β to α. Due to non ...
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 mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups.
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.
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...
An example in topology is the composition of paths, where the identity and association conditions hold only up to reparameterization, and hence up to homotopy, which is the 2-isomorphism for this 2-category. These n-isomorphisms must well behave between hom-sets and expressing this is the difficulty in the definition of weak n-categories.