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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.
A 1 homotopy theory is founded on a category called the A 1 homotopy category ().Simply put, the A 1 homotopy category, or rather the canonical functor (), is the universal functor from the category of smooth -schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line A 1 becomes contractible.
The space of ordered configuration of two points in is homeomorphic to the product of the Euclidean 3-space with a circle, i.e. (). [ 2 ] More generally, the configuration space of two points in R n {\displaystyle \mathbf {R} ^{n}} is homotopy equivalent to the sphere S n − 1 {\displaystyle S^{n-1}} .
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.For most spaces that are considered in practice, namely for all those that satisfy the T 0 separation axiom, this preorder is even a partial order (called the specialization order).
The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a ...
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology ), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...
Path (topology) Fundamental group; Homotopy group; Seifert–van Kampen theorem; Pointed space; Winding number; Simply connected. Universal cover; Monodromy; Homotopy lifting property; Mapping cylinder; Mapping cone (topology) Wedge sum; Smash product; Adjunction space; Cohomotopy; Cohomotopy group; Brown's representability theorem; Eilenberg ...
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space [note 1] is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer . A connected topological space X is called an Eilenberg–MacLane space of type K ( G , n ) {\displaystyle K(G,n)} , if it has n -th homotopy group π n ...