Ad
related to: what is a homotopy in math meaning dictionary chart printable templateeducation.com has been visited by 100K+ users in the past month
This site is a teacher's paradise! - The Bender Bunch
- Worksheet Generator
Use our worksheet generator to make
your own personalized puzzles.
- Printable Workbooks
Download & print 300+ workbooks
written & reviewed by teachers.
- Interactive Stories
Enchant young learners with
animated, educational stories.
- Education.com Blog
See what's new on Education.com,
explore classroom ideas, & more.
- Worksheet Generator
Search results
Results from the WOW.Com Content Network
Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology , but nowadays is learned as an independent discipline.
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) [1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces:. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of ...
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
The homotopy group functors assign to each path-connected topological space the group () of homotopy classes of continuous maps . Another construction on a space X {\displaystyle X} is the group of all self-homeomorphisms X → X {\displaystyle X\to X} , denoted H o m e o ( X ) . {\displaystyle {\rm {Homeo}}(X).}
This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy. One says that a topological space X is an H-space if there exists e and μ such that the triple (X, e, μ) is an H-space as in the above definition. [3]
Ad
related to: what is a homotopy in math meaning dictionary chart printable templateeducation.com has been visited by 100K+ users in the past month
This site is a teacher's paradise! - The Bender Bunch