How do you show homotopy equivalence?
Two spaces X and Y are said to be homotopy equivalent (written X ≃ Y ) if there is a homotopy equivalence f : X → Y . Remark 2.4. By Remark 2.2, X ∼ = Y =⇒ X ≃ Y.
What are analogous and homologous structures?
Structures with similar anatomy, morphology, embryology and genetics but dissimilar functions are known as homologous structures. Structures that are superficially similar but anatomical dissimilar doing the same function are known as analogous structures.
Who invented homotopy analysis method?
Professor Shijun Liao
4.1 Introduction. The homotopy analysis method (HAM), developed by Professor Shijun Liao (1992, 2012), is a powerful mathematical tool for solving nonlinear problems. The method employs the concept of homotopy from topology to generate a convergent series solution for nonlinear systems.
What is a homotopy invariant?
A functor on spaces (e.g. some cohomology functor) is called “homotopy invariant” if it does not distinguish between a space X and the space X×I, where I is an interval; equivalently if it takes the same value on morphisms which are related by a (left) homotopy.
What is the main difference between homologous and analogous features?
Homologous structures are structures that evolve in living organisms that have a common ancestor. Analogous structures are those that evolve independently in different living organisms but have a similar or the same function.
Is connectedness a homotopy invariant?
Path connectedness is a homotopy invariant.
What are the difference of homologous and analogous?
Homologous structures share a similar embryonic origin; analogous organs have a similar function. For example, the bones in the front flipper of a whale are homologous to the bones in the human arm. These structures are not analogous.
What is a homotopy equivalence between X and Y?
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.
What are homotopy equivalences in topology?
The homotopy equivalences are the weak equivalences in the Strøm model structure on topological spaces. The homotopy category resulting from inverting all homotopy equivalences in Top is the same as that resulting from identifying homotopic maps. Sometimes an apparently stronger form of homotopy equivalence is needed.
What is the difference between weak and homotopy equivalence?
Any homotopy equivalence is also a weak homotopy equivalence. Conversely, in the context of topological spaces, any weak homotopy equivalence between m-cofibrant spaces (spaces that are homotopy equivalent to CW complexes) is a homotopy equivalence. This is the Whitehead theorem.
What is the homotopy category?
The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent.