## What is the difference between homotopy and homology?

In topology|lang=en terms the difference between homotopy and homology. is that homotopy is (topology) a system of groups associated to a topological space while homology is (topology) a theory associating a system of groups to each topological space.

## What is null homotopy?

A continuous map. between topological spaces is said to be null-homotopic if it is homotopic to a constant map. If a space has the property that , the identity map on , is null-homotopic, then. is contractible.

**What is homotopic in complex analysis?**

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps.

### Is homotopy continuous?

Two continuous functions from one topological space to another are called homo- topic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. More precisely, we have the following definition.

### Is homotopy equivalence Bijective?

When two spaces X and X′ become homotopy equivalent after one suspension, the homotopy equivalence between the suspensions will induce a bijection between the towers {[ΣX, Y(n)]} and {[ΣX′, Y(n)]}, with the two properties just mentioned. This bijection is not necessarily a group isomorphism however.

**Is a retraction a homotopy equivalence?**

Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are homotopy equivalent if and only if they are both homeomorphic to deformation retracts of a single larger space. Any topological space that deformation retracts to a point is contractible and vice versa.

#### Who introduced homotopy?

Introduction In recent years, the homotopy perturbation method (HPM), first proposed by Dr. Ji Huan He [1], [2], has successfully been applied to solve many types of linear and nonlinear functional equations.

#### What is the meaning of homotopy perturbation method?

The homotopy perturbation method is a powerful and efficient technique for finding solutions of nonlinear equations without the need of a linearization process. The method was first introduced by He in 1998 [1], [2]. HPM is a combination of the perturbation and homotopy methods.

**Is a circle homotopy equivalent to a point?**

Disclaimer: OBVIOUSLY a circle is NOT homotopic to a point. However, if we just pick one point x0 from S1, then the mapping cylinder looks just like a cone, with F(x,0)=x, F(x0,t)=x0 (which is a line going down the cone from the circle to the bottom apex), and F(x,1)=x0.

## When do you use the term pointed homotopy?

Also, if g is a retraction from X to K and f is the identity map, this is known as a strong deformation retract of X to K . When K is a point, the term pointed homotopy is used.

## When is a homotopy called a nullhomotopic?

A map is called nullhomotopicwhen it is homotopic to a constant map. A space is called contractiblewhen the identity map from the space to itself is nullhomotopic. For example, each real topological vector space is contractible – a suitable homotopy is given by F(x,t) = (1-t)x.

**When do two spaces have the same homotopy type?**

Given two spaces X and Y, we say they are homotopy equivalent, or of the same homotopy type, if there exist 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. The maps f and g are called homotopy equivalences in this case.

### Which is the 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.