In topology, two continuous functions from one topological space to another are called homotopic , Now my question here is :Why Homotopy so interesting in Mathematics and what is its relationship with Manifolds?
Dear Dr. Rafik Zeraoulia
Greating!
The following links may be useful to answer
https://www.maths.ed.ac.uk/~v1ranick/papers/browder3.pdf
https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/homotopy-theory-2015/description/
http://faculty.sites.uci.edu/rstern/files/2011/03/8_HomologyTransverality_Inventiones.pdf
https://www.maths.ed.ac.uk/~v1ranick/papers/cohenhom.pdf
https://web.math.rochester.edu/people/faculty/doug/otherpapers/pont4.pdf
With best wishes
Dear Rafik Zeraoulia,
Homotopy Theory is one of the essential topics in Mathematics.
It helps to study and classify curves, surfaces, and manifolds of any dimension. One of the major issues tackled by Homotopy Theory is to
compute the Poincare group ( First Fundamental group) of the given topological space. One needs to read advanced courses in algebra and topology as a prerequisite of the Homotopy Theory.
We said that two continuous maps f,g: X→Y are homotopic,
denoted f∼g ( f is homotopic to g ),
if there exists a continuous map H(x,t): X×I→Y, I=[0,1]
satisfies : H(x,0)=f(x) and H(x,1)=g(x).
Roughly speaking:
One can prove that X = triangle and the Y = circle have the same homotopy type and they have the same Poincare group Z.
They said to have the same homotopy type.
But the Poincare group of the segment [a,b] and the disc ( open or closed)is zero.
In brief, it is some sort of non-quantum geometry.
One of the main results:
Any two equivalent topological spaces have the same homotopy type.
For a serious study to our universe, we need to consider the homotopy type, where all convinced that it is not Euclidean space.
Also, to study the knots of the veins and arteries knots, DNA, etc., inside our body, the tools of homotopy theory are strongly needed.
Best regards
homotopy theory is in topology. That is, it is a continuous deformation of topological spaces. The category of manifolds and smooth maps is ofcourse more stronger also topologicaly equivalency is stronger than homotopic equivalency. It means that if we have a diffeomorphism between manifolds, then they are already homotopic each other. That is, one is obtained from other by a continuous deformation. Also there are smooth homotopic maps which are called isotopy. But the homotopic maps between infinite dimensional Hilbert manifolds can be deformed to a diffeomorphism. It is proved by ]James Eels in 1960-1970.
Homotopy does also have application in differential equations. The Homotopy Analysis Method (HAM) is one of the recently introduced methods for solving DEs.
Homotopy is a principal part of algebraic topology in which the technique of algebra especially group theory is used to convert a topologicl problem to algebraic one. It has important applications in pure and applied mathematics. For example in knot theory which includes manifolds and even in quantum field theory and gravitation.
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