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- Explain homotopy to me - Mathematics Stack Exchange
A homotopy is a map between maps in the following sense: fix f, g: X → Y continous maps, and h: X × I → Y the homotopy We can see h as the map that takes t ∈ I to ht ∈C(X, Y), where ht is defined by ht(x) = h(x, t) Going up a bit ahead, we'll say that a path in a topological space is a map defined on I taking values in the space
- What is the relation between homotopy groups and homology?
But there are some specific homotopy groups, if only outside the stable range, which are not computable by those homological methods Thus the relation between homotopy groups and homology is a very complicated one, with much still to explore
- Isotopy and Homotopy - Mathematics Stack Exchange
What is the difference between homotopy and isotopy at the intuitive level Some diagrammatic explanation will be helpful for me
- What is the difference between homotopy and homeomorphism?
What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic?
- On the definition of homotopy. - Mathematics Stack Exchange
The definition of homotopy you give is correct This is a fully general definition, that works for all spaces and continuous mappings between them When looking at something like the fundamental group of a space X X, as you correctly point out, it is not interesting to look at whether or not two mappings [0, 1] → X [0, 1] → X are homotopic In fact, you'll find that the equivalence classes
- general topology - Homotopy equivalence between spaces intuition . . .
Ok, so homotopy equivalence is enough, but why is it better than homeomorphism? The answer is because it makes computations easier It is much easier to show that two spaces are homotopy equivalent than to show they are homeomorphic, and with this new (weaker) notion of equivalence, we can compare the homology of spaces that aren't homeomorphic
- Approximation and homotopy - Mathematics Stack Exchange
In Differential Topology the usual strategy to discuss homotopy classes by differential means is approximation, because a manifold M M is a euclidean neigborhood retract (ENR) and any good approximation of a map f: X → M f: X → M is homotopic to f f
- Homotopy equivalence of pairs - Mathematics Stack Exchange
Homotopy equivalence of pairs Ask Question Asked 7 years ago Modified 1 year ago
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