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- 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
- What is the difference between homotopy and homeomorphism?
Anyways, homotopy equivalence is weaker than homeomorphic Counterexample to your claim: the 2-dimensional cylinder and a Möbius strip are both 2-dimensional manifolds and homotopy equivalent, but not homeomorphic
- Explain homotopy to me - Mathematics Stack Exchange
I have been struggling with general topology and now, algebraic topology is simply murder Some people seem to get on alright, but I am not one of them unfortunately Please, the answer I need is i
- 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
- Non-homotopic spaces with the same homology groups
As for spaces with the same $\pi_*$ but not homeomorphic , this is easy : just take any non-singleton contractible space (such as $\mathbb R$) and more generally homotopy-equivalent spaces that aren't homeomorphic
- 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
- About Homotopy lifting property - Mathematics Stack Exchange
The homotopy lifting property shows that any path homotopy $h_t : f \simeq g$ lifts to a homotopy $\tilde h_t$ starting with $\tilde f$ But it is not a priori clear that that $\tilde h_t$ is a path homotopy $\tilde f \simeq \tilde g$
- Why is $S^1\times \ {1\}$ homotopy equivalent to the solid torus $T^2 . . .
I am currently self-studying the basics of algebraic topology and i just learned the definitions of retract, deformationretract and homotopy equivalence Now in my book there is an example of a
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