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There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that you think are beautiful and illuminating to the study of homotopy theory.

Following is what comes to my mind at first place:

  • Sphere eversion (Regular homotopy)
  • Poincaré's homology sphere (Poincaré conjecture)
  • Hopf fibration (homotopy groups of n-spheres)
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    $\begingroup$ I have the feeling that it would have been better to edit this question than to close it. $\endgroup$ – alvarezpaiva Apr 15 '14 at 16:28
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    $\begingroup$ I'd never dream about such question. To give it full justice though, you may rephrase your question in full generality: Beautiful constructions in Mathematics that facilitate one's understanding of Algebra. That would be really something! $\endgroup$ – Włodzimierz Holsztyński Apr 16 '14 at 2:17
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Links of singularities of complex algebraic varieties are fascinating geometrical and topological objects.

For example Brieskorn manifolfs. A Brieskorn manifold is the intersection of a small sphere around the origin with the singular complex hypersurface $$z_0^{k_1}+\cdots+z_n^{k_n}=0$$
they are closed smooth oriented $(n−2)$-connected $(2n−1)$-manifolds. They give examples of exotic spheres.

The study of Brieskorn manifold and exotic spheres motivate the study of the surgerical classification of manifolds, stable homotopy groups of spheres (J-homomorphism, Adams' conjecture, Kervaire invariant.....), etc

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Borsuk-Ulam theorem and related results. See the very nice book by Matousek, and also this really cool paper by M. G. Dobbins. (the latter uses more equivariant methods, but the same spirit).

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