The best answers are voted up and rise to the top. Why do the homology groups capture holes in a space better than the homotopy groups? Ask Question. Asked 11 years, 6 months ago. Active 4 years, 4 months ago. Viewed 10k times. Improve this question. Community Bot 1 2 2 silver badges 3 3 bronze badges. Akela Akela 3, 2 2 gold badges 28 28 silver badges 40 40 bronze badges. For example, a connected space with simple nonabelian fundamental group will have trivial first homology, but I would claim that it has a nontrivial one dimensional hole.
This shows that a 3-dimensional sphere can wind around, in a non-trivial way, a 2-dimensional "hole" using your terminology. So one reason that homotopy groups are complicated is that higher dimensional spheres can wind around lower dimensional holes. I thought it best reflects the intuitive picture. Show 9 more comments. Active Oldest Votes. Improve this answer. Allen Hatcher Allen Hatcher The Cech cohomology of this Hawaiian pearl is what you'd expect, right?
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