This fact is related to: homotopy groups
In terms of the Hurewicz homomorphism: absolute version
and moreover, all the reduced homology groups up to are zero. In particular, and for .
is surjective and has kernel precisely the derived subgroup of , so is isomorphic to the abelianization of .
In terms of first non-vanishing member: absolute version
Suppose is a Path-connected space (?) that is simply connected. In particular, and are both trivial (the one-point set and the trivial group respectively). Then:
- The smallest for which is nontrivial is the same as the smallest for which is nontrivial.
- Both of these groups are isomorphic, and the Hurewicz homomorphism gives an isomorphism.
In the case that we are only given that is a path-connected space, and the Hurewicz homomorphism descends to this natural identification.
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