The fundamental group (first homotopy group) is an algebraic invariant that helps me to determine if two topological spaces are not homeomorphic, in particular the fundamental group allow me to classify Some surfaces...Do you know of any similar utility of higher order homotopy groups?
Utility of higher order homotopy groups?
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algebraic-topology
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0Sometimes the fundamental groups are the same, but the spaces aren't homeomorphic. If you get lucky, a higher homology group will tell you. – 2017-02-07
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The $\pi_n$ are also homotopy invariants, so they can perform similar classification. Furthermore, the classification above is complete in following sense (Whitehead's theorem): If $f:X \to Y$ is a map of CW-complexes (if you're not familiar with them, think of a reasonable submanifold of $\mathbb{R}^k$) that induces an isomorphism $f_*:\pi_n(X) \to \pi_n(Y)$ for all $n$, then $f$ is a homotopy equivalence. Homotopy equivalence is not equivalent to homeomorphism, but it's the best we can expect; the $\pi_*$ are invariant under homotopy equivalence, and there are spaces even in the finite CW category that are homotopy equivalent but not homeomorphic.
There are lots of other applications (Eilenberg-Maclane spaces, the Hurewicz theorem, classifying spaces, the Pontyragin-Thom construction, charges in physics as elements of $\pi_n$, etc.), but I'll stop here for space.
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Higher homotopy groups have the same "utility," as you put it. Homeomorphic spaces have isomorphic homotopy groups of all orders, so by the contrapositive, if two spaces have non-isomorphic $n$th homotopy groups, then they are not homeomorphic.
Actually, a weaker condition is necessary: any two homotopy equivalent spaces have isomorphic homotopy groups of all orders. Homotopy equivalent is implied by homoemorphic.