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Homotopy groups are notoriously difficult to compute, in contrast with homology and cohomology groups, where are generally computable: even the higher homotopy groups of spheres are not fully understood.  Even small homotopy groups surprising turn out to be nontrivial: the group <math>\pi_3(S^2)</math> is isomorphic to the group of integers, generated by the [[Hopf fibration]].

Homotopy groups are notoriously difficult to compute, in contrast with homology and cohomology groups, where are generally computable: even the higher homotopy groups of spheres are not fully understood.  Even small homotopy groups surprising turn out to be nontrivial: the group <math>\pi_3(S^2)</math> is isomorphic to the group of integers, generated by the [[Hopf fibration]].

A famous [[conjecture]] stated in terms of homotopy groups is the recently-proven Poincare conjecture, which states that any manifold homotopy equivalent to a [[sphere]] actually is a sphere.  The precise formulation depends on whether one works in the category of smooth, piecewise-linear, or topological manifolds.

A famous [[conjecture]] stated in terms of homotopy groups is the recently-proven [[Poincare conjecture]], which states that any manifold homotopy equivalent to a [[sphere]] actually is a sphere.  The precise formulation depends on whether one works in the category of smooth, piecewise-linear, or topological manifolds.

[[category: Topology]]

[[category: Topology]]