| 11. | Hence the only non-trivial homotopy group is \ pi _ 1 ( X)
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| 12. | A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups.
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| 13. | This property makes fibrant objects the " correct " objects on which to define homotopy groups.
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| 14. | Another way is to examine the type of topological singularity at a point with the homotopy group.
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| 15. | For stable homotopy groups there are more precise results about " p "-torsion.
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| 16. | On homotopy groups, where ? denotes the loop functor and'" denotes the smash product.
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| 17. | In the two examples above all the maps between homotopy groups are applications of the suspension functor.
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| 18. | As the third homotopy group of S ^ 3 has been found to be the set of integers,
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| 19. | The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory.
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| 20. | It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general.
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