Date: 2024-05-19 | Article ID: 61813 |

Homotopy groups are examples of topological invariants: topologically equivalent spaces have the same homotopy groups. Roughly, the nth **homotopy group** of a topological space M expresses the inequivalent ways an n-sphere can be continuously mapped into M, regarding two such maps as equivalent if one can be continuously morphed into the other. The homotopy group with n=1 is called the **fundamental group**. This article introduces homotopy groups and the related concept of a **covering space**. A covering space E of M is like M but "unwrapped" so that E's fundamental group is only part of M's fundamental group.

**2024-05-19**(added a reference, moved some content from the section about homotopy sets to a new article)**2024-05-02**(cited two more references, added a missing word "connected")**2024-04-25**(added a phrase with "basepoint" to a sentence in the first paragraph about higher homotopy groups, added a section about homotopy sets with and without basepoints, removed one sphere example after finding a discrepancy in the literature, added two more sphere examples, moved a footnote that was tied to the wrong sentence)**2024-04-20**(separated the cartesian-product example into its own section and added more results, added more sphere examples)**2024-04-16**(added a cross-reference for the direct product of two groups, updated publication info for one reference)**2024-03-24**(moved one section to a better place in the sequence, added citations to new references, changed equal -> isomorphic in a few statements about homotopy groups)**2024-02-27**(removed unused references, added a missing reference)**2024-02-25**(first version)

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