homotopy group

mathematics

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July 1, 2026

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In mathematics, a homotopy group is a concept within algebraic topology. It is used to classify geometric regions by studying different types of paths within them. Two paths are considered "homotopic" if one can be continuously deformed into the other without leaving the region and keeping the endpoints fixed.1

More formally, homotopy involves mapping points from an interval to a region. A homotopy map shows this gradual deformation. When paths start and end at the same point, the collection of paths homotopic to each other forms a "homotopy class." The set of these classes can form an algebraic structure called a group, known as the fundamental group of the region.1

Higher-dimensional homotopy groups can also be defined for spaces in more than three dimensions. While homology groups are often easier to compute and are topological invariants, homotopy groups are more delicate and provide deeper information about a space's structure, though they are generally more difficult to determine.12

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mathematics
Algebraic topology
…the branch of topology called homotopy theory. Being more delicate, it is both more basic and more difficult. There are usually standard methods for computing homology and cohomology groups, and they are completely known for many spaces. In contrast, there is scarcely an interesting class of spaces for which all…
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…the branch of topology called homotopy theory. Being more delicate, it is both more basic and more difficult. There are usually standard methods for computing homology and cohomology groups, and they are completely known for many spaces. In contrast, there is scarcely an interesting class of spaces for which all…
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topology
Fundamental group
…what are known as the homotopy groups of a topological space. These groups, as well as another class of groups called homology groups, are actually invariant under mappings called homotopy retracts, which include homeomorphisms. Homotopy theory and homology theory are among the many specializations within algebraic topology.
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…what are known as the homotopy groups of a topological space. These groups, as well as another class of groups called homology groups, are actually invariant under mappings called homotopy retracts, which include homeomorphisms. Homotopy theory and homology theory are among the many specializations within algebraic topology.
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