Prompt: In the field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other, being examples of topological invariants (which reflect, in algebraic terms, the structure of spheres viewed as topological spaces). The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other. (The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n1.): For 0<i<n, any mapping from Si to Sn is homotopic (meaning continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. Therefore the homotopy group is the trivial group. When i=n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn(Sn) with the group of integers under addition. The most interesting and surprising results occur when i>n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping. The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of algebraic topology and stable homotopy theory were devised to compute more and more of the stable stems of such.
Prompt: In the field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other, being examples of topological invariants (which reflect, in algebraic terms, the structure of spheres viewed as topological spaces). The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other. (The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n1.): For 0<i<n, any mapping from Si to Sn is homotopic (meaning continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. Therefore the homotopy group is the trivial group. When i=n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn(Sn) with the group of integers under addition. The most interesting and surprising results occur when i>n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping. The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of algebraic topology and stable homotopy theory were devised to compute more and more of the stable stems of such.
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Prompt:
In the field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other, being examples of topological invariants (which reflect, in algebraic terms, the structure of spheres viewed as topological spaces). The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other. (The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n1.): For 0<i<n, any mapping from Si to Sn is homotopic (meaning continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. Therefore the homotopy group is the trivial group. When i=n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn(Sn) with the group of integers under addition. The most interesting and surprising results occur when i>n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping. The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of algebraic topology and stable homotopy theory were devised to compute more and more of the stable stems of such.
Dream Level: is increased each time when you "Go Deeper" into the dream. Each new level is harder to achieve and
takes more iterations than the one before.
Rare Deep Dream: is any dream which went deeper than level 6.
Deep Dream
You cannot go deeper into someone else's dream. You must create your own.
Deep Dream
Currently going deeper is available only for Deep Dreams.
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