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In the late 19th century Camille Jordan introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory (O'Connor & Robertson 2001). A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced (O'Connor & Robertson 1996).Higher homotopy groups were first defined by Eduard Čech in 1932 (Čech 1932, p. 203). (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.) Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper and also for the Hurewicz theorem which can be used to calculate some of the groups (May 1999a). An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result waswasHans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results (May 1999a). In 1953 George W. Whitehead showed that there is a metastable range for the homotopy groups of spheres. Jean-Pierre Serre used spectral sequences to show that most of these groups are finite, the exceptions being πn(Sn) and π4n−1(S2n). Others who worked in this area included José Adem, Hiroshi Toda, Frank Adams and J. Peter May. The stable homotopy groups πn+k(Sn) are known for k up to 64, and, as of 2007, unknown for larger k (Hatcher 2002, Stable homotopy groups, pp. 385–393).

In the late 19th century Camille Jordan introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory (O'Connor & Robertson 2001). A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced (O'Connor & Robertson 1996).Higher homotopy groups were first defined by Eduard Čech in 1932 (Čech 1932, p. 203). (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.) Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper and also for the Hurewicz theorem which can be used to calculate some of the groups (May 1999a). An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was

History

In the late 19th century Camille Jordan introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory (O'Connor & Robertson 2001). A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced (O'Connor & Robertson 1996).

The question of computing the homotopy group πn+k (S n) for positive k turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups πn+k (S n) are independent of n for n ≥ k + 2. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for n < k + 2) are more erratic; nevertheless, they have been tabulated for k < 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.

The problem of determining πi (S n) falls into three regimes, depending on whether i is less than, equal to, or greater than n:|

For 0 < i < n, any mapping from S i to S n is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of S i to a single point of S n. Therefore the homotopy group is the trivial group.

When i = n, every map from S n to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn(S n) with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping.

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 S 3 around the usual sphere S 2 in a non-trivial fashion, and so is not equivalent to a one-point mapping.

The i-th homotopy group πi (S n) summarizes the different ways in which the i-dimensional sphere S i can be mapped continuously into the n-dimensional sphere S n. This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.

The problem of determining πi (S n) falls into three regimes, depending on whether i is less than, equal to, or greater than n:

The Hopf fibration is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.

This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.

The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as S n — generalizes the familiar circle (S1) and the ordinary sphere (S 2). The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin.

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

Maestlin and Kepler communicated through a series of letters about Kepler's book the Mysterium Cosmographim in which Maestlin added his appendix “On the Dimensions of the Heavenly Circles and Spheres, according to the Prutenic tables after the theory of Nicolaus Copernicus” This appendix contained a set of planetary distances in addition to a method of deriving them from the Prutenic tables. Maestlin also added his own understanding of Nicolaus Copernicus’ geometry to Kepler's book. When. Maestlin and Kepler were communicating through letters regarding Kepler's book the discussed such topics such as the inaccuracy of the values that Copernicus used when calculating the spheres of the cosmos With the help of Maeslin in 1595, Kepler believed that he had discovered the relationship between the planets period and the distance from the sun. He did so by first assuming equal velocity of each planet and then observing that the planets did not just revolve just according to the length of their radii. Kepler observed that the sun exerted a force that was progressively attenuated as planets are farther and farther away from the sun itself. Maeslin even provided the geometry to help visualize Kepler's theory of the sun force and its effects of the other planets

Maestlin and Kepler communicated through a series of letters about Kepler's book the Mysterium Cosmographim in which Maestlin added his appendix “On the Dimensions of the Heavenly Circles and Spheres, according to the Prutenic tables after the theory of Nicolaus Copernicus” This appendix contained a set of planetary distances in addition to a method of deriving them from the Prutenic tables. Maestlin also added his own understanding of Nicolaus Copernicus’ geometry to Kepler's book. When. Maestlin and Kepler were communicating through letters regarding Kepler's book the discussed such topics such as the inaccuracy of the values that Copernicus used when calculating the spheres of the cosmos

The preface in the 1596 republication of Rheticus' Narratio Prima was also written by Maestlin. This preface was an introduction to the work of Copernicus. Additionally, Maestlin made many contributions to tables and diagrams in Kepler's Mysterium Cosmographicum. Kepler's publication of Mysterium Cosmographicum was supervised by Maestlin, in which he added his own appendix to the publication over Copernican planetary theory with help from Erasmus Reinhold's Prutenic Tables.[20] The information Maestlin used for his appendix from the Prutenic Tables, by Erasmus Reinhold, was used to help readers that were not well educated in astronomy to be able to read Johannes Kepler's Mysterium Cosmographicum as not much information over the basics was included. A discussion of the great sphere and the lunar sphere, as well as more discussion and conclusions to his descriptions of the Copernican planetary theory was also added by Maestlin in Kepler's book.[20] Maestlin's appendix was written more than once, and in the final version he wrote the appendix in correspondence to “the needs of a hypothetical educated reader.” However, Maestlin also answered the questions Kepler had while writing the Mysterium Cosmographicum in his appendix.

The preface in the 1596 republication of Rheticus' Narratio Prima was also written by Maestlin. This preface was an introduction to the work of Copernicus. Additionally, Maestlin made many contributions to tables and diagrams in Kepler's Mysterium Cosmographicum. Kepler's publication of Mysterium Cosmographicum was supervised by Maestlin, in which he added his own appendix to the publication over Copernican planetary theory with help from Erasmus Reinhold's Prutenic Tables.[20] The information Maestlin used for his appendix from the Prutenic Tables, by Erasmus Reinhold, was used to help readers that were not well educated in astronomy to be able to read Johannes Kepler's Mysterium Cosmographicum as not much information over the basics was included.