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Stable Homotopy Theory

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Part of the Lecture Notes in Mathematics book series (LNM,volume 3)

Abstract

There is a number of phenomena in homotopy theory which are independent of the precise dimensions considered, provided that the dimensions are large enough. For example, nn+1(Sn) = Z2 for n > 2. Such phenomena, in general, are called stable. One can also point to more complicated theorems (e.g. about spectral sequences) such that each clause of the theorem is true for sufficiently large n, but there is no n which makes all the clauses of the theorem true at once. In proving such a theorem, if you don’t take care, you rapidly find yourself carrying a large number of explicit conditions n > N(p, q, r, ...), which are not only tedious but basically irrelevant. What we want is a standard convention that we are only considering what happens for sufficiently large n . One anproach is to work in a suitably constructed category, in which the objects are not spaces but “stable objects” of some sort. For example, the S-theory of Spanier and Whitehead is such a category.

Keywords

  • Exact Sequence
  • Stable Complex
  • Spectral Sequence
  • Homotopy Theory
  • Stable Object

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1966 Springer-Verlag Berlin Heidelberg

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Adams, J.F. (1966). Stable Homotopy Theory. In: Stable Homotopy Theory. Lecture Notes in Mathematics, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-15905-7_3

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  • DOI: https://doi.org/10.1007/978-3-662-15905-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-15907-1

  • Online ISBN: 978-3-662-15905-7

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