homotopy theory, (∞,1)-category theory, homotopy type theory
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For $Y \to Z$ a morphism of pointed ∞-groupoids and $X \to Y$ its homotopy fiber, there is a long exact sequence of homotopy groups
In terms of presentations this means:
for $Y \to Z$ a fibration in the classical model structure on topological spaces or in the classical model structure on simplicial sets, and for $X \to Y$ the ordinary fiber of topological spaces or simplicial sets, respectively, we have such a long exact sequence.
For background and details see fibration sequence.
Given a tower of homotopy fibers such as a Whitehead tower or Adams resolution, the long exact sequences of homotopy groups for each stage combine to yield an exact couple. The corresponding spectral sequence is the Adams spectral sequence.
The observation of long exact sequences of homotopy groups for homotopy fiber sequences originates (according to Switzer 75, p. 35) in
The first exhaustive study of these is due to
whence the terminology Puppe sequences.
Textbook accounts:
Norman Steenrod, Thm. 17.4 in: The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)
Robert Switzer, around 2.59 in: Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Stanley Kochmann, Corollary 3.2.7 in: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Tammo tom Dieck, Thm. 6.1.2 in: Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)
See also:
In the generality of categorical homotopy groups in an $(\infty,1)$-topos:
Last revised on August 17, 2021 at 12:53:04. See the history of this page for a list of all contributions to it.