nLab long exact sequence of homotopy groups

Contents

Context

Homotopy theory

Idea
Properties
Related concepts
References

Idea

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

\cdots \to \pi_{n+1}(Z) \to \pi_n(X) \to \pi_n(Y) \to \pi_n(Z) \to \pi_{n-1}(X) \to \cdots \,.

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.

Properties

Proposition

(fundamental crossed module of a fibration)
Given a Serre fibration $p \,\colon\, T \longrightarrow B$ of pointed topological spaces, with fiber $\iota \,\colon\,F \longrightarrow T$ , the induced homomorphism of fundamental groups

\pi_1(F) \xrightarrow{\phantom{--} \iota_\ast \phantom{--}} \pi_1(T)

constitutes a crossed module with respect to the canonical group action $\alpha \,\colon\, \pi_1(T) \times \pi_1(F) \longrightarrow \pi_1(F)$ .

This is discussed in Brown, Higgins & Sivera 2011, §2.6, there attributed to Daniel Quillen.

Proposition

Given a Serre fibration $p \,\colon\, T \longrightarrow B$ of pointed topological spaces, with fiber $\iota \,\colon\,F \longrightarrow T$ , the image of the first connecting homomorphism of the corresponding long exact sequence of homotopy groups

\pi_2(B) \longrightarrow \pi_1(F)

is in the center $Z\big(\pi_1(F)\big) \hookrightarrow \pi_1(F)$ .

Proof

By exactness, the image in question is the kernel of $\pi_1(F) \to \pi_1(T)$ , which is the homomorphism of a crossed module by Prop. , and the kernels of such homomorphisms are central (by this Prop.).

Instead, textbooks usually state an analogous property for the LES of relative homotopy groups: For $(X,A)$ a “pair” — hence a subspace inclusion $A \hookrightarrow X$ (here: of pointed topological spaces) — there is a long exact sequence of the form

\cdots \to \pi_{n+1}(X,A) \longrightarrow \pi_n(A) \longrightarrow \pi_n(X) \longrightarrow \pi_n(X,A)

such that

\pi_2(A) \longrightarrow \pi_2(X,A)

factors through the center (cf. Whitehead 1978 Cor 3.5 on p. 166, tom Dieck 2008 Cor 6.2.7).

With a little care, this translates to the statement here by specializing $A \coloneqq T$ and taking $X \coloneqq Cyl(T \to B)$ to be the mapping cylinder of the fibration.

Related concepts

long exact sequence in homology
- long exact sequence in chain homology
- long exact sequence in generalized homology
fiber sequence
Gysin sequence
Serre long exact 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.

References

The observation of long exact sequences of homotopy groups for homotopy fiber sequences originates (according to Switzer 75, p. 35) in

M. G. Barratt, Track groups I, II. Proc. London Math. Soc. 5, 71-106, 285-329 (1955).

The first exhaustive study of these is due to

Dieter Puppe: Homotopiemengen und ihre induzierten Abbildungen I, Math. Z. 69 (1958) 299-344 [doi:10.1007/BF01187411, eudml:169745, pdf]

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 212 Springer (1975) [doi:10.1007/978-3-642-61923-6]
George W. Whitehead, section IV.2 in: Elements of Homotopy Theory, Springer (1978) [doi:10.1007/978-1-4612-6318-0]
Stanley Kochmann, Corollary 3.2.7 in: Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7 American Mathematical Society (1996) [ams:fim-7, cds:2264210]
Allen Hatcher, p. 344 of: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Tammo tom Dieck, Thm. 6.1.2 in: Algebraic topology, European Mathematical Society (2008) [doi:10.4171/048, pdf]
Anatoly Fomenko, Dmitry Fuchs, section 9.8 of: Homotopical Topology, Graduate Texts in Mathematics 273, Springer (2016) [doi:10.1007/978-3-319-23488-5, pdf]