# Puppe sequence

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In [mathematics](/source/mathematics), the '''Puppe sequence''' is a construction of [homotopy theory](/source/homotopy_theory), so named after [Dieter Puppe](/source/Dieter_Puppe). It comes in two forms: a [long exact sequence](/source/long_exact_sequence), built from the [mapping fibre](/source/homotopy_fiber) (a [fibration](/source/fibration)), and a long coexact sequence, built from the [mapping cone](/source/mapping_cone_(topology)) (which is a [cofibration](/source/cofibration)).<ref name=rotman>[Joseph J. Rotman](/source/Joseph_J._Rotman), ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag {{isbn|0-387-96678-1}} ''(See Chapter 11 for construction.)''</ref> Intuitively, the Puppe sequence allows us to think of [homology theory](/source/homology_theory) as a [functor](/source/functor) that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of [relative homotopy group](/source/relative_homotopy_group)s.

== Exact Puppe sequence ==

A sequence of pointed spaces and pointed maps <math display="block">\dots \to X_{n+1} \to X_n \to X_{n-1} \to \dots</math> is called exact if the induced sequence <math display="block">\dots \to [Z, X_{n+1}] \to [Z, X_n] \to [Z, X_{n-1}] \to \dots</math> is exact as a sequence of pointed sets (taking the kernel of a map to be those elements mapped to the basepoint) for every pointed space <math>Z</math>.

Let <math>f\colon (X,x_0)\to(Y,y_0)</math> be a [continuous map](/source/continuous_map) between [pointed space](/source/pointed_space)s and let <math>Mf</math> denote the [mapping fibre](/source/homotopy_fiber) (the [fibration](/source/fibration) dual to the [mapping cone](/source/mapping_cone_(topology))). One then obtains an exact sequence:

:<math>Mf\to X \to Y</math>

where the mapping fibre is defined as:<ref name=rotman/>

:<math>Mf = \{(x,\omega) \in X\times Y^I : \omega(0)=y_0 \mbox{ and } \omega(1)=f(x) \}</math>

Observe that the [loop space](/source/loop_space) <math>\Omega Y</math> injects into the mapping fibre: <math>\Omega Y \to Mf</math>, as it consists of those maps that both start and end at the basepoint <math>y_0</math>.  One may then show that the above sequence extends to the longer sequence

:<math>\Omega X \to \Omega Y \to Mf\to X \to Y</math>

The construction can then be iterated to obtain the exact Puppe sequence

:<math>\cdots \to \Omega^2(Mf) \to \Omega^2 X \to \Omega^2 Y \to \Omega(Mf) \to \Omega X \to \Omega Y \to Mf\to X \to Y</math>

The exact sequence is often more convenient than the coexact sequence in practical applications, as [Joseph J. Rotman](/source/Joseph_J._Rotman) explains:<ref name=rotman/>
:''(the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.''

==Examples==
===Example: Relative homotopy===
As a special case,<ref name=rotman/> one may take ''X'' to be a subspace ''A'' of ''Y'' that contains the basepoint ''y''<sub>0</sub>, and ''f'' to be the inclusion <math>i:A\hookrightarrow Y</math> of ''A'' into ''Y''.  One then obtains an exact sequence in the [category of pointed spaces](/source/category_of_pointed_spaces):

:<math>\begin{align}
\cdots &\to \pi_{n+1}(A) \to \pi_{n+1}(Y) \to \left [S^0,\Omega^n(Mi) \right ]\to \pi_n(A) \to \pi_n(Y)\to\cdots \\
\cdots &\to \pi_1(A) \to \pi_1(Y) \to \left [S^0,Mi \right ]\to \pi_0(A) \to \pi_0(Y)
\end{align}</math>

where the <math>\pi_n</math> are the [homotopy group](/source/homotopy_group)s, <math>S^0</math> is the zero-sphere (i.e. two points) and <math>[U,W]</math> denotes the [homotopy equivalence](/source/homotopy_equivalence) of maps from ''U'' to ''W''.  Note that <math>\pi_{n+1}(X)=\pi_1(\Omega^n X)</math>.  One may then show that 

:<math>\left [S^0,\Omega^n(Mi) \right ]= \left [S^n,Mi \right ]=\pi_n(Mi)</math>

is in [bijection](/source/bijection) to the relative homotopy group <math>\pi_{n+1}(Y,A)</math>, thus giving rise to the '''relative homotopy sequence of pairs'''

:<math>\begin{align}
\cdots &\to \pi_{n+1}(A) \to \pi_{n+1}(Y) \to \pi_{n+1}(Y,A) \to \pi_n(A) \to \pi_n(Y)\to\cdots \\
\cdots &\to \pi_1(A) \to \pi_1(Y) \to \pi_1(Y,A)\to \pi_0(A) \to \pi_0(Y)
\end{align}</math>

The object <math>\pi_n(Y,A)</math> is a group for <math>n\ge 2</math> and is abelian for <math>n\ge 3</math>.

===Example: Fibration===
As a special case,<ref name=rotman/> one may take ''f'' to be a [fibration](/source/fibration) <math>p:E\to B</math>. Then the [mapping fiber](/source/homotopy_fiber) ''Mp'' has the [homotopy lifting property](/source/homotopy_lifting_property) and it follows that ''Mp'' and the fiber <math>F=p^{-1}(b_0)</math> have the same [homotopy type](/source/homotopy_type). It follows trivially that maps of the sphere into ''Mp'' are homotopic to maps of the sphere to ''F'', that is, 

:<math>\pi_n(Mp) = \left [S^n,Mp \right ] \simeq \left [S^n, F \right ] = \pi_n(F).</math>

From this, the Puppe sequence gives the '''homotopy sequence of a fibration''':

:<math>\begin{align}
\cdots &\to \pi_{n+1}(E) \to \pi_{n+1}(B) \to \pi_n(F) \to \pi_n(E) \to \pi_n(B)\to\cdots \\
\cdots &\to \pi_1(E) \to \pi_1(B) \to \pi_0(F)\to \pi_0(E) \to \pi_0(B)
\end{align}</math>

===Example: Weak fibration===
[Weak fibrations](/source/Serre_fibration) are strictly weaker than fibrations, however, the main result above still holds, although the proof must be altered. The key observation, due to [Jean-Pierre Serre](/source/Jean-Pierre_Serre), is that, given a weak fibration <math>p\colon E\to B</math>, and the fiber at the basepoint given by <math>F=p^{-1}(b_0)</math>, that there is a bijection

:<math>p_*\colon \pi_n(E,F)\to\pi_n(B,b_0)</math>.

This bijection can be used in the relative homotopy sequence above, to obtain the '''homotopy sequence of a weak fibration''', having the same form as the fibration sequence, although with a different connecting map.

== Coexact Puppe sequence ==
Let <math>f\colon A \to B</math> be a [continuous map](/source/continuous_map) between [CW complex](/source/CW_complex)es and let <math>C(f)</math> denote a [mapping cone](/source/Mapping_cone_(topology)) of ''f'', (i.e., the cofiber of the map ''f''), so that we have a (cofiber) sequence:

:<math>A\to B\to C(f)</math>.

Now we can form <math>\Sigma A</math> and <math>\Sigma B,</math> [suspensions](/source/suspension_(topology)) of ''A'' and ''B'' respectively, and also <math>\Sigma f \colon \Sigma A \to \Sigma B</math> (this is because [suspension](/source/suspension_(topology)) might be seen as a [functor](/source/functor)), obtaining  a sequence:

:<math>\Sigma A \to \Sigma B \to C(\Sigma f)</math>.

Note that suspension preserves cofiber sequences.

Due to this powerful fact we know that <math>C(\Sigma f)</math> is [homotopy equivalent](/source/homotopy_equivalent) to <math>\Sigma C(f).</math> By collapsing <math>B\subset C(f)</math> to a point, one has a natural map <math>C(f) \to \Sigma A.</math> Thus we have a sequence:

:<math>A\to B\to C(f) \to \Sigma A \to \Sigma B \to \Sigma C(f).</math> 

Iterating this construction, we obtain the Puppe sequence associated to <math>A\to B</math>:

:<math>A\to B\to C(f) \to \Sigma A \to \Sigma B \to \Sigma C(f) \to \Sigma^2 A \to \Sigma^2 B \to \Sigma^2 C(f) \to \Sigma^3 A \to \Sigma^3 B \to \Sigma^3 C(f) \to \cdots </math>

==Some properties and consequences==
It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form:

: <math>X\to Y\to C(f)</math>.

By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in the [homotopy category](/source/homotopy_category).

If one is now given a [topological half-exact functor](/source/topological_half-exact_functor), the above property implies that, after acting with the functor in question on the Puppe sequence associated to <math>A\to B</math>, one obtains a long [exact sequence](/source/exact_sequence).

A result, due to [John Milnor](/source/John_Milnor),<ref>[John Milnor](/source/John_Milnor) "Construction of Universal Bundles I" (1956) ''[Annals of Mathematics](/source/Annals_of_Mathematics)'', '''63''' pp. 272-284.</ref> is that if one takes the [Eilenberg–Steenrod axioms](/source/Eilenberg%E2%80%93Steenrod_axioms) for [homology theory](/source/homology_theory), and replaces excision by the exact sequence of a [weak fibration](/source/Serre_fibration) of pairs, then one gets the homotopy analogy of the [Eilenberg–Steenrod theorem](/source/Eilenberg%E2%80%93Steenrod_theorem): there exists a unique sequence of functors <math>\pi_n\colon P\to\bf{Sets}</math> with ''P'' the category of all pointed pairs of topological spaces.

==Remarks==
As there are two "kinds" of [suspension](/source/suspension_(topology)), [unreduced and reduced](/source/suspension_(topology)), one can also consider unreduced and reduced Puppe sequences (at least if dealing with [pointed space](/source/pointed_space)s, when it's possible to form reduced suspension).

==References==
<references/>
* [Edwin Spanier](/source/Edwin_Spanier), ''Algebraic Topology'', Springer-Verlag (1982) ''Reprint, McGraw Hill (1966)''

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Category:Homotopy theory

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Adapted from the Wikipedia article [Puppe sequence](https://en.wikipedia.org/wiki/Puppe_sequence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Puppe_sequence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.