In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration).[1] Intuitively, the Puppe sequence allows us to think of homology theory as a 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 groups.
A sequence of pointed spaces and pointed maps is called exact if the induced sequence 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
Z
Let
f\colon(X,x0)\to(Y,y0)
Mf
Mf\toX\toY
where the mapping fibre is defined as:[1]
Mf=\{(x,\omega)\inX x YI:\omega(0)=y0and\omega(1)=f(x)\}
\OmegaY
\OmegaY\toMf
y0
\OmegaX\to\OmegaY\toMf\toX\toY
The construction can then be iterated to obtain the exact Puppe sequence
… \to\Omega2(Mf)\to\Omega2X\to\Omega2Y\to\Omega(Mf)\to\OmegaX\to\OmegaY\toMf\toX\toY
The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:[1]
(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.
As a special case,[1] one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion
i:A\hookrightarrowY
\begin{align} … &\to\pin+1(A)\to\pin+1(Y)\to\left[S0,\Omegan(Mi)\right]\to\pin(A)\to\pin(Y)\to … \\ … &\to\pi1(A)\to\pi1(Y)\to\left[S0,Mi\right]\to\pi0(A)\to\pi0(Y) \end{align}
where the
\pin
S0
[U,W]
\pin+1
n | |
(X)=\pi | |
1(\Omega |
X)
\left[S0,\Omegan(Mi)\right]=\left[Sn,Mi\right]=\pin(Mi)
is in bijection to the relative homotopy group
\pin+1(Y,A)
\begin{align} … &\to\pin+1(A)\to\pin+1(Y)\to\pin+1(Y,A)\to\pin(A)\to\pin(Y)\to … \\ … &\to\pi1(A)\to\pi1(Y)\to\pi1(Y,A)\to\pi0(A)\to\pi0(Y) \end{align}
The object
\pin(Y,A)
n\ge2
n\ge3
p:E\toB
F=p-1(b0)
\pin(Mp)=\left[Sn,Mp\right]\simeq\left[Sn,F\right]=\pin(F).
From this, the Puppe sequence gives the homotopy sequence of a fibration:
\begin{align} … &\to\pin+1(E)\to\pin+1(B)\to\pin(F)\to\pin(E)\to\pin(B)\to … \\ … &\to\pi1(E)\to\pi1(B)\to\pi0(F)\to\pi0(E)\to\pi0(B) \end{align}
Weak fibrations 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, is that, given a weak fibration
p\colonE\toB
F=p-1(b0)
p*\colon\pin(E,F)\to\pin(B,b0)
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.
Let
f\colonA\toB
C(f)
A\toB\toC(f)
Now we can form
\SigmaA
\SigmaB,
\Sigmaf\colon\SigmaA\to\SigmaB
\SigmaA\to\SigmaB\toC(\Sigmaf)
Note that suspension preserves cofiber sequences.
Due to this powerful fact we know that
C(\Sigmaf)
\SigmaC(f).
B\subsetC(f)
C(f)\to\SigmaA.
A\toB\toC(f)\to\SigmaA\to\SigmaB\to\SigmaC(f).
Iterating this construction, we obtain the Puppe sequence associated to
A\toB
A\toB\toC(f)\to\SigmaA\to\SigmaB\to\SigmaC(f)\to\Sigma2A\to\Sigma2B\to\Sigma2C(f)\to\Sigma3A\to\Sigma3B\to\Sigma3C(f)\to …
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:
X\toY\toC(f)
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.
If one is now given a topological half-exact functor, the above property implies that, after acting with the functor in question on the Puppe sequence associated to
A\toB
A result, due to John Milnor,[2] is that if one takes the Eilenberg–Steenrod axioms for homology theory, and replaces excision by the exact sequence of a weak fibration of pairs, then one gets the homotopy analogy of the Eilenberg–Steenrod theorem: there exists a unique sequence of functors
\pin\colonP\to\bf{Sets}
As there are two "kinds" of suspension, unreduced and reduced, one can also consider unreduced and reduced Puppe sequences (at least if dealing with pointed spaces, when it's possible to form reduced suspension).