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). 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.
Exact Puppe sequence A sequence of pointed spaces and pointed maps ⋯ → X n + 1 → X n → X n − 1 → … {\displaystyle \dots \to X_{n+1}\to X_{n}\to X_{n-1}\to \dots } is called exact if the induced sequence ⋯ → [ Z , X n + 1 ] → [ Z , X n ] → [ Z , X n − 1 ] → … {\displaystyle \dots \to [Z,X_{n+1}]\to [Z,X_{n}]\to [Z,X_{n-1}]\to \dots } 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 {\displaystyle Z} . Let f : ( X , x 0 ) → ( Y , y 0 ) {\displaystyle f\colon (X,x_{0})\to (Y,y_{0})} be a continuous map between pointed spaces and let M f {\displaystyle Mf} denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:
M f → X → Y {\displaystyle Mf\to X\to Y}
where the mapping fibre is defined as:
M f = { ( x , ω ) ∈ X × Y I : ω ( 0 ) = y 0 and ω ( 1 ) = f ( x ) } {\displaystyle Mf=\{(x,\omega )\in X\times Y^{I}:\omega (0)=y_{0}{\mbox{ and }}\omega (1)=f(x)\}}
Observe that the loop space Ω Y {\displaystyle \Omega Y} injects into the mapping fibre: Ω Y → M f {\displaystyle \Omega Y\to Mf} , as it consists of those maps that both start and end at the basepoint y 0 {\displaystyle y_{0}} . One may then show that the above sequence extends to the longer sequence
Ω X → Ω Y → M f → X → Y {\displaystyle \Omega X\to \Omega Y\to Mf\to X\to Y}
The construction can then be iterated to obtain the exact Puppe sequence
⋯ → Ω 2 ( M f ) → Ω 2 X → Ω 2 Y → Ω ( M f ) → Ω X → Ω Y → M f → X → Y {\displaystyle \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}
The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:
(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, one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion i : A ↪ Y {\displaystyle i:A\hookrightarrow Y} of A into Y. One then obtains an exact sequence in the category of pointed spaces:
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