Puppe cofiber sequence

Intuition [Beginner]

Given a continuous map $f:A\to X$ between spaces, you can form a new space called the cofibre (or mapping cone) $C_f$ by taking $X$ and gluing a cone on $A$ along the map $f$. This is a geometric construction: attach the cone $CA=A\times [0,1]/A\times \{1\}$ to $X$ by identifying $(a,0)$ with $f(a)$ for each $a\in A$.

The Puppe cofibre sequence says that from $f$, you get an infinite sequence of spaces and maps:

$$A\stackrel{f}{\to }X\to C_f\to \Sigma A\to \Sigma X\to \Sigma C_f\to {\Sigma }^{2}A\to \cdots$$

where $\Sigma$ denotes suspension (doubling the cone). Applying homotopy groups to this sequence produces a long exact sequence, which is a powerful computational tool. Each arrow carries information about the map $f$, and the sequence lets you extract homotopy-group data step by step.

Visual [Beginner]

A horizontal sequence of spaces connected by arrows. On the left, $A$ (a circle) maps via $f$ into $X$ (a cylinder). Next, the mapping cone $C_f$: $X$ with a cone on $A$ attached. Next, $\Sigma A$ (a suspension, drawn as a diamond shape). Next, $\Sigma X$ (a larger diamond). Each space is connected to the next by a continuous map.

The pattern repeats: cone, suspend, cone, suspend. Each step lifts the information one suspension higher.

Worked example [Beginner]

The inclusion $S^1\hookrightarrow D^2$. Let $f:S^1\to D^2$ be the inclusion of the boundary circle into the disk. The mapping cone $C_f$ is obtained by gluing a cone on $S^1$ to $D^2$ along the boundary. Since $D^2$ is already a cone on $S^1$, attaching another cone produces $S^2$ (the result is a 2-sphere, formed by gluing two disks along their boundaries).

The Puppe sequence gives: $S^1\to D^2\to S^2\to \Sigma S^1=S^2\to \Sigma D^2\to \Sigma S^2=S^3\to \cdots$. On homotopy groups this yields ${\pi }_n(S^1)\to {\pi }_n(D^2)\to {\pi }_n(S^2)\to {\pi }_{n-1}(S^1)\to \cdots$, which for $n=2$ gives $0\to 0\to {\pi }_2(S^2)\to {\pi }_1(S^1)\to 0$, confirming ${\pi }_2(S^2)\cong \mathbb{Z}$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Mapping cone). Let $f:A\to X$ be a continuous map. The mapping cone (or homotopy cofibre) of $f$ is the pushout:

$$C_f=X{\cup }_{f}CA=(X\bigsqcup CA)/(f(a)\sim (a,0))$$

where $CA=A\times [0,1]/A\times \{1\}$ is the cone on $A$. The inclusion $X\hookrightarrow C_f$ is always a cofibration.

Definition (Puppe cofibre sequence). Given $f:A\to X$, the Puppe cofibre sequence is the infinite sequence:

$$A\stackrel{f}{\to }X\stackrel{i}{\to }{C}_f\stackrel{q}{\to }\Sigma A\stackrel{-\Sigma f}{\to }\Sigma X\stackrel{-\Sigma i}{\to }\Sigma C_f\stackrel{-\Sigma q}{\to }{\Sigma }^{2}A\to \cdots$$

where $i:X\hookrightarrow C_f$ is the inclusion, $q:C_f\to \Sigma A$ collapses $X$ to a point (and maps the cone $CA$ to $\Sigma A=CA/A$), and the negative signs indicate reversal of the suspension coordinate. Each map in the sequence is the "connecting map" from the previous mapping cone.

Theorem (Long exact sequence on homotopy groups). For any based space $Y$, applying $[-,Y]$ to the Puppe cofibre sequence produces a long exact sequence of pointed sets (groups from $\Sigma A$ onward):

$$\cdots \to [\Sigma C_f,Y]\to [\Sigma X,Y]\to [\Sigma A,Y]\to [C_f,Y]\to [X,Y]\to [A,Y]$$

Key theorem with proof [Intermediate+]

Theorem (Cofibre sequence exactness). The Puppe cofibre sequence is exact: the image of each map equals the kernel (i.e., the set of null-homotopic classes) of the next map, at every stage.

Proof sketch. For the first three terms: $[A,Y]\leftarrow [X,Y]\leftarrow [C_f,Y]$, exactness at $[X,Y]$ means $i^{\ast }[g]=0$ iff $g$ extends over $C_f$. If $g\circ i$ is null-homotopic via $H:X\times I\to Y$ with $H(-,0)=g\circ i$ and $H(-,1)=\ast$, then $g$ and $H$ together define a map $C_f=X{\cup }_{f}CA\to Y$, i.e., $g$ extends. The converse is immediate: if $g$ extends to $\tilde{g}:C_f\to Y$, then $\tilde{g}\circ i=g$ composed with the null-homotopy of the cone gives $g\circ i\simeq \ast$.

The higher stages follow by suspension: $[{\Sigma }^n{C}_f,Y]=[C_f,{\Omega }^{n}Y]$ and the same argument applies by iterating. $\square$

Bridge. The cofibre sequence is the dual of the fibre sequence that arises from [topology.fibration]; where fibrations pull back along maps and produce long exact sequences in homotopy groups going down, cofibrations push forward and produce long exact sequences going up. The mapping cone $C_f$ plays the same role in the cofibre world that the homotopy fibre $F_f$ plays in the fibre world, and the suspension $\Sigma A$ that appears at the third term connects back to the CW-attachment language of 03.12.26 via the identification $\Sigma A\simeq CA/A$, a quotient of the cone that is the cofibre of $A\to \ast$.

Exercises [Intermediate+]

Advanced results [Master]

Coaction on the cofibre. The cofibre $C_f$ of a map $f:A\to X$ of based spaces carries a coaction of $\Sigma A$, i.e., a map $C_f\to C_f\vee \Sigma A$ making the sequence $C_f\to \Sigma A\to \Sigma X$ a cofibre sequence in the based category. This coaction is the dual of the action of $\Omega B$ on the fibre $F$ of a fibration.

Eckmann-Hilton duality. The Puppe cofibre sequence and the Puppe fibre sequence 03.12.28 are related by Eckmann-Hilton duality, which exchanges: injective $\leftrightarrow$ surjective, cofibration $\leftrightarrow$ fibration, cone $\leftrightarrow$ path, suspension $\leftrightarrow$ loop. This duality is not a theorem but a guiding analogy; it becomes precise in the stable category where suspension is invertible.

Synthesis. The cofibre sequence is the first step in stable homotopy theory; it iterates the mapping-cone construction of [topology.cofibration] to produce an infinite exact sequence that, upon passage to stable homotopy groups, becomes the exact triangle in the triangulated category of spectra. The CW-approximation of 03.12.26 ensures that any map can be replaced by a cofibration between CW complexes before forming the cofibre sequence, and the suspension $\Sigma$ that drives the iteration is the same construction that appears in the Freudenthal suspension theorem emerging from [topology.blakers-massey]. The coaction on the cofibre dualises the action of $\Omega B$ on the fibre from 03.12.28, completing the picture of how homotopy theory organises maps into exact sequences.

Full proof set [Master]

Proposition (Iterated cofibre). Let $f:A\to X$ be a map of based spaces. Let $i:X\to C_f$ be the inclusion and $q:C_f\to \Sigma A$ the quotient map. Then the homotopy cofibre of $q$ is $\Sigma X$, and the homotopy cofibre of $-\Sigma f:\Sigma A\to \Sigma X$ is $\Sigma C_f$.

Proof. The cofibre of $q$ is $C_q=C_f{\cup }_{q}C(C_f)$. But $C(C_f)=C(X{\cup }_{f}CA)=CX{\cup }_{f\times \mathrm{id}}C(CA)$. Since $C(CA)=CA\times I/CA\times \{1\}$ deformation-retracts onto $CA\times \{1\}/\sim$ which is a cone on $\Sigma A$, and collapsing $C_f$ to a point leaves $CX/X\cong \Sigma X$. A careful rearrangement of the pushout squares shows $C_q\simeq \Sigma X$. The second claim follows by suspension of the first: $\Sigma C_f=C_{\Sigma f}$ by naturality of the cone construction. $\square$

Connections [Master]

The CW approximation 03.12.26 ensures that any map can be replaced by a cellular cofibration before forming the cofibre sequence, guaranteeing good homotopical behaviour.

The Puppe fibre sequence 03.12.28 is the Eckmann-Hilton dual: where the cofibre sequence uses cones and suspensions, the fibre sequence uses paths and loops.

Cofibrations [topology.cofibration] and the homotopy extension property are the structural foundation on which the cofibre sequence is built; the mapping cone of a cofibration is homotopy equivalent to the quotient.

Bibliography [Master]

@article{puppe1958,
  author = {Puppe, Dieter},
  title = {Homotopiemengen und ihre induzierten Abbildungen},
  journal = {Math. Z.},
  volume = {69},
  pages = {299--344},
  year = {1958}
}

@book{may-concise,
  author = {May, J. Peter},
  title = {A Concise Course in Algebraic Topology},
  publisher = {University of Chicago Press},
  year = {1999}
}

@book{hatcher2002,
  author = {Hatcher, Allen},
  title = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year = {2002}
}

@book{whitehead-elements,
  author = {Whitehead, George W.},
  title = {Elements of Homotopy Theory},
  publisher = {Springer},
  year = {1978}
}

Historical & philosophical context [Master]

Dieter Puppe introduced the cofibre and fibre sequences in his 1958 paper "Homotopiemengen und ihre induzierten Abbildungen" ^{[Puppe 1958]}, which established the exactness properties that bear his name. The cofibre sequence formalises a pattern that had been used implicitly since the work of J.H.C. Whitehead on CW complexes.

The philosophical significance of the cofibre sequence is that it converts geometric information about a map (how $A$ sits inside $X$) into algebraic information (a long exact sequence of homotopy groups). This geometry-to-algebra translation is the central theme of algebraic topology. The iteration through suspensions connects to stable homotopy theory, where the cofibre sequence becomes an exact triangle in a triangulated category, the foundation for modern chromatic homotopy theory.