Puppe fiber sequence

Intuition [Beginner]

Given a continuous map $f:X\to Y$, the fibre of $f$ over a basepoint $y_0$ is the set of points in $X$ that map to $y_0$. But for homotopy theory, we use a better version called the homotopy fibre $F_f$: instead of just taking the inverse image, we take all points in $X$ together with a path in $Y$ from $f(x)$ to $y_0$.

The Puppe fibre sequence says that from $f:X\to Y$, you get an infinite sequence:

$$\cdots \to {\Omega }^{2}Y\to \Omega F_f\to \Omega X\to \Omega Y\to F_f\to X\stackrel{f}{\to }Y$$

where $\Omega$ denotes the based loop space (all loops at $y_0$). Applying homotopy groups produces a long exact sequence. This is the Eckmann-Hilton dual of the cofibre sequence: where the cofibre sequence uses cones and suspensions (going up in dimension), the fibre sequence uses paths and loops (going down in dimension).

Visual [Beginner]

A vertical sequence of spaces. At the top, $Y$ (a target space). Below it, $X$ mapping into $Y$ via $f$. Below $X$, the homotopy fibre $F_f$ (drawn as the subspace of $X$ whose images are connected to $y_0$ by paths in $Y$). Below that, $\Omega Y$ (loops at $y_0$). Below that, $\Omega X$, then $\Omega F_f$, then ${\Omega }^{2}Y$.

The pattern iterates by taking loop spaces. Each loop-space level drops the dimension of the homotopy groups by one.

Worked example [Beginner]

The Hopf fibration. The Hopf map $f:S^3\to S^2$ has fibre $S^1$. The Puppe fibre sequence gives:

$$\cdots \to \Omega S^3\to \Omega S^2\to S^1\to S^3\to S^2$$

Applying ${\pi }_3$ (using ${\pi }_3(S^1)=0$ and ${\pi }_2(S^1)=0$) yields: $0\to {\pi }_3(S^2)\to {\pi }_2(S^1)\to {\pi }_2(S^3)\to {\pi }_2(S^2)\to {\pi }_1(S^1)\to {\pi }_1(S^3)\to {\pi }_1(S^2)$. Substituting known values: $0\to {\pi }_3(S^2)\to 0\to 0\to \mathbb{Z}\to \mathbb{Z}\to 0\to 0$. This gives ${\pi }_3(S^2)\cong \mathbb{Z}$, the classic result discovered by Hopf in 1931.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Homotopy fibre). Let $f:X\to Y$ be a map of based spaces with $f(x_0)=y_0$. The homotopy fibre of $f$ is the pullback:

$$F_f=X{\times }_{Y}PY=\{(x,\gamma )\in X\times Y^I:f(x)=\gamma (0),\gamma (1)=y_0\}$$

where $PY=\{\gamma \in Y^I:\gamma (1)=y_0\}$ is the based path space. The projection $F_f\to X$ sends $(x,\gamma )\mapsto x$.

Definition (Puppe fibre sequence). Given $f:X\to Y$, the Puppe fibre sequence is:

$$\cdots \to {\Omega }^{2}X\to {\Omega }^{2}Y\to \Omega F_f\to \Omega X\stackrel{\Omega f}{\to }\Omega Y\to F_f\stackrel{j}{\to }X\stackrel{f}{\to }Y$$

where $j:F_f\to X$ is the projection and the connecting map $\Omega Y\to F_f$ sends a loop $\omega$ to $(\ast ,\omega )\in F_f$ (the basepoint of $X$ together with the loop as the path in $Y$).

Theorem (Long exact sequence on homotopy groups). The Puppe fibre sequence induces a long exact sequence of groups (and a pointed set at the end):

$$\cdots \to {\pi }_n(F_f)\to {\pi }_n(X)\stackrel{f_{\ast }}{\to }{\pi }_n(Y)\stackrel{\partial }{\to }{\pi }_{n-1}(F_f)\to \cdots \to {\pi }_0(F_f)\to {\pi }_0(X)\to {\pi }_0(Y)$$

Key theorem with proof [Intermediate+]

Theorem (Action of $\Omega Y$ on $F_f$ and exactness). The loop space $\Omega Y$ acts on the homotopy fibre $F_f$ by concatenation: for $(x,\gamma )\in F_f$ and $\omega \in \Omega Y$, define $(x,\gamma )\cdot \omega =(x,\gamma \ast \omega )$. This action makes the sequence $\Omega Y\to F_f\to X$ into a fibre sequence, and for $n\ge 1$, the image of ${\pi }_n(\Omega Y)\to {\pi }_n(F_f)$ is the kernel of ${\pi }_n(F_f)\to {\pi }_n(X)$, giving exactness.

Proof. The action $(x,\gamma )\cdot \omega =(x,\gamma \ast \omega )$ is continuous and satisfies the axioms of a (right) group action up to homotopy. For exactness at ${\pi }_n(Y)$: the boundary map $\partial :{\pi }_n(Y)\to {\pi }_{n-1}(F_f)$ sends a class $[\alpha :S^n\to Y]$ to the class of the lift $\tilde{\alpha }:S^{n-1}\to F_f$ obtained by viewing $S^n$ as the mapping cone of $S^{n-1}$ and reading off the homotopy fibre data. The composition $f_{\ast }\circ j_{\ast }$ is zero because $f\circ j$ factors through the contractible path space $PY$. Conversely, if $f_{\ast }[\beta ]=0$ for $[\beta ]\in {\pi }_n(X)$, then $f\circ \beta$ is null-homotopic, and the null-homotopy provides a path lifting $\beta$ to a map $S^n\to F_f$, so $[\beta ]=j_{\ast }[\tilde{\beta }]$. $\square$

Bridge. The fibre sequence is the Eckmann-Hilton dual of the cofibre sequence 03.12.27; where that sequence builds cones and suspensions going upward in dimension, this sequence builds path spaces and loop spaces going downward. The homotopy fibre $F_f$ is the pullback of the path fibration $PY\to Y$ studied in [topology.fibration], and the action of $\Omega Y$ on $F_f$ mirrors the coaction of $\Sigma A$ on the cofibre $C_f$, connecting the two sequences through the loop-suspension adjunction that also underlies [topology.compact-open-topology].

Exercises [Intermediate+]

Advanced results [Master]

Fibre sequence from a fibration. If $p:E\to B$ is a Hurewicz fibration with fibre $F=p^{-1}(b_0)$, then $F\simeq F_p$ (the homotopy fibre is weakly equivalent to the actual fibre). This is the practical content: for genuine fibrations, the Puppe fibre sequence uses the literal fibre, not the homotopy fibre.

Loop-space recognition. The Puppe fibre sequence is the primary tool for recognising loop spaces. If $F\to E\to B$ is a fibre sequence with $E$ contractible, then $F\simeq \Omega B$. This connects directly to the classifying-space construction $BG$ and the bar resolution for simplicial groups 03.12.39.

Synthesis. The fibre sequence converts the homotopy-lifting property of [topology.fibration] into exact algebraic data, dualising the cofibre sequence 03.12.27 by replacing cones with path spaces and suspensions with loops. The action of $\Omega Y$ on $F_f$ is the fibre-level manifestation of the homotopy-group long exact sequence from [topology.fibration], the loop-space recognition result connects to the bar construction 03.12.39 where $G\simeq \Omega BG$, and the iteration through loop spaces mirrors the iteration through suspensions in stable homotopy theory. Together, the cofibre and fibre sequences provide the two pillars on which all of homotopy theory rests.

Full proof set [Master]

Proposition (Fibre of the inclusion $j:F_f\to X$). The homotopy fibre of the projection $j:F_f\to X$ is $\Omega Y$, and the sequence $\Omega Y\to F_f\to X$ is a fibre sequence.

Proof. The homotopy fibre of $j$ is $F_j=F_f{\times }_{X}PX$. An element of $F_j$ is a triple $((x,\gamma ),\omega )$ where $(x,\gamma )\in F_f$ (so $f(x)=\gamma (0)$ and $\gamma (1)=y_0$) and $\omega :I\to X$ is a path with $\omega (0)=x$ and $\omega (1)=x_0$. But since $PX$ is contractible, the projection $F_j\to \Omega Y$ sending $((x,\gamma ),\omega )\mapsto f\circ \omega \ast {\gamma }^{-1}$ (concatenate $f\circ \omega$ with the reverse of $\gamma$) is a homotopy equivalence. The inverse sends a loop $\lambda \in \Omega Y$ to $((x_0,c_{y_0}),c_{x_0},\lambda )$ where $c$ denotes constant paths. Hence $F_j\simeq \Omega Y$. $\square$

Connections [Master]

The cofibre sequence 03.12.27 is the Eckmann-Hilton dual; the two sequences together give a complete picture of how maps produce exact sequences, and they converge to the same stable exact triangles.

Fibrations [topology.fibration] and the homotopy lifting property are the starting point: the Puppe fibre sequence extends the long exact sequence of a fibration to an infinite sequence.

The loop-space recognition principle connects to the W-bar construction 03.12.39, where $G\simeq \Omega BG$ is proved via the fibre sequence of the universal bundle $EG\to BG$.

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's 1958 paper ^{[Puppe 1958]} introduced both the cofibre and fibre sequences simultaneously, recognising the duality between them. The fibre sequence formalises a pattern that had been used in the theory of fibrations since the work of Hurewicz and Steenrod in the 1940s.

The philosophical significance is twofold. First, the fibre sequence shows that every continuous map, not just fibrations, generates long exact sequences in homotopy groups (via the homotopy fibre replacement). Second, the duality between fibre and cofibre sequences reveals a deep structural symmetry in homotopy theory: the same algebraic machinery (exact sequences) captures both "pullback" geometry (fibrations) and "pushout" geometry (cofibrations). This duality becomes an equivalence in the stable category, where suspension is invertible.