Long Exact Sequence of a Fibration

⚡ Key Takeaways

A fibration $F \hookrightarrow E \xrightarrow{p} B$ induces a long exact sequence relating the homotopy groups of the fiber $F$ , total space $E$ , and base $B$ :

\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots

Key result: The connecting homomorphism $\partial$ encodes how loops in the base "unwind" into paths in the fiber, making homotopy groups computable by induction on fibrations. Applies to: algebraic topology, computation of homotopy groups of spheres, classifying spaces, and gauge theory. Origin: Hurewicz, Hopf, and Serre (1930s–1950s).

💡 The Core Insight

A fibration is a map with the homotopy lifting property: homotopies in the base can be lifted to homotopies in the total space, given a starting lift. This single property forces the fiber inclusion $i : F \to E$ and projection $p : E \to B$ to behave, on homotopy groups, like a short exact sequence that fails to close up—and the failure is measured precisely by a connecting map $\partial : \pi_n(B) \to \pi_{n-1}(F)$ . Geometrically, $\partial$ takes a sphere in the base, lifts it to a disk in $E$ , and reads off the boundary sphere lying in $F$ .

DefinitionFibration

A continuous map $p : E \to B$ is a Serre fibration if it has the homotopy lifting property with respect to all CW complexes (equivalently all disks $D^n$ ): given a homotopy $H : X \times I \to B$ and an initial lift $\tilde{h}_0 : X \to E$ with $p \circ \tilde{h}_0 = H(-, 0)$ , there exists a lift $\tilde{H} : X \times I \to E$ with $p \circ \tilde{H} = H$ and $\tilde{H}(-, 0) = \tilde{h}_0$ . If the lifting property holds for all topological spaces $X$ , $p$ is a Hurewicz fibration. The fiber over a basepoint $b_0 \in B$ is $F = p^{-1}(b_0)$ , with a chosen basepoint $e_0 \in F$ .

DefinitionThe Maps

Fix basepoints $b_0 \in B$ , $e_0 \in F \subseteq E$ . The inclusion $i : F \hookrightarrow E$ and projection $p : E \to B$ induce homomorphisms

1

\quad i_* : \pi_n(F, e_0) \to \pi_n(E, e_0), \qquad p_* : \pi_n(E, e_0) \to \pi_n(B, b_0).

Derivation hint: functoriality of $\pi_n$ applied to the maps $i$ and $p$ . The connecting homomorphism is defined on a class $[\alpha] \in \pi_n(B, b_0)$ represented by $\alpha : (D^n, \partial D^n) \to (B, b_0)$ by lifting $\alpha$ to $\tilde{\alpha} : D^n \to E$ (using HLP, with $\partial D^n$ mapping into $F$ ) and setting

2

\quad \partial[\alpha] = [\tilde{\alpha}|_{\partial D^n}] \in \pi_{n-1}(F, e_0).

Derivation hint: a sphere in $B$ lifts to a disk in $E$ ; its boundary lands in $F = p^{-1}(b_0)$ since $\alpha(\partial D^n) = b_0$ .

TheoremLong Exact Sequence of a Fibration

Let $p : E \to B$ be a Serre fibration with fiber $F = p^{-1}(b_0)$ , and assume $B$ is path-connected. Then there is an exact sequence of groups (and pointed sets at the low end):

3

\quad \cdots \to \pi_n(F) \xrightarrow{i_*} \pi_n(E) \xrightarrow{p_*} \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \xrightarrow{i_*} \cdots \to \pi_0(E) \to \pi_0(B).

Derivation hint: identify the sequence with the LES of the pair $(E, F)$ via the isomorphism $p_* : \pi_n(E, F) \xrightarrow{\sim} \pi_n(B)$ . The maps are homomorphisms for $n \geq 1$ ; the tail $\pi_1(B) \to \pi_0(F) \to \pi_0(E) \to \pi_0(B)$ is an exact sequence of pointed sets. The group $\pi_1(B)$ acts on $\pi_0(F)$ .

ProofConstruction via the Long Exact Sequence of a Pair

Step 1 — The relative LES. For any pair $(E, F)$ there is the long exact sequence of relative homotopy groups (a standard result, see Long Exact Sequence of a Pair):

4

\quad \cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(E, F) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots

Derivation hint: exactness is built into the definitions of $\pi_n(E,F)$ via the boundary map restricting maps $(D^n, \partial D^n, *) \to (E, F, *)$ to $\partial D^n$ .

Step 2 — The key isomorphism. The fibration property forces

5

\quad p_* : \pi_n(E, F) \xrightarrow{\ \cong\ } \pi_n(B, b_0) = \pi_n(B).

Derivation hint: a relative class is a map $(D^n, \partial D^n) \to (E, F)$ ; projecting gives $(D^n, \partial D^n) \to (B, b_0)$ . HLP gives a unique-up-to-homotopy lift back, establishing a bijection.

Step 3 — Surjectivity of $p_*$ on relative groups. Given $\alpha : (D^n, \partial D^n) \to (B, b_0)$ , apply the homotopy lifting property to the constant homotopy contracting $D^n$ to a point, lifting $\alpha$ to $\tilde\alpha : D^n \to E$ with $\tilde\alpha(\partial D^n) \subseteq F$ . Then $p_*[\tilde\alpha] = [\alpha]$ .

Step 4 — Injectivity of $p_*$ . If $p_*[\tilde\alpha] = p_*[\tilde\beta]$ , a homotopy of the projected maps lifts (again by HLP) to a homotopy of relative classes, so $[\tilde\alpha] = [\tilde\beta]$ in $\pi_n(E, F)$ .

Step 5 — Substitute. Replacing $\pi_n(E, F)$ by $\pi_n(B)$ in [4] via the isomorphism [5] yields the sequence [3]. The connecting map $\partial : \pi_n(E,F) \to \pi_{n-1}(F)$ composed with $(p_*)^{-1}$ gives exactly the geometric $\partial$ of [2]. $\blacksquare$

∎

🧠 Intuition

Think of the fibration as a "bundle of fibers" sitting over the base. A class in $\pi_n(B)$ is an $n$ -sphere drawn in the base. To understand it via $E$ , you try to lift this sphere up into the total space. The homotopy lifting property always lets you lift a disk (a filled-in hemisphere), but the lift need not close up into a sphere—its boundary $(n-1)$ -sphere may wander around inside the fiber $F$ . That residual boundary sphere is exactly $\partial[\alpha]$ . If $\partial[\alpha] = 0$ , the boundary is contractible, the lift closes up, and the original base sphere comes from a genuine sphere in $E$ (i.e., lies in $\operatorname{im}\, p_*$ ). Exactness is the bookkeeping statement: "a base sphere lifts to a total-space sphere iff its fiber-boundary is trivial."

Figure

A base sphere lifting to a disk in E with boundary in the fiber F

WH

Witold Hurewicz

1904–1956

Hurewicz introduced higher homotopy groups in 1935 and almost immediately recognized their connection to fiber spaces. His fiber space axioms (the covering homotopy property) crystallized the structure now called a fibration. Working at the intersection of point-set and algebraic topology, he proved early forms of the exact sequence and the Hurewicz theorem connecting $\pi_n$ to $H_n$ . Hurewicz died in a fall at an archaeological site in Mexico while attending a symposium; his work on fibrations was foundational to the subsequent explosion of computations by Serre and others.

ExampleHopf Fibration

S^1 \hookrightarrow S^3 \to S^2

Given: The Hopf map $p : S^3 \to S^2$ with fiber $F = S^1$ . We compute $\pi_n(S^2)$ for small $n$ .

Step 1. Write the relevant segment of [3]:

6

\quad \pi_n(S^1) \to \pi_n(S^3) \to \pi_n(S^2) \xrightarrow{\partial} \pi_{n-1}(S^1) \to \pi_{n-1}(S^3).

Derivation hint: substitute $F=S^1$ , $E=S^3$ , $B=S^2$ into [3].

Step 2 — Case $n = 2$ . Since $\pi_2(S^1) = 0$ and $\pi_1(S^1) = \mathbb{Z}$ , $\pi_1(S^3) = 0$ , the segment becomes

0 \to \pi_2(S^3) \to \pi_2(S^2) \xrightarrow{\partial} \mathbb{Z} \to 0.

With $\pi_2(S^3) = 0$ , $\partial$ is an isomorphism: $\boxed{\pi_2(S^2) \cong \mathbb{Z}}$ .

Step 3 — Case $n = 3$ . Now $\pi_3(S^1) = 0$ and $\pi_2(S^1) = 0$ , so

0 \to \pi_3(S^3) \xrightarrow{p_*} \pi_3(S^2) \to 0,

giving $p_*$ an isomorphism: $\boxed{\pi_3(S^2) \cong \pi_3(S^3) \cong \mathbb{Z}}$ , generated by the Hopf class.

Answer: $\pi_2(S^2) = \mathbb{Z}$ , $\pi_3(S^2) = \mathbb{Z}$ (dimensionless homotopy groups). The Hopf map is the generator of $\pi_3(S^2)$ .

ExamplePath-Loop Fibration and

\pi_n(\Omega X)

Given: For a pointed space $X$ , the path fibration $\Omega X \hookrightarrow PX \xrightarrow{p} X$ , where $PX = \{ \gamma : I \to X : \gamma(0) = x_0 \}$ is the contractible path space and $\Omega X$ the loop space (the fiber over $x_0$ via $\gamma \mapsto \gamma(1)$ ).

Step 1. Since $PX$ is contractible, $\pi_n(PX) = 0$ for all $n$ . The LES [3] gives

7

\quad 0 = \pi_n(PX) \to \pi_n(X) \xrightarrow{\partial} \pi_{n-1}(\Omega X) \to \pi_{n-1}(PX) = 0.

Derivation hint: contractibility kills the total-space terms, forcing $\partial$ to be an isomorphism.

Step 2. Exactness forces $\partial$ to be an isomorphism:

\boxed{\pi_n(X) \cong \pi_{n-1}(\Omega X)} \quad \text{for all } n \geq 1.

Answer: This loop-space shift $\pi_n(X) \cong \pi_{n-1}(\Omega X)$ underlies the whole theory of Spectra and infinite loop spaces. For $X = S^2$ it recovers $\pi_3(S^2) \cong \pi_2(\Omega S^2)$ .

⚠️ Common mistake

The sequence is exact, not split. Exactness gives $\operatorname{im} = \ker$ at each spot, but does not imply $\pi_n(E) \cong \pi_n(F) \oplus \pi_n(B)$ . Splitting requires extra structure (e.g., a section, or a product bundle). The Hopf fibration shows the failure: $\pi_3(S^3) = \mathbb{Z}$ but $\pi_3(S^1) \oplus \pi_3(S^2) = 0 \oplus \mathbb{Z}$ —coincidentally equal here, but $\pi_2$ : $\pi_2(S^3)=0$ while $\pi_2(S^2)=\mathbb{Z}\ne \pi_2(S^1)\oplus\pi_2(S^2)$ requires care.

$\partial$ is not induced by a continuous map. The connecting homomorphism lowers degree and is defined by a lifting construction, not by functoriality of a single map.

Low-degree terms are pointed sets, not groups. Below $\pi_1$ , "exactness" means $\operatorname{im} = \ker$ as preimages of the basepoint; talking about quotient groups there is meaningless.

Counterexample

A quasifibration that is not a fibration. Consider a map $p : E \to B$ that fails the homotopy lifting property at some point. The naive sequence built from $i_*, p_*, \partial$ need not be exact. For instance, take $B = [0,1]$ with the map collapsing $E = \{0\} \cup ([\tfrac12,1]\times S^1)$ in a way that the fiber over $0$ is a point but over $t>0$ is $S^1$ . The "fiber" changes homotopy type, the HLP fails, and $p_*$ on $\pi_1$ does not fit into an exact sequence with a well-defined connecting map. Moral: exactness genuinely requires the lifting axiom; mere surjective maps with varying fibers do not suffice. (Dold–Thom quasifibrations are precisely the weaker class where one recovers a usable LES.)

connection

The fibration LES sits in a web of homotopy-theoretic machinery. It is the homotopy analogue of the Long Exact Sequence of a Pair, obtained by the substitution $\pi_n(E,F) \cong \pi_n(B)$ . Iterating the construction over the path-loop fibration produces the Puppe Sequence and motivates Spectra. When one upgrades from groups to a multiplicative filtration, the fibration LES is the "first page" intuition behind the Serre Spectral Sequence, which computes homology of the total space from that of base and fiber. The connecting homomorphism is the bridge to Obstruction Theory: $\partial$ measures the obstruction to extending sections. For principal $G$ -bundles $G \hookrightarrow EG \to BG$ , the LES with $EG$ contractible yields $\pi_n(BG) \cong \pi_{n-1}(G)$ , linking to Classifying Spaces and Characteristic Classes.

🔬 Where this appears

⏳ History

- 1935 — Hurewicz introduces higher homotopy groups

\pi_n

and the notion of fiber spaces, recognizing the covering homotopy property.

- 1937–1941 — Hurewicz and Steenrod formalize fiber bundles and prove the covering homotopy theorem, the prototype HLP.

- 1941 — Hopf's earlier (1931) discovery of the Hopf map

S^3\to S^2

is reinterpreted: the LES yields

\pi_3(S^2)=\mathbb Z

, shocking the community by showing

\pi_n(S^m)\ne 0

for

n>m

.

- 1944 — Eckmann and Eilenberg develop the relative homotopy sequence machinery.

- 1950 — J.-P. Serre, in his thesis, introduces Serre fibrations (HLP only for CW complexes) and combines the LES with the spectral sequence, computing vast ranges of homotopy groups of spheres.

- 1951 — Serre proves the finiteness of

\pi_n(S^m)

for

n\ne m

(and

n\ne 2m-1

for even spheres), wielding the fibration LES and spectral sequence together; awarded the Fields Medal (1954).

- 1952 — Dold and (later) Dold–Thom develop quasifibrations, extending the LES to maps with non-constant fibers, used in the Dold–Thom theorem

\pi_n(SP(X))\cong\tilde H_n(X)

.

- 1955–1958 — Bott uses the LES on Lie group fibrations to prove the Bott periodicity theorems, computing

\pi_k(U)

and

\pi_k(O)

.

- 1962 — Puppe formalizes the cofiber/fiber sequence duality, placing the LES inside the broader Puppe sequence framework.

open-problem

The LES reduces homotopy computations to understanding the connecting map $\partial$ , but $\partial$ itself can be extraordinarily subtle. Computing the homotopy groups of spheres $\pi_{n+k}(S^n)$ in the stable range remains open in general; the LES alone is insufficient and must be combined with the Serre Spectral Sequence, Adams Spectral Sequence, and chromatic methods. A persistent difficulty is that exactness pins down a group only up to extension: even with all maps known, $\pi_n(E)$ may be one of several groups consistent with the sequence (the extension problem). Determining which extension occurs—e.g., distinguishing $\mathbb Z/4$ from $\mathbb Z/2\oplus\mathbb Z/2$ —requires finer invariants (Toda brackets, secondary operations). For non-simply-connected bases the $\pi_1(B)$ -action on the fiber's homotopy introduces twisted coefficients, and the very definition of exactness at low degrees (pointed sets, not groups) limits algebraic manipulation. The general structure of the unstable homotopy groups, governed at each stage by such connecting maps, is widely regarded as one of the deepest unsolved problems in mathematics.

Formal Statement

Why It Holds / Derivation

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also