03.12.52 · modern-geometry / homotopy

Relative homotopy group $π_{n} (X, A, x_{0})$

shipped3 tiersLean: none

Anchor (Master): Hatcher §4.1 (compression criterion, action of $\pi_1$, the LES); Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §B.1.6 and §2.1; J.H.C. Whitehead 1949-50 *Combinatorial homotopy I, II*; G. W. Whitehead *Elements of Homotopy Theory* Ch. IV

Intuition Beginner

The absolute homotopy group $π_{n} (X, x_{0})$ counts the ways an $n$ -sphere can sit inside a space, up to deformation. The relative homotopy group $π_{n} (X, A, x_{0})$ asks a slightly different question, one that pays attention to a chosen subspace $A$ . Instead of a closed sphere, we use a disk whose boundary is required to land in $A$ . The disk can roam freely through $X$ , but its rim must stay home in $A$ at all times.

The new measurement records disks that cannot be slid entirely back into $A$ while keeping their rim inside $A$ . If a disk can be retracted into $A$ without ever lifting its rim out, it counts as the identity. The disks that resist this retraction are the interesting classes. They detect the difference between living in the big space $X$ and living in the smaller subspace $A$ .

The reason this matters is that the relative groups stitch the homotopy of $A$ and the homotopy of $X$ into one exact sequence. They are the connective tissue that lets a computation in one space flow into the other.

Visual Beginner

A picture of a space $X$ with a shaded subspace $A$ sitting inside it, and a basepoint $x_{0}$ marked on $A$ . Over this sits a disk $D^{n}$ drawn as a filled circle, with its outer rim coloured to match $A$ and an arrow showing that the rim maps into $A$ while the inside of the disk maps anywhere in $X$ . One marked point of the rim is pinned to $x_{0}$ . A second arrow shows the disk being slid back toward $A$ ; a green check marks a disk that retracts into $A$ and a red cross marks a disk that refuses to retract.

The picture captures the message: a relative class is a disk anchored to $A$ by its rim, and the class is the identity exactly when the disk can be pulled into $A$ without releasing its rim.

Worked example Beginner

Compute the relative group $π_{2} (D^{2}, S^{1}, s_{0})$ for the disk relative to its boundary circle.

Step 1. A class is a map of the square's data: a disk $D^{2}$ whose boundary circle lands in $S^{1}$ , with one boundary point pinned to $s_{0}$ . Since $X = D^{2}$ is the whole disk, the map of the inside is unconstrained, but the rim must wrap the circle $S^{1}$ some number of times.

Step 2. The integer that classifies such a map is the winding number of the rim around $S^{1}$ . A rim that wraps once gives one class; a rim that wraps $k$ times gives another. The inside of the disk always fills in, because $D^{2}$ is filled, so the only freedom is in the rim.

Step 3. The boundary operation reads off the rim and lands it in $π_{1} (S^{1}) = Z$ . Reading the rim's winding number gives a matching between $π_{2} (D^{2}, S^{1})$ and $π_{1} (S^{1}) = Z$ .

Step 4. So $π_{2} (D^{2}, S^{1}, s_{0}) = Z$ , with the generator being the disk whose rim wraps the circle once. This is the simplest non-vanishing relative group, and it shows the boundary rim is where the information lives.

What this tells us: when the big space $X$ has no homotopy of its own (the disk is contractible), the entire relative group comes from the boundary of the subspace. The boundary map carries the relative group onto the homotopy of $A$ .

Check your understanding Beginner

Formal definition Intermediate+

Fix a topological space $X$ , a subspace $A \subseteq X$ , and a basepoint $x_{0} \in A$ . The standard model uses the $n$ -cube $I^{n} = [0, 1]^{n}$ with two distinguished pieces of its boundary. Write $$ J^{n-1} = \overline{\partial I^n \setminus (I^{n-1} \times {0})}, $$ so that $\partial I^{n} = (I^{n - 1} \times {0}) \cup J^{n - 1}$ , where $I^{n - 1} \times {0}$ is one chosen face and $J^{n - 1}$ is the union of the remaining faces. A representative of a relative class is a continuous map of triples $$ f : (I^n, I^{n-1} \times {0}, J^{n-1}) \to (X, A, x_0), $$ that is, $f$ carries the cube into $X$ , the distinguished face into $A$ , and all of $J^{n - 1}$ to the single point $x_{0}$ . Collapsing $J^{n - 1}$ identifies this with a map of triples $(D^{n}, S^{n - 1}, s_{0}) \to (X, A, x_{0})$ : the disk into $X$ , its boundary sphere into $A$ , and a basepoint $s_{0} \in S^{n - 1}$ to $x_{0}$ .

The relative homotopy set $π_{n} (X, A, x_{0})$ is the set of homotopy classes of such maps, where homotopies are through maps of triples. For $n \geq 2$ the group operation is concatenation along the first cube coordinate: $$ (f + g)(t_1, t_2, \ldots, t_n) = \begin{cases} f(2t_1, t_2, \ldots, t_n) & t_1 \leq \tfrac{1}{2}, \ g(2t_1 - 1, t_2, \ldots, t_n) & t_1 \geq \tfrac{1}{2}. \end{cases} $$ This is well-defined because the gluing wall $t_{1} = \frac{1}{2}$ lies in $J^{n - 1}$ and so is sent to $x_{0}$ by both halves. For $n \geq 3$ the group is abelian by an Eckmann-Hilton interchange between the first and second coordinates.

The boundary homomorphism is restriction to the distinguished face: $$ \partial : \pi_n(X, A, x_0) \to \pi_{n-1}(A, x_0), \qquad \partial[f] = [f|{I^{n-1} \times {0}}]. $$ The restriction lands in $\pi{n-1}(A) $b ec a u se t h e f a ce ma p s in t o$ A $an d i t so w nb o u n d a r y l i es in$ J^{n-1} $, w hi c h g oes t o$ x_0 $. T h e ma p$ \partial $i s ah o m o m or p hi s m f or$ n \geq 2$.

The action of $π_{1} (A, x_{0})$ sends a loop $γ$ in $A$ and a class $[f]$ to a class $γ \cdot [f]$ obtained by dragging the basepoint of $f$ around $γ$ : shrink the cube into a smaller concentric cube, then fill the collar with the loop $γ$ applied on the $J^{n - 1}$ -faces. This makes $π_{n} (X, A, x_{0})$ a module over $Z [π_{1} (A, x_{0})]$ for $n \geq 2$ .

Counterexamples to common slips

$π_{1} (X, A, x_{0})$ is not a group. It is a pointed set: a class is a path in $X$ from a point of $A$ to $x_{0}$ , and there is no coordinate to glue two such paths into a third. Treating it as a group is the most common slip.
The boundary map $\partial$ is not injective in general. For $(D^{2}, S^{1})$ it happens to be an isomorphism, but for a pair where $X$ carries its own homotopy, $\partial$ has a kernel equal to the image of $π_{n} (X) \to π_{n} (X, A)$ .
The group $π_{2} (X, A, x_{0})$ need not be abelian. The action of $π_{1} (A)$ obstructs commutativity, and the precise package is a crossed module, not an abelian group. Only from $n \geq 3$ onward is commutativity forced.
The action of $π_{1} (A)$ is on the relative group, with the loop living in $A$ , not in $X$ . Using a loop in $X$ instead gives the action on the absolute group $π_{n} (X)$ and a different structure.

Key theorem with proof Intermediate+

Theorem (long exact sequence of a pair; Hatcher Theorem 4.3). Let $(X, A)$ be a pair with basepoint $x_{0} \in A$ . The inclusion-induced maps $i_ : \pi_n(A, x_0) \to \pi_n(X, x_0) $an d$ j_* : \pi_n(X, x_0) \to \pi_n(X, A, x_0) $t o g e t h er w i t h t h e b o u n d a r y ma p$ \partial$ assemble into a long exact sequence* $$ \cdots \to \pi_n(A, x_0) \xrightarrow{i_*} \pi_n(X, x_0) \xrightarrow{j_*} \pi_n(X, A, x_0) \xrightarrow{\partial} \pi_{n-1}(A, x_0) \to \cdots $$ ending in $π_{0} (A) \to π_{0} (X)$ , with exactness of pointed sets at the last terms and of groups above.

Proof. We prove exactness at the relative term $π_{n} (X, A, x_{0})$ ; the remaining two exactness statements are recorded in the Full proof set. Here $j_{*}$ is the map sending an absolute class $[g : (D^{n}, s_{0}) \to (X, x_{0})]$ to its class as a map of triples, where the boundary sphere $S^{n - 1}$ maps to the single point $x_{0} \in A$ .

Composite is the identity. Take $[g] \in π_{n} (X, x_{0})$ . Its image $j_{*} [g]$ has representative the same map $g$ regarded relative to $A$ , with $g (S^{n - 1}) = {x_{0}}$ . The boundary $\partial j_{*} [g]$ restricts $g$ to the distinguished face, which is the constant map at $x_{0}$ . The constant map represents the identity of $π_{n - 1} (A, x_{0})$ , so $\partial \circ j_{*} = 0$ , giving $im j_{*} \subseteq ker \partial$ .

Kernel lies in the image. Suppose $[f] \in π_{n} (X, A, x_{0})$ satisfies $\partial [f] = 0$ . By definition $\partial [f] = [f ∣_{I^{n - 1} \times {0}}]$ , so the restriction of $f$ to the distinguished face is null-homotopic in $A$ rel its boundary. Let $H : (I^{n - 1} \times {0}) \times I \to A$ be such a null-homotopy, with $H_{0} = f ∣_{I^{n - 1} \times {0}}$ and $H_{1}$ the constant map at $x_{0}$ , keeping the face boundary at $x_{0}$ throughout. The face $I^{n - 1} \times {0}$ together with the homotopy parameter $I$ forms a collar $I^{n - 1} \times [0, 1]$ glued onto $I^{n}$ along that face. Extend $f$ across this collar by $H$ . Since the cube $I^{n}$ with a collar attached to one face is homeomorphic to $I^{n}$ rel the opposite face structure, the extended map is a homotopy of $f$ , through maps of triples, to a map $f^{'}$ whose distinguished face is now constant at $x_{0}$ . A map of triples whose distinguished face is constant at $x_{0}$ sends all of $\partial I^{n}$ to $x_{0}$ , so it is exactly an absolute class $[g^{'}] \in π_{n} (X, x_{0})$ with $j_{*} [g^{'}] = [f^{'}] = [f]$ . Hence $ker \partial \subseteq im j_{*}$ , completing exactness at $π_{n} (X, A, x_{0})$ . $□$

Bridge. The long exact sequence builds toward every later computation that trades homotopy of a subspace against homotopy of the whole space, and the same machine appears again in the relative Hurewicz theorem and the Blakers-Massey excision range. The foundational reason exactness holds is that the relative group is the homotopy of a single geometric object — the homotopy fibre of the inclusion $A ↪ X$ — and a fibre sequence always unrolls into a long exact sequence; this is exactly the statement that $π_{n} (X, A) = π_{n - 1} (hofib (A \to X))$ , so the boundary map $\partial$ is the connecting map of a fibration in disguise. The central insight is that the three families $π_{n} (A)$ , $π_{n} (X)$ , $π_{n} (X, A)$ are not independent: pinning down any two pins the third through exactness. This generalises the snake lemma of homological algebra to the non-abelian, non-additive setting of homotopy, and the compression criterion is dual to the lifting criterion for the homotopy fibre. Putting these together, the relative group is the precise obstruction to compressing a map of $X$ into $A$ , and the long exact sequence is the bookkeeping that turns that obstruction into a computation; the bridge is the recognition that exactness here is the homotopy shadow of a fibre sequence, which appears again in the Postnikov-tower and Whitehead-tower constructions.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

State and justify the compression criterion: $[f] \in π_{n} (X, A, x_{0})$ is the identity if and only if $f$ is homotopic rel $S^{n - 1}$ to a map into $A$ .

Hint

The identity of the relative group is represented by the constant map into $A$ ; deforming into $A$ rel the rim is the same as deforming to the constant element.

Answer

A representative $f : (D^{n}, S^{n - 1}, s_{0}) \to (X, A, x_{0})$ is the identity class precisely when it is homotopic through maps of triples to the constant map at $x_{0}$ . If $f$ deforms rel $S^{n - 1}$ to a map $g$ landing entirely in $A$ , then $g$ is a map of the disk into $A$ ; since $D^{n}$ is contractible rel a boundary point and $A$ receives the whole disk, $g$ contracts within $A$ to the constant map, and composing gives a triple-homotopy of $f$ to the identity. Conversely a triple-homotopy of $f$ to the constant map at $x_{0}$ is in particular a homotopy rel $S^{n - 1}$ landing in the point $x_{0} \in A$ . Rubric: full credit for both directions and the use of contractibility of $D^{n}$ rel a point.

Exercise 7 (hard, short-answer).

Prove that $π_{n} (X, A, x_{0})$ is abelian for $n \geq 3$ by an Eckmann-Hilton argument.

Hint

For $n \geq 3$ there are at least two free interior coordinates, the $t_{1}$ and $t_{2}$ directions, each carrying a concatenation operation, and the two operations interchange.

Answer

For $n \geq 3$ the cube model has the distinguished face $I^{n - 1} \times {0}$ and at least two coordinate directions, $t_{1}$ and $t_{2}$ , along which the gluing wall lands in $J^{n - 1}$ . This gives two concatenation operations $+_{1}$ and $+_{2}$ on $π_{n} (X, A, x_{0})$ , both with the same identity (the constant class). The two operations satisfy the interchange law $$ (a +_1 b) +_2 (c +_1 d) = (a +_2 c) +_1 (b +_2 d), $$ since the four-quadrant subdivision of the $t_{1} t_{2}$ -square can be read in either order. By the Eckmann-Hilton lemma, two unital operations on a set that interchange and share a unit coincide and are commutative. Hence $+_{1} = +_{2}$ is abelian. For $n = 2$ only the $t_{1}$ direction is free (the $t_{2} = 0$ face is the distinguished one mapping to $A$ ), so a single operation survives and commutativity can fail. Rubric: full credit for exhibiting two interchanging operations and invoking Eckmann-Hilton; partial for stating the law without the $n = 2$ contrast.

Exercise 8 (hard, short-answer).

Describe the action of $π_{1} (A, x_{0})$ on $π_{2} (D^{2} \cup_{φ} e^{2}, S^{1})$ for a $2$ -cell attached to the circle along a degree- $d$ map $φ$ , and identify the orbit structure.

Hint

This is the prototype of Whitehead's free crossed module. The relative $π_{2}$ is generated by the attached cell, and $π_{1} (A) = Z$ acts by re-tracing the basepoint around the circle.

Answer

Let $A = S^{1}$ with $π_{1} (A) = Z = ⟨ t ⟩$ , and attach one $2$ -cell along $φ$ of degree $d$ , giving $X = S^{1} \cup_{φ} e^{2}$ . Whitehead's theorem on the second relative group says $π_{2} (X, S^{1})$ is the free crossed $π_{1} (A)$ -module on the single generator $c$ corresponding to the cell, so as a group it is the free $Z [π_{1} (A)]$ -permutation-like module generated by the translates $t^{k} \cdot c$ . The boundary $\partial : π_{2} (X, S^{1}) \to π_{1} (S^{1})$ sends the generator $c$ to $φ$ , an element of degree $d$ , i.e. $\partial (c) = t^{d}$ , and is $π_{1} (A)$ -equivariant: $\partial (t^{k} \cdot c) = t^{k} t^{d} t^{- k} = t^{d}$ since $π_{1} (S^{1})$ is abelian. The action of $π_{1} (A) = Z$ permutes the generators $t^{k} \cdot c$ by translation of the exponent, so the orbit of $c$ is the full $Z$ -indexed family ${t^{k} \cdot c}$ . Rubric: full credit for identifying the free crossed module, the equivariant boundary, and the $Z$ -translation orbit; partial for the boundary computation alone.

Lean formalization Intermediate+

Mathlib ships the absolute homotopy groups $π_{n} (X, x_{0})$ through Topology.Homotopy.HomotopyGroup as classes of maps of the $n$ -cube fixing the whole boundary, but it does not yet ship the relative homotopy groups $π_{n} (X, A, x_{0})$ . The intended formalisation would read schematically:

import Mathlib.Topology.Homotopy.HomotopyGroup
import Mathlib.Topology.Homotopy.Path

/-- A representative of a relative homotopy class: a map of the cube sending
the distinguished face into `A` and the rest of the boundary to `x₀`. -/
structure RelGenLoop (n : ℕ) (X : TopCat) (A : Set X) (x₀ : X) where
  toFun   : (Fin n → I) → X
  face    : ∀ p, p (0 : Fin n) = 0 → toFun p ∈ A     -- distinguished face → A
  jBdry   : ∀ p, (∃ i, i ≠ 0 ∧ (p i = 0 ∨ p i = 1)) → toFun p = x₀

/-- The relative homotopy group, classes under triple-homotopy. -/
def RelativeHomotopyGroup (n : ℕ) (X : TopCat) (A : Set X) (x₀ : X) : Type :=
  Quotient (relHomotopic n X A x₀)

/-- The boundary homomorphism `∂ : πₙ(X, A) → πₙ₋₁(A)`. -/
noncomputable def relBoundary (n : ℕ) (X : TopCat) (A : Set X) (x₀ : X) :
    RelativeHomotopyGroup (n + 1) X A x₀ →+ HomotopyGroup n A x₀ :=
  sorry  -- restrict to the distinguished face

/-- The long exact sequence of the pair, as exactness at the relative term. -/
theorem pairLES_exact_relative
    (n : ℕ) (X : TopCat) (A : Set X) (x₀ : X) :
    Function.Exact
      (relInclusion n X A x₀)        -- jₙ : πₙ(X) → πₙ(X, A)
      (relBoundary n X A x₀) :=
  sorry  -- compression: ∂[f] = 0 ↔ f deforms rel S^{n-1} into an absolute class

The proof gap is substantive. Mathlib needs the distinguished-face cube model, the concatenation operation along the first coordinate with the $J^{n - 1}$ wall mapping to $x_{0}$ , the proof that the operation is abelian for $n \geq 3$ via Eckmann-Hilton on two interior coordinates, the boundary homomorphism as restriction, and the three exactness statements of the long exact sequence. The action of $π_{1} (A)$ as a $Z [π_{1} (A)]$ -module structure, and the Whitehead crossed module $Π_{2} (X, A, x_{0}) = (\partial : π_{2} (X, A) \to π_{1} (A))$ satisfying the Peiffer identity, are further targets that build toward the formalised Higher Homotopy van Kampen theorem.

Advanced results Master

Theorem (compression criterion; Hatcher Lemma 4.6). A map $f : (D^{n}, S^{n - 1}, s_{0}) \to (X, A, x_{0})$ represents the identity of $π_{n} (X, A, x_{0})$ if and only if it is homotopic rel $S^{n - 1}$ to a map with image in $A$ .

The compression criterion is the geometric heart of the relative groups: the relative group measures the failure to compress a disk into the subspace, and a class is the identity precisely when no obstruction is present. It is the criterion that powers the cellular approximation arguments in the relative setting and the obstruction-theoretic reading of the relative groups.

Theorem (relative groups as a fibre homotopy group). Let $p : P X \to X$ be the path fibration with fibre $Ω X$ , and let $F = hofib (A ↪ X)$ be the homotopy fibre of the inclusion. Then there is a natural isomorphism $π_{n} (X, A, x_{0}) ≅ π_{n - 1} (F)$ for $n \geq 1$ , under which the boundary map $\partial$ becomes the map induced by $F \to A$ .

This reformulation explains the long exact sequence as the homotopy exact sequence of the fibration $F \to A \to X$ . The relative group is not a new invariant but the absolute homotopy of the homotopy fibre shifted by one. This identification appears again in the construction of Postnikov and Whitehead towers, where successive homotopy fibres carry the relative information one stage at a time.

Theorem (the second relative group is a crossed module; Whitehead 1949). The boundary map $\partial : π_{2} (X, A, x_{0}) \to π_{1} (A, x_{0})$ , together with the action of $π_{1} (A, x_{0})$ on $π_{2} (X, A, x_{0})$ , is a crossed module: it satisfies CM1, $\partial (γ \cdot α) = γ \partial (α) γ^{- 1}$ , and CM2, the Peiffer identity $\partial (α) \cdot β = α β α^{- 1}$ .

This is the structural fact that the relative $π_{2}$ is not merely an abelian group with an action but a genuinely non-abelian algebraic object. The Peiffer identity is what records the non-commutativity that the abelianness of $π_{n}$ for $n \geq 3$ erases. Write $Π_{2} (X, A, x_{0})$ for this crossed module. For a relative CW pair $X = A \cup (2-cells)$ , Whitehead proved $Π_{2} (X, A, x_{0})$ is the free crossed module on the attaching maps of the $2$ -cells over $π_{1} (A)$ , which is the engine of the two-dimensional van Kampen theorem of Brown and Higgins.

Higher Whitehead products via crossed complexes

Theorem (Whitehead product on relative groups; G. W. Whitehead, Ch. X). For classes $α \in π_{p} (X, A, x_{0})$ and $β \in π_{q} (X, A, x_{0})$ there is a relative Whitehead product $[α, β] \in π_{p + q - 1} (X, A, x_{0})$ , bilinear up to the $π_{1} (A)$ -action, graded-commutative up to sign, and satisfying a graded Jacobi identity. The absolute Whitehead product is the special case $A = {x_{0}}$ .

The relative Whitehead products organise the higher non-abelian structure that the long exact sequence alone cannot see. The right home for this structure is the crossed complex of a filtered space: the sequence $$ \cdots \to \pi_n(X_n, X_{n-1}, x_0) \xrightarrow{\partial} \pi_{n-1}(X_{n-1}, X_{n-2}, x_0) \to \cdots \to \pi_2(X_2, X_1, x_0) \xrightarrow{\partial} \pi_1(X_1, x_0), $$ where each $\partial \circ \partial = 0$ , each term for $n \geq 2$ is a $π_{1} (X_{1})$ -module, and the bottom map $π_{2} \to π_{1}$ is a crossed module. This is the crossed complex $Π (X_{*})$ of the filtration, the central object of the Brown-Higgins-Sivera programme. The Whitehead products on the relative groups become the structure constants of a graded bracket on $Π (X_{*})$ , and the higher products are encoded by the tensor product of crossed complexes. This subsection builds toward the dedicated treatment in the crossed-complex unit 03.12.55, where the bracket is realised as a map out of the tensor product $Π (X_{*}) \otimes Π (Y_{*})$ .

Theorem (action determines the homomorphism part). For $n \geq 2$ the relative group $π_{n} (X, A, x_{0})$ is generated as a $Z [π_{1} (A)]$ -module by the images of the absolute classes $j_\pi_n(X) $t o g e t h er w i t h t h ee l e m e n t s w h ose b o u n d a r y d oes n o t v ani s h, an d t h e a c t i o n o f$ \pi_1(A) $i s t h e u ni q u es t r u c t u r e mak in g$ \partial$ equivariant.*

This is the module-theoretic shadow of the crossed-module structure in degree two and its higher analogue: the relative group is controlled by the action of $π_{1} (A)$ , and the Whitehead products supply the remaining bilinear structure that the module structure alone misses.

Synthesis. The relative homotopy group is the foundational reason the homotopy of a subspace and the homotopy of the whole space fit into one exact sequence, and the central insight is that it is the absolute homotopy of a homotopy fibre shifted by one degree: $π_{n} (X, A) = π_{n - 1} (hofib (A \to X))$ . Putting these together, the long exact sequence of the pair is the homotopy exact sequence of a fibration, the boundary map is its connecting map, and the compression criterion is the geometric reading of exactness. This is exactly the structure that the relative Hurewicz theorem 03.12.19 abelianises in the bottom dimension, that the Blakers-Massey theorem 03.12.21 bounds the excision range of, and that the crossed complex 03.12.55 promotes from a sequence of groups to a non-abelian chain complex carrying a graded Whitehead bracket. The relative $π_{2}$ is dual to the abelianised picture: where $π_{n}$ for $n \geq 3$ is forced abelian by Eckmann-Hilton, $π_{2}$ retains the Peiffer non-commutativity as a crossed module, and this is the central insight that the Brown-Higgins-Sivera programme exploits. The bridge from a single relative group to a multi-dimensional algebraic invariant is the filtration: the relative groups of consecutive skeleta assemble into the crossed complex $Π (X_{*})$ , the Whitehead products become its structure constants, and the higher products generalise to a map out of the tensor product of crossed complexes that this pattern recurs throughout the higher van Kampen theory.

Full proof set Master

Proposition (exactness at $π_{n} (X, x_{0})$ ). In the long exact sequence of the pair $(X, A)$ , the sequence is exact at the absolute term: $\operatorname{im} i_ = \ker j_*$.*

Proof. The map $i_{*} : π_{n} (A) \to π_{n} (X)$ is induced by inclusion and $j_{*} : π_{n} (X) \to π_{n} (X, A)$ regards an absolute class as relative.

Composite vanishes. Take $[σ] \in π_{n} (A, x_{0})$ with representative $σ : (D^{n}, s_{0}) \to (A, x_{0})$ , all of $D^{n}$ landing in $A$ . Then $i_{*} [σ]$ is the same map viewed in $X$ , and $j_{*} i_{*} [σ]$ is that map regarded relative to $A$ . Since $σ (D^{n}) \subseteq A$ , the representative compresses into $A$ at once, so by the compression criterion $j_{*} i_{*} [σ]$ is the identity of $π_{n} (X, A)$ . Hence $im i_{*} \subseteq ker j_{*}$ .

Kernel lies in the image. Suppose $[g] \in π_{n} (X, x_{0})$ satisfies $j_{*} [g] = 0$ . Regarding $g$ as a map of triples with $g (S^{n - 1}) = {x_{0}} \subseteq A$ , the vanishing $j_{*} [g] = 0$ means, by the compression criterion, that $g$ is homotopic rel $S^{n - 1}$ to a map $g^{'}$ with image in $A$ . The homotopy fixes $S^{n - 1}$ at $x_{0}$ , so $g^{'}$ is a based map $(D^{n}, S^{n - 1}, s_{0}) \to (A, x_{0}, x_{0})$ , that is, $g^{'} : (S^{n}, s_{0}) \to (A, x_{0})$ after collapsing the boundary. This $g^{'}$ represents a class $[σ] \in π_{n} (A, x_{0})$ with $i_{*} [σ] = [g^{'}] = [g]$ in $π_{n} (X)$ , since the compressing homotopy lives in $X$ . Hence $ker j_{*} \subseteq im i_{*}$ . $□$

Proposition (exactness at $π_{n - 1} (A, x_{0})$ ). The sequence is exact at the subspace term: $\operatorname{im} \partial = \ker i_$.*

Proof. Composite vanishes. Take $[f] \in π_{n} (X, A, x_{0})$ . Then $\partial [f] = [f ∣_{I^{n - 1} \times {0}}]$ is the restriction to the distinguished face, a class in $π_{n - 1} (A)$ . Applying $i_{*}$ includes this face-map into $X$ . But the face is the boundary of the cube $f$ , and the whole cube $f$ provides a null-homotopy of that boundary inside $X$ : contracting the cube toward the opposite face $J^{n - 1}$ , which maps to $x_{0}$ , exhibits $i_{*} \partial [f]$ as null-homotopic in $X$ . Hence $i_{*} \partial = 0$ and $im \partial \subseteq ker i_{*}$ .

Kernel lies in the image. Suppose $[τ] \in π_{n - 1} (A, x_{0})$ satisfies $i_{*} [τ] = 0$ , so $τ : (D^{n - 1}, s_{0}) \to (A, x_{0})$ becomes null-homotopic in $X$ . Let $G : D^{n - 1} \times I \to X$ be a null-homotopy with $G_{0} = τ$ landing in $A$ , $G_{1}$ the constant map at $x_{0}$ , and $G (s_{0}, t) = x_{0}$ throughout. View $D^{n - 1} \times I$ as the $n$ -cube $I^{n}$ with $D^{n - 1} \times {0}$ the distinguished face. Then $G$ is a map of triples $(I^{n}, I^{n - 1} \times {0}, J^{n - 1}) \to (X, A, x_{0})$ : the distinguished face carries $τ$ into $A$ , the top $D^{n - 1} \times {1}$ and the side ${s_{0}} \times I$ map to $x_{0}$ , so all of $J^{n - 1}$ goes to $x_{0}$ . Thus $G$ represents a class $[f] \in π_{n} (X, A, x_{0})$ with $\partial [f] = [τ]$ , giving $ker i_{*} \subseteq im \partial$ . $□$

Proposition (the boundary map is a homomorphism for $n \geq 2$ ). The map $\partial : π_{n} (X, A, x_{0}) \to π_{n - 1} (A, x_{0})$ satisfies $\partial ([f] + [g]) = \partial [f] + \partial [g]$ .

Proof. The sum $[f] + [g]$ is concatenation along the first coordinate $t_{1}$ . Restricting the concatenated cube to the distinguished face $I^{n - 1} \times {0}$ restricts each half to its own distinguished face along the same $t_{1}$ -direction, since the $t_{1} = \frac{1}{2}$ wall meets the distinguished face in a copy of $J^{n - 2}$ that maps to $x_{0}$ . The restriction is therefore the concatenation of $f ∣_{face}$ and $g ∣_{face}$ along $t_{1}$ , which is the sum in $π_{n - 1} (A, x_{0})$ . Hence $\partial$ respects the operation. The same computation shows $\partial$ is $π_{1} (A)$ -equivariant, since dragging the basepoint of the concatenation around a loop drags each summand's face simultaneously. $□$

Connections Master

Crossed module of a pair 03.12.53. The boundary map $\partial : π_{2} (X, A, x_{0}) \to π_{1} (A, x_{0})$ with the $π_{1} (A)$ -action is exactly Whitehead's crossed module, and the relative group developed here is its underlying group. The crossed-module unit takes this structure as primitive and proves the Peiffer identity and the free-crossed-module theorem for relative CW pairs; the relative group is the object on which the entire crossed-module algebra is built, and the boundary map is the structure map of the crossed module.
Crossed complex of a filtered space 03.12.55. The relative groups of consecutive skeleta $π_{n} (X_{n}, X_{n - 1}, x_{0})$ assemble, via the boundary maps with $\partial \circ \partial = 0$ , into the crossed complex $Π (X_{*})$ . The higher Whitehead products of this unit become the graded bracket on $Π (X_{*})$ , realised through the tensor product of crossed complexes. The crossed-complex unit is where the multi-dimensional non-abelian invariant assembled from these relative groups receives its full algebraic treatment and its Higher Homotopy van Kampen theorem.
Higher Homotopy van Kampen theorem 03.12.56. The classical van Kampen theorem computes $π_{1}$ of a union from the fundamental groupoid; its higher analogue computes the relative groups and crossed complexes of a union of filtered spaces. The relative homotopy groups of this unit are the raw material that the higher van Kampen theorem glues, and the gluing is possible precisely because the relative groups, unlike absolute $π_{n}$ for $n \geq 2$ , retain enough non-abelian structure to satisfy a colimit description.
Homotopy and homotopy group 03.12.01. The absolute homotopy groups are the special case $π_{n} (X, x_{0}) = π_{n} (X, {x_{0}}, x_{0})$ of the relative groups, and the long exact sequence of a pair degenerates to the absolute groups when $A = {x_{0}}$ . The relative groups are the genuine extension that lets absolute homotopy of a subspace and of the whole space interact, and the boundary map is the connecting homomorphism that the absolute theory alone cannot supply.
Hurewicz theorem 03.12.19. The relative Hurewicz theorem identifies $π_{n} (X, A) ≅ H_{n} (X, A)$ for an $(n - 1)$ -connected pair with $A$ simply connected, abelianising the relative group in its bottom non-vanishing dimension. The crossed-module structure of $π_{2} (X, A)$ is exactly what the relative Hurewicz theorem must abelianise away, and the requirement that $A$ be simply connected is the requirement that the $π_{1} (A)$ -action be the identity action so that the relative group is already abelian.
Blakers-Massey theorem 03.12.21. The Blakers-Massey theorem bounds the range in which the inclusion-induced map on relative homotopy groups $π_{k} (A, C) \to π_{k} (X, B)$ of a triad is an isomorphism, so it is a quantitative excision statement about precisely the relative groups defined here. The relative groups are the objects whose excision Blakers-Massey controls, and the long exact sequence of the pair is the tool the Blakers-Massey proof unrolls at every stage.

Historical & philosophical context Master

The relative homotopy groups were introduced by Witold Hurewicz in his 1935-36 series on the topology of deformations, alongside the absolute higher homotopy groups, with the long exact sequence of a pair appearing as the organising device that related a space to its subspaces. The structure was sharpened by J.H.C. Whitehead in his Combinatorial homotopy papers of 1949-50 ^{[Whitehead 1949]}, where the second relative group $π_{2} (X, A, x_{0})$ was identified as a crossed module over $π_{1} (A)$ — the first recognition that the relative groups carry genuinely non-abelian structure beyond what the long exact sequence records. Whitehead's free-crossed-module theorem for the relative $π_{2}$ of a CW pair is the precise two-dimensional analogue of the way the fundamental group of a graph is free, and it is the technical foundation on which the later non-abelian theory rests.

The philosophical content is the recognition that the relative groups are not an independent invariant but the homotopy of a homotopy fibre, shifted by one degree. This reframing, due in spirit to the fibration-sequence viewpoint of the 1950s and made precise by G. W. Whitehead ^{[Whitehead 1978]} and others, dissolves the apparent novelty of the relative groups into the single principle that every map has a homotopy fibre and every fibration has a long exact sequence. What remains genuinely new is the non-abelian structure in low degrees: the crossed module of $π_{2}$ and the Whitehead products. Ronald Brown, Philip Higgins, and Rafael Sivera, in their 2011 Nonabelian Algebraic Topology ^{[Brown-Higgins-Sivera 2011]}, built an entire reorganisation of algebraic topology around the relative groups of a filtered space, replacing the abelian chain complex of singular homology with the non-abelian crossed complex assembled from the relative homotopy groups of skeleta, and proving a Higher Homotopy van Kampen theorem that computes these crossed complexes by colimits. Their thesis is that the abelianisation forced on $π_{n}$ for $n \geq 3$ discards information that the relative and filtered picture retains, and that the correct foundational object is the crossed complex rather than the chain complex.

Bibliography Master

@article{WhiteheadCombinatorialII,
  author  = {Whitehead, J. H. C.},
  title   = {Combinatorial homotopy II},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {55},
  year    = {1949},
  pages   = {453--496}
}

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

@book{BrownHigginsSivera,
  author    = {Brown, Ronald and Higgins, Philip J. and Sivera, Rafael},
  title     = {Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids},
  series    = {EMS Tracts in Mathematics},
  volume    = {15},
  publisher = {European Mathematical Society},
  year      = {2011}
}

@book{GWhiteheadElements,
  author    = {Whitehead, George W.},
  title     = {Elements of Homotopy Theory},
  series    = {Graduate Texts in Mathematics},
  volume    = {61},
  publisher = {Springer-Verlag},
  year      = {1978}
}

@book{MayConcise,
  author    = {May, J. Peter},
  title     = {A Concise Course in Algebraic Topology},
  series    = {Chicago Lectures in Mathematics},
  publisher = {University of Chicago Press},
  year      = {1999}
}

@article{EckmannHilton1962,
  author  = {Eckmann, Beno and Hilton, Peter J.},
  title   = {Group-like structures in general categories I: Multiplications and comultiplications},
  journal = {Mathematische Annalen},
  volume  = {145},
  year    = {1962},
  pages   = {227--255}
}

@article{Hurewicz1935,
  author  = {Hurewicz, Witold},
  title   = {Beitr\"age zur Topologie der Deformationen},
  journal = {Proceedings Koninklijke Akademie van Wetenschappen Amsterdam},
  volume  = {38},
  year    = {1935},
  pages   = {112--119, 521--528}
}

Prerequisites

03.12.01
03.12.08
03.12.10

Used in

03.12.53
03.12.55

Tier anchors

beginner: Hatcher *Algebraic Topology* §4.1 informal opening on maps of pairs and the relative groups
intermediate: Hatcher §4.1 (relative homotopy groups, the boundary map, the long exact sequence 4.3); May *A Concise Course in Algebraic Topology* Ch. 9
master: Hatcher §4.1 (compression criterion, action of $\pi_1$, the LES); Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §B.1.6 and §2.1; J.H.C. Whitehead 1949-50 *Combinatorial homotopy I, II*; G. W. Whitehead *Elements of Homotopy Theory* Ch. IV

References

Hatcher — Algebraic Topology · §4.1 the relative homotopy groups $\pi_n(X, A, x_0)$, the boundary map $\partial$, the compression criterion (Lemma 4.6), the action of $\pi_1(A)$, and the long exact sequence of a pair (Theorem 4.3)
Brown-Higgins-Sivera — Nonabelian Algebraic Topology · §B.1.6 the relative homotopy groups and the homotopy long exact sequence; §2.1 the crossed module $\Pi_2(X, A, x_0)$ from $\partial : \pi_2(X, A) \to \pi_1(A)$; free at https://groupoids.org.uk/pdffiles/NAT-book.pdf
Whitehead, J.H.C. — Combinatorial homotopy II · Bull. Amer. Math. Soc. 55 (1949) 453-496; introduces crossed modules and the second relative homotopy group as a crossed module over $\pi_1$
Whitehead, G.W. — Elements of Homotopy Theory · Springer GTM 61, Ch. IV §2-§3 relative homotopy groups, the boundary operator, and the exact sequence of a pair; Ch. X Whitehead products
May — A Concise Course in Algebraic Topology · Ch. 9 the long exact sequence of a pair and the relative homotopy groups via the homotopy fibre
Eckmann-Hilton — Group-like structures in general categories · Math. Ann. 145 (1962); the interchange argument forcing commutativity of $\pi_n(X, A)$ for $n \geq 3$

Estimated time

beginner: 14m
intermediate: 34m
master: 62m