Homotopy group
In algebraic topology, homotopy groups are algebraic invariants associated to a pointed topological space (X, x_0), generalizing the fundamental group \pi_1(X, x_0) to higher dimensions by classifying continuous maps from the n-sphere S^n to X that send a basepoint of S^n to x_0, up to basepoint-preserving homotopy equivalence, with the group operation defined by concatenating maps along the equator of the sphere.[1] The nth homotopy group, denoted \pi_n(X, x_0), captures obstructions to contracting n-dimensional spheres in X, thereby detecting higher-dimensional "holes" in the space.[2] For n = 1, \pi_1(X, x_0) is the fundamental group, which may be non-abelian and classifies loops based at x_0 up to homotopy. For n \geq 2, \pi_n(X, x_0) is always abelian, and the groups are independent of the choice of basepoint if X is path-connected.[1] These groups are homotopy invariants, meaning they remain unchanged under weak homotopy equivalences, making them powerful tools for classifying topological spaces up to homotopy type.[2] The concept was first proposed by Eduard Čech in 1932[2] but was rigorously defined and developed by Witold Hurewicz in 1935, who showed that \pi_n(X) vanishes if and only if X is n-connected and established their abelian nature for n \geq 2.[3] Hurewicz also introduced the Hurewicz homomorphism relating homotopy groups to homology groups, which is an isomorphism under certain connectivity conditions, linking homotopy theory to singular homology.[1] Homotopy groups play a central role in modern algebraic topology, appearing in the study of fibrations via long exact sequences—such as \dots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \dots for a fibration F \to E \to B—and in theorems like Whitehead's, which characterizes weak homotopy equivalences between simply connected spaces.[1] Computing homotopy groups is notoriously difficult, especially for spheres, where \pi_n(S^k) is trivial for n < k but highly nontrivial and finite for n > k, with ongoing research revealing their structure through spectral sequences and stable homotopy theory.[2]Fundamentals
Introduction
Homotopy groups form a sequence of algebraic structures associated to a topological space, serving as invariants that capture essential features of its topology, particularly the presence of "holes" in various dimensions. These groups generalize the intuitive notion of connectivity, providing a framework to distinguish spaces that cannot be continuously deformed into one another—known as homotopy equivalence—and thus play a central role in the classification of topological spaces within algebraic topology. By quantifying obstructions to contracting maps from spheres into the space, homotopy groups reveal deformation properties that simpler invariants, like homology groups, may overlook. The development of homotopy groups traces back to the early 20th century, building on Henri Poincaré's introduction of the fundamental group in his 1895 paper "Analysis Situs," where he analyzed 1-dimensional loops to detect basic holes in manifolds.[4] This laid the groundwork for homotopy theory, but it was Witold Hurewicz who, building on Eduard Čech's 1932 proposal, rigorously defined higher-dimensional homotopy groups in his 1935 paper "Beiträge zur Topologie der Deformationen I. Höherdimensionale Homotopiegruppen," extending the concept to classify more intricate topological features beyond mere path-connectedness.[5][6] Hurewicz's innovation shifted the focus from qualitative descriptions to algebraic invariants, profoundly influencing subsequent advances in the field. A key distinction in homotopy groups is their algebraic behavior: the first homotopy group π₁ is non-abelian, reflecting the non-commutative nature of loop compositions in a space, whereas higher homotopy groups πₙ for n ≥ 2 are abelian, allowing for simpler group-theoretic analysis.[5] In many significant examples, such as the homotopy groups of spheres, these invariants are finitely generated abelian groups, highlighting their computational tractability while underscoring their importance in probing the subtle, higher-dimensional structures that define a space's homotopy type.[7]Definition
A homotopy group is defined in the context of pointed topological spaces. A pointed topological space is a pair (X, x_0), where X is a topological space and x_0 \in X is a designated basepoint.[8] The higher homotopy groups were first proposed by Eduard Čech in 1932 and rigorously defined by Witold Hurewicz in 1935, building on the fundamental group.[5][6] The loop space \Omega X of a pointed space (X, x_0) consists of all based loops in X, defined as the set of continuous maps f: (I, \partial I) \to (X, x_0), where I = [0,1] is the unit interval and \partial I = \{0,1\}.[8] Two based loops f, g: I \to X are homotopic, denoted f \simeq g, if there exists a continuous homotopy H: I \times I \to X such that H(s, 0) = f(s), H(s, 1) = g(s), and H(0, t) = H(1, t) = x_0 for all s, t \in I.[8] The first homotopy group, or fundamental group, is \pi_1(X, x_0) = \pi_0(\Omega X), the set of path components (homotopy classes) of the loop space \Omega X, equipped with the group operation induced by concatenation of loops: for homotopic classes and, the product is [f \cdot g], where (f \cdot g)(s) = f(2s) for s \in [0, 1/2] and (f \cdot g)(s) = g(2s - 1) for s \in [1/2, 1].[8] This group structure arises naturally from the topological properties of loop concatenation, with the constant loop at x_0 serving as the identity.[5] For n \geq 2, the nth homotopy group \pi_n(X, x_0) is defined as the set of pointed homotopy classes of continuous maps [S^n, X]_*, where S^n is the n-sphere with basepoint (the north pole, say), and maps send the basepoint of S^n to x_0.[8] Equivalently, \pi_n(X, x_0) = \pi_0(\Omega^n X), the path components of the n-fold iterated loop space, where \Omega^1 X = \Omega X and \Omega^{k+1} X = \Omega(\Omega^k X) for k \geq 1.[8] The group operation on \pi_n(X, x_0) for n \geq 2 is induced by the cogroup structure on S^n, given by the pinch map that combines two maps via the equatorial decomposition of S^n, resulting in an abelian group.[8] Hurewicz showed that these groups are abelian for n \geq 2.[5] If X is path-connected, the homotopy groups \pi_n(X, x_0) are independent of the choice of basepoint x_0, up to canonical isomorphism induced by paths connecting basepoints.[8] More generally, if pointed spaces (X, x_0) and (Y, y_0) are homotopy equivalent via a pointed map f: (X, x_0) \to (Y, y_0) with pointed homotopy inverse, then f induces group isomorphisms \pi_n(X, x_0) \cong \pi_n(Y, y_0) for all n \geq 1.[8]Geometric and Algebraic Properties
Relation to holes
Homotopy groups provide a geometric means to detect and classify "holes" in topological spaces, where a hole in dimension n corresponds to the inability to continuously deform an n-dimensional sphere embedded in the space to a point. The nth homotopy group \pi_n(X, x_0) at basepoint x_0 \in X captures these features by classifying pointed maps from the n-sphere S^n into X up to homotopy, with non-trivial elements indicating the presence of such holes.[8] The 0th homotopy set \pi_0(X), which is the set of path components of X, detects 0-dimensional holes, manifesting as disconnectedness in the space. Each component represents a distinct "piece" that cannot be connected by continuous paths, thus quantifying the space's overall fragmentation.[8] In dimension 1, the fundamental group \pi_1(X, x_0) identifies 1-dimensional holes through loops based at x_0 that cannot be contracted to a point. For instance, the circle S^1 has \pi_1(S^1) \cong \mathbb{Z}, generated by the loop winding once around the circle, reflecting the single 1-dimensional hole inherent to its topology.[8] For n \geq 2, the higher homotopy groups \pi_n(X, x_0) reveal n-dimensional holes via maps S^n \to X that resist homotopy to a constant map. These groups are always abelian and detect more subtle voids; notably, the 2-sphere S^2 has trivial \pi_1(S^2) but \pi_2(S^2) \cong \mathbb{Z}, indicating no 1-dimensional holes yet a fundamental 2-dimensional hole filled by the identity map on S^2. In general, the n-sphere S^n exhibits \pi_n(S^n) \cong \mathbb{Z}, capturing its own n-dimensional hole, while lower homotopy groups vanish for i < n.[8] A concrete example is the torus T^2, which has \pi_1(T^2) \cong \mathbb{Z} \oplus \mathbb{Z}, corresponding to two independent 1-dimensional holes (one along each generating loop), while all higher homotopy groups \pi_n(T^2) = 0 for n \geq 2, indicating no higher-dimensional holes.[8] Vanishing theorems further illuminate this detection: a space X is simply connected if it is path-connected and \pi_1(X) = 0, meaning no 1-dimensional holes obstruct loop contractions. More generally, X is k-connected if \pi_i(X) = 0 for all i \leq k, signifying the absence of holes in dimensions up to k.[8]Basic algebraic structure
Homotopy groups possess a rich algebraic structure, beginning with their functorial nature. A continuous map f: (X, x_0) \to (Y, y_0) between pointed topological spaces induces a group homomorphism f_*: \pi_n(X, x_0) \to \pi_n(Y, y_0) for each n \geq 1, preserving the group operation and satisfying (f \circ g)_* = f_* \circ g_* and \mathrm{id}_* = \mathrm{id}.[9] This makes the assignment (X, x_0) \mapsto \pi_n(X, x_0) a functor from the category of pointed spaces to the category of groups, with composition and identities preserved.[9] Homotopy invariance ensures that the algebraic structure is robust under deformation. If two maps f, g: (X, x_0) \to (Y, y_0) are homotopic relative to the basepoint, then f_* = g_* on all homotopy groups \pi_n.[9] Consequently, homotopy equivalences between pointed spaces induce isomorphisms on all \pi_n, providing a way to classify spaces up to homotopy via their homotopy groups.[9] In particular, the fundamental group \pi_1 may be non-abelian, reflecting complex looping behaviors, whereas higher homotopy groups \pi_n for n \geq 2 are always abelian.[9] Exact sequences arise in specific contexts, such as covering spaces, illustrating the interplay between homotopy and group extensions. For a path-connected regular covering space projection p: (\tilde{X}, \tilde{x}_0) \to (X, x_0) with discrete fiber, the induced map p_*: \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0) is injective, and there is a short exact sequence $1 \to \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0) \to G \to 1, where G is the deck transformation group acting freely and properly discontinuously on \tilde{X}.[9] For n \geq 2, p_* induces isomorphisms \pi_n(\tilde{X}, \tilde{x}_0) \cong \pi_n(X, x_0), showing that higher homotopy detects the same "holes" in base and cover.[9] The Hurewicz homomorphism provides a bridge to homology theory, encoding homotopy information algebraically. For a path-connected pointed space X, the n-th Hurewicz map h_n: \pi_n(X, x_0) \to H_n(X; \mathbb{Z}) sends a homotopy class represented by a map f: (S^n, *) \to (X, x_0) to the homology class of its image, using the fundamental class of the sphere.[9] This map is surjective for all n \geq 1, and if X is simply connected, it is an isomorphism for n = 2; more generally, for an (n-1)-connected space, h_n is an isomorphism onto the first nonzero homology group.[9] The Whitehead theorem ties homotopy groups to the global topology of CW-complexes. A map f: X \to Y between connected CW-complexes that induces isomorphisms f_*: \pi_n(X, x_0) \to \pi_n(Y, y_0) for all n \geq 1 is a homotopy equivalence.[9] Equivalently, a weak homotopy equivalence between such spaces is a genuine homotopy equivalence, emphasizing the role of CW-structure in detecting homotopy types through algebraic invariants.[9]Exact Sequences and Fibrations
Long exact sequence of a fibration
In algebraic topology, a Serre fibration is a continuous map p: E \to B between topological spaces that satisfies the homotopy lifting property with respect to all inclusions \partial I^n \hookrightarrow I^n, where I^n denotes the n-dimensional cube (or equivalently, the n-disk D^n) and \partial I^n its boundary, for all n \geq 0. Specifically, for any map f: \partial I^n \to E and homotopy H: I^n \times I \to B such that p \circ f is homotopic to the restriction of H to \partial I^n \times \{0\} relative to \partial I^n \times \{0\}, there exists a lift \tilde{H}: I^n \times I \to E extending f on \partial I^n \times \{0\} and projecting to H via p. This property ensures that fibrations behave well with respect to homotopy, allowing local properties of the fiber to relate to global homotopy invariants of the total space and base. Given a Serre fibration p: E \to B with basepoint b_0 \in B and fiber F = p^{-1}(b_0), assuming all spaces are path-connected and pointed appropriately, the homotopy groups of E, B, and F are connected by a long exact sequence. This sequence takes the form \cdots \to \pi_{n+1}(B) \xrightarrow{\partial} \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots \to \pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F) \to \pi_0(E) \to \pi_0(B) \to 0, which is exact at each term for n \geq 1, where exactness means the image of each map equals the kernel of the next. For n=0, the sequence terminates with \pi_0(B) \to 0, but \pi_0 groups are pointed sets rather than groups, so exactness is interpreted set-theoretically: the map \pi_0(F) \to \pi_0(E) is surjective onto the preimage under \pi_0(E) \to \pi_0(B) of the basepoint component. The maps \pi_n(F) \to \pi_n(E) and \pi_n(E) \to \pi_n(B) are induced by the inclusion F \hookrightarrow E and p: E \to B, respectively. The boundary map \partial: \pi_{n+1}(B) \to \pi_n(F) is the connecting homomorphism, which captures how homotopy classes in the base relate to those in the fiber via the fibration structure. A sketch of its derivation relies on the path-lifting property of Serre fibrations: given a representative [ \alpha ] \in \pi_{n+1}(B) based at b_0, lift the corresponding (n+1)-sphere in B (or its cell decomposition) to a homotopy in E whose boundary traces a loop in F, yielding an element in \pi_n(F); the exactness follows from composing lifts and homotopies uniquely up to homotopy in E, B, and F. This construction ensures the sequence is long and exact, providing a tool to compute homotopy groups recursively from fiber, total space, and base. A special case arises for the trivial fibration p: F \times B \to B given by projection onto the second factor, where the fiber is F \times \{b_0\} \cong F. Here, the long exact sequence splits naturally, with \pi_n(E) \cong \pi_n(F) \oplus \pi_n(B) for all n \geq 1 via the product structure, and the boundary map \partial vanishes, reflecting the direct product decomposition. This splitting highlights how non-trivial fibrations introduce interactions between the homotopy of F and B through \partial.Exact sequences for pairs and cofibrations
In algebraic topology, relative homotopy groups provide a way to capture the homotopy information of a space X relative to a subspace A \subset X. For a pair of pointed spaces (X, A) with basepoint in A, the nth relative homotopy group \pi_n(X, A) is defined as the set of homotopy classes of pointed maps (I^n, \partial I^n, J^{n-1}) \to (X, A, *), where I^n is the n-dimensional cube, \partial I^n is its boundary, and J^{n-1} = \partial I^{n-1} \times I \cup I^{n-1} \times \{0\} is the relevant face for the basepoint; these classes form a group under pointwise concatenation for n \geq 2, which is abelian for n \geq 3. Equivalently, \pi_n(X, A) can be viewed as the (n-1)th homotopy group of the path space of maps from the basepoint to A. A key tool for relating absolute and relative homotopy groups is the notion of a cofibration, which ensures that homotopies defined on a subspace extend well-behavedly to the whole space. A continuous map i: A \to X is a cofibration if it has the homotopy extension property: for every topological space Y, every continuous map u: X \to Y, and every continuous homotopy H: A \times I \to Y such that H(a, 0) = u(i(a)) for all a \in A, there exists a continuous homotopy \tilde{H}: X \times I \to Y such that \tilde{H}|_{A \times I} = H and \tilde{H}(x, 0) = u(x) for all x \in X. This property holds, for example, for cell inclusions in CW-complexes. In categories such as compactly generated Hausdorff spaces, cofibrations are equivalent to the pair (X, A) being a neighborhood deformation retract pair, where there exists an open neighborhood V of A in X and a deformation retraction of V onto A. When i: A \to X is a cofibration, there is a long exact sequence in homotopy groups: \cdots \to \pi_n(A) \xrightarrow{i_*} \pi_n(X) \xrightarrow{j_*} \pi_n(X, A) \xrightarrow{\partial} \pi_{n-1}(A) \to \cdots, where i_* and j_* are induced by the inclusion and quotient maps, respectively, and the boundary map \partial is defined by composing a relative class with the quotient map X/A \simeq X \cup_A CA (the mapping cone on A) and restricting to the top face of the cube. This sequence is exact at each term, meaning the image of each map equals the kernel of the next, and it arises from the cofiber sequence A \to X \to X/A. The exactness implies that relative homotopy groups measure the "new" homotopy classes introduced by adjoining X to A. The cofiber sequence provides a dual perspective: for a cofibration i: A \to X, the cofiber C_i = X \cup_A CA, where CA = A \times I / A \times \{1\} is the cone on A, fits into the exact sequence of homotopy groups \pi_*(A) \to \pi_*(X) \to \pi_*(C_i) \to \pi_{*-1}(A) \to \cdots, which is a segment of the long exact sequence for the pair (C_i, X), with C_i / X \simeq \Sigma A (the suspension of A). This connects relative homotopy to absolute homotopy of quotient spaces and is particularly useful for computations involving cell attachments. To preserve exactness under natural transformations or maps between such sequences, the five lemma applies: given a commutative diagram of abelian groups with exact rows \begin{CD} 0 @>>> A_1 @>f_1>> B_1 @>g_1>> C_1 @>>> 0 \\ @. @VVV @VVf_2V @VVg_2V @. \\ 0 @>>> A_2 @>>h_2> B_2 @>>k_2> C_2 @>>> 0, \end{CD} if f_1 and g_2 are isomorphisms (or more generally, if the outer maps are iso and middles mono/epi in certain combinations), then f_2 and g_1 are isomorphisms; this lemma extends to the ends of long exact sequences in homotopy groups, ensuring isomorphisms in relative settings like excisions.[10][11]Key Examples and Applications
Homotopy groups of spheres
The homotopy groups of spheres, denoted \pi_n(S^k), vanish for n < k, are isomorphic to \mathbb{Z} when n = k, and become increasingly complex for n > k, capturing essential features of higher-dimensional topology. These groups are fundamental in algebraic topology, as spheres serve as building blocks for more general spaces, and their computation reveals patterns of stability under suspension.[12][13] In low dimensions, explicit computations yield \pi_1(S^1) \cong \mathbb{Z}, \pi_2(S^2) \cong \mathbb{Z}, \pi_3(S^2) \cong \mathbb{Z}, \pi_3(S^3) \cong \mathbb{Z}, and \pi_4(S^3) \cong \mathbb{Z}_2. The Hopf fibration S^1 \to S^3 \to S^2 provides a key example, where the long exact sequence of the fibration implies that the generator of \pi_3(S^2) corresponds to the Hopf map, establishing \pi_3(S^2) \cong \mathbb{Z}.[12][14] The suspension map \Sigma: \pi_n(S^k) \to \pi_{n+1}(S^{k+1}) induces isomorphisms in a range determined by the Freudenthal suspension theorem, which states that it is an isomorphism for n < 2k - 1 and surjective for n = 2k - 1, assuming k \geq 2. This theorem highlights the stabilization phenomenon, where homotopy groups become independent of the base dimension in sufficiently high ranges.[13][15] The stable homotopy groups of spheres are defined as \pi_n^s = \lim_{k \to \infty} \pi_{n+k}(S^k), capturing the behavior for n \geq k + 1. These groups are finite abelian for n > 0 (by the Serre finiteness theorem) and exhibit intricate torsion structures; for instance, \pi_1^s \cong \mathbb{Z}_2, \pi_2^s \cong \mathbb{Z}_2, and \pi_3^s \cong \mathbb{Z}_{24}. Computations of these groups up to dimension 90 rely on spectral sequences and motivic methods, underscoring their computational complexity beyond low dimensions.[16][17][18] The following table lists selected low-dimensional homotopy groups \pi_n(S^m) for m = 1 to $7 and n \leq 10, focusing on non-trivial cases (trivial groups are indicated as 0); values are drawn from classical computations, with infinite cyclic groups denoted \mathbb{Z} and finite ones by their standard decompositions.[19][12]| n \setminus m | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| 1 | \mathbb{Z} | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | \mathbb{Z} | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | \mathbb{Z} | \mathbb{Z} | 0 | 0 | 0 | 0 |
| 4 | 0 | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z} | 0 | 0 | 0 |
| 5 | 0 | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z} | 0 | 0 |
| 6 | 0 | \mathbb{Z}_{12} | \mathbb{Z}_{12} | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z} | 0 |
| 7 | 0 | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z} \oplus \mathbb{Z}_{12} | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z} |
| 8 | 0 | \mathbb{Z}_2 | \mathbb{Z}_2 | \mathbb{Z}_2 \oplus \mathbb{Z}_2 | \mathbb{Z}_{24} | \mathbb{Z}_2 | \mathbb{Z}_2 |
| 9 | 0 | \mathbb{Z}_3 | \mathbb{Z}_3 | 0 | 0 | \mathbb{Z}_{24} | \mathbb{Z}_2 |
| 10 | 0 | \mathbb{Z}_{15} \oplus \mathbb{Z}_2 | \mathbb{Z}_{15} \oplus \mathbb{Z}_2 | \mathbb{Z}_2 | 0 | 0 | \mathbb{Z}_{24} |