Wedge sum

In algebraic topology, the wedge sum (also known as the one-point union) of two pointed topological spaces (X, x_0) and (Y, y_0) is the quotient space obtained from the disjoint union X \sqcup Y by identifying the basepoints x_0 and y_0 to a single point, with the resulting basepoint being this identified point.^[1] Denoted X \vee Y, this construction is the coproduct in the category of pointed topological spaces and basepoint-preserving continuous maps, satisfying the universal property that any pair of basepoint-preserving maps f: X \to Z and g: Y \to Z (for a pointed space Z) induces a unique basepoint-preserving map h: X \vee Y \to Z such that h \circ i_X = f and h \circ i_Y = g, where i_X and i_Y are the inclusion maps.^[1] This operation extends naturally to finite or infinite families of pointed spaces \{X_\alpha\}_{\alpha \in A}, yielding \bigvee_{\alpha \in A} X_\alpha as the quotient of the disjoint union \sqcup_{\alpha \in A} X_\alpha by identifying all basepoints x_\alpha to one point.^[1] The wedge sum plays a central role in homotopy theory and related areas by allowing the decomposition of complex spaces into simpler components attached at basepoints, facilitating computations of algebraic invariants.^[1] For path-connected pointed spaces X and Y, the fundamental group satisfies \pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y), the free product of the individual fundamental groups, as established by the Seifert–van Kampen theorem (assuming open neighborhoods of the basepoints in X and Y deformation retract to the basepoints and their intersection is path-connected).^[1] This result extends to infinite wedges of path-connected spaces, yielding \pi_1\left(\bigvee_\alpha X_\alpha\right) \cong {*_\alpha} \pi_1(X_\alpha), the free product over the index set, provided the spaces satisfy suitable local contractibility conditions near basepoints.^[1] Notable examples include the wedge of n circles S^1 \vee \cdots \vee S^1 (n times), whose fundamental group is the free group on n generators, modeling the topology of a "bouquet" or "rose" with n petals.^[1] For higher homotopy groups, the wedge sum exhibits a direct sum decomposition when n \geq 2: if X and Y are path-connected and well-pointed (i.e., the inclusions of basepoint neighborhoods are cofibrations), then \pi_n(X \vee Y) \cong \pi_n(X) \oplus \pi_n(Y) for all n \geq 2.^[1] This isomorphism holds more generally for wedges \bigvee_\alpha X_\alpha of countably many such spaces, with \pi_n\left(\bigvee_\alpha X_\alpha\right) \cong \bigoplus_\alpha \pi_n(X_\alpha) for n \geq 2, though infinite wedges may require compactness assumptions for uncountable indices.^[1] In homology theory, the reduced singular homology groups of a wedge sum satisfy \tilde{H}_n\left(\bigvee_\alpha X_\alpha\right) \cong \bigoplus_\alpha \tilde{H}_n(X_\alpha) for all n \geq 0, reflecting the additivity of homology functors on coproducts in the pointed category.^[1] These properties make the wedge sum indispensable for constructing models like bouquets of spheres \bigvee_\alpha S^k_\alpha, whose homotopy and homology groups provide free abelian structures that underpin calculations in stable homotopy theory and spectral sequences.^[1]

Definition and Construction

Topological Wedge Sum

In topology, the wedge sum provides a means to combine pointed topological spaces by identifying their basepoints. Consider two pointed topological spaces (X, x_0) and (Y, y_0), where X and Y are topological spaces equipped with distinguished points x_0 \in X and y_0 \in Y. The wedge sum X \vee Y is defined as the quotient space obtained from the disjoint union X \sqcup Y, which consists of all pairs (p, 0) with p \in X and (q, 1) with q \in Y, by imposing the equivalence relation that identifies x_0 with y_0, specifically (x_0, 0) \sim (y_0, 1), while leaving all other points distinct. This quotient space inherits the quotient topology from the disjoint union topology on X \sqcup Y, where open sets are unions of open sets from X and Y respectively. The resulting space X \vee Y has a natural basepoint, denoted *, which is the equivalence class of the identified points.^[1] More explicitly, the underlying set of X \vee Y is the set (X \sqcup Y) / \sim, where \sim is the smallest equivalence relation containing the pair (x_0, 0) \sim (y_0, 1), and the topology is the finest one making the quotient map q: X \sqcup Y \to (X \sqcup Y)/\sim continuous. This construction ensures that the inclusions i_X: X \to X \vee Y and i_Y: Y \to X \vee Y, defined by p \mapsto [(p, 0)] and q \mapsto [(q, 1)], are continuous maps sending basepoints to *. The notation X \vee Y is standard, with the basepoint * understood implicitly in the pointed category. For a family of pointed spaces \{(X_\alpha, x_\alpha) \mid \alpha \in A\}, the wedge sum generalizes to \bigvee_{\alpha \in A} X_\alpha, formed similarly as the quotient of \coprod_{\alpha \in A} X_\alpha by identifying all x_\alpha to a single point.^[1] The wedge sum satisfies a universal property in the category of pointed topological spaces, denoted \mathbf{Top}_*, which consists of pointed spaces and basepoint-preserving continuous maps. Specifically, X \vee Y is the pushout (colimit) of the diagram

\begin{CD} * @>>> X \\ @VVV @VV{i_X}V \\ Y @>>i_Y> X \vee Y, \end{CD}

where the maps from the terminal pointed space * (a singleton) to X and Y send the point to the respective basepoints. This means that for any pointed space Z and basepoint-preserving maps f: X \to Z, g: Y \to Z, there exists a unique basepoint-preserving map h: X \vee Y \to Z such that h \circ i_X = f and h \circ i_Y = g. In the broader context, this positions the wedge sum as the coproduct in \mathbf{Top}_*.^[2]^[3]

Categorical Wedge Sum

In the category of pointed sets, denoted \mathrm{Set}_*, the wedge sum of two pointed sets (X, x_0) and (Y, y_0) is the coproduct, constructed as the disjoint union X \sqcup Y with the basepoints identified via the equivalence x_0 \sim y_0. This operation equips the resulting object with a unique basepoint and satisfies the universal property of coproducts in \mathrm{Set}_*. The universal property of the wedge sum asserts that for any pointed set Z, the set of pointed morphisms \mathrm{Hom}_*(X \vee Y, Z) is naturally isomorphic to the fiber product \mathrm{Hom}_*(X, Z) \times_{\mathrm{Hom}_*(*, Z)} \mathrm{Hom}_*(Y, Z), where the fiber product is taken over morphisms that fix the basepoint, and * denotes the terminal pointed set (a singleton). This isomorphism ensures that any pair of pointed morphisms from X and Y to Z that agree on the basepoints factors uniquely through the wedge sum X \vee Y. This construction extends to any category with coproducts where objects are equipped with basepoints, such as the category of pointed topological spaces \mathrm{Top}_* or the category of pointed simplicial sets s\mathrm{Set}_*. In these settings, the wedge sum X \vee Y serves as the coproduct, with inclusions i_X: X \to X \vee Y and i_Y: Y \to X \vee Y that are cofibrations in the model category structure. The notation X \vee Y is consistently used to denote this categorical coproduct across such pointed categories.^[1]

Examples

Basic Examples

One of the simplest examples of the wedge sum is the space S^1 \vee S^1, formed by taking two circles, each pointed at a basepoint, and identifying those basepoints to a single point. This construction yields a topological space known as the figure-eight or infinity symbol (\infty), consisting of two loops joined at their common intersection point.^[1] Another basic illustration is the wedge sum of two closed intervals, [0,1] \vee [0,1], where each interval is pointed at its endpoint 0. Identifying the two basepoints 0 results in a space topologically equivalent to a V-shaped graph, with the vertex at the glued point and the free endpoints at 1 for each interval. This forms a simple 1-dimensional complex.^[1] For discrete spaces, consider the wedge sum of two pointed singletons, \{a\} \vee \{b\}, where each is a discrete point serving as its own basepoint. The identification of a and b collapses the space to a single point, though in the context of bouquets or multiple such attachments, it simplifies to a central point with attached trivial "legs" that do not alter the topology beyond the discrete union.^[1] In all these cases, the topology near the basepoint is determined by the union of neighborhoods from each summand, excluding the basepoint itself, which ensures that open sets around the wedge point combine the local structures without additional identifications. This gluing preserves the path components away from the basepoint while creating a unique junction at the identified point.^[1] To build intuition for algebraic invariants, the fundamental group of S^1 \vee S^1 is the free group on two generators, \mathbb{Z} * \mathbb{Z}, corresponding to the independent loops around each circle; this arises intuitively from the disjoint paths in each circle, amalgamated only at the basepoint.^[1]

Advanced Examples

One advanced construction is the infinite wedge sum \bigvee_{n \in \mathbb{N}} X_n of countably many pointed spaces (X_n, x_n), formed by taking the quotient of their disjoint union under the identification of all basepoints x_n to a single point. A representative example is the countable infinite wedge of circles \bigvee_{n=1}^\infty S^1, whose fundamental group is the free group on countably infinitely many generators.^[1] The Hawaiian earring provides a pathological variant of an infinite wedge of circles, constructed as the subspace of \mathbb{R}^2 consisting of circles of radius $1/n centered at (1/n, 0) for n \in \mathbb{N}, all passing through the origin; unlike the standard CW-complex wedge sum, this embedding induces a subspace topology that yields an uncountable fundamental group, highlighting subtleties in infinite constructions.^[1] The wedge sum S^n \vee S^m of spheres of distinct dimensions n \neq m (with n, m \geq 1) has reduced homology groups \tilde{H}_k(S^n \vee S^m) \cong \mathbb{Z} for k = n, m and zero otherwise, so its homotopy type is not equivalent to that of a single sphere S^k for any k.^[1] In the category of simplicial sets, the wedge sum of pointed simplicial sets is their coproduct, and geometric realization preserves this operation; for instance, the wedge \bigvee_{n=1}^\infty \Sigma S^1_n of countably many simplicial circles (where \Sigma denotes suspension) models the free product of their homotopy groups, facilitating computations of higher homotopy groups in combinatorial topology.^[3] A pathological case arises in the uncountable wedge sum \bigvee_{\alpha \in A} I_\alpha of uncountably many closed intervals I_\alpha = [0,1] (pointed at 0), which, while Hausdorff if each I_\alpha is, fails to be locally compact at the basepoint since no neighborhood of the basepoint is contained in a compact subset, in contrast to the countable case where local compactness is preserved. Wedge sums play a role in constructing Moore spaces M(\mathbb{Z}/m\mathbb{Z}, n), which are cell complexes with prescribed homology \tilde{H}_n(M(\mathbb{Z}/m\mathbb{Z}, n)) \cong \mathbb{Z}/m\mathbb{Z} and vanishing elsewhere; these are built by attaching an (n+1)-cell to a wedge of n-spheres via a degree-m map.^[1] Similarly, in CW-complexes, the quotient of the n-skeleton by the (n-1)-skeleton is homotopy equivalent to a wedge of n-spheres, one per n-cell, aiding the decomposition of suspensions.^[1]

Properties

Algebraic Properties

The wedge sum is the coproduct in the category of pointed topological spaces (Top_*), rendering it a functorial operation on pointed objects that respects the categorical structure, including the preservation of colimits where defined in the underlying category.^[4] As such, continuous basepoint-preserving maps f: X \to X' and g: Y \to Y' induce a canonical map f \vee g: X \vee Y \to X' \vee Y' on the wedge sum.^[1] The operation is associative up to canonical homeomorphism: for pointed spaces X, Y, and Z, there exists a basepoint-preserving homeomorphism (X \vee Y) \vee Z \cong X \vee (Y \vee Z).^[1] It is also commutative: X \vee Y \cong Y \vee X via a basepoint-preserving homeomorphism that swaps the components after identification of basepoints.^[1] The terminal pointed space, consisting of a single point *, acts as the neutral (identity) element under wedge sum: X \vee * \cong X for any pointed space X, with the isomorphism induced by the inclusion of the basepoint.^[1] In the category of pointed sets, the wedge sum X \vee Y is the quotient of the disjoint union X \sqcup Y by the identification of the basepoints, yielding a cardinality |X \vee Y| = |X| + |Y| - 1 for finite pointed sets X and Y.^[4] Unlike the coproduct in the unpointed category of topological spaces, which distributes over products, the wedge sum does not distribute over the categorical product of pointed spaces. However, in the context of pointed topological spaces, the smash product distributes over the wedge sum: for pointed spaces X, Y, and Z, there is a canonical homeomorphism X \wedge (Y \vee Z) \cong (X \wedge Y) \vee (X \wedge Z).^[5]

Homotopy Properties

The wedge sum operation preserves homotopy equivalences between basepoint-preserving maps. Specifically, if f: X \to X' and g: Y \to Y' are basepoint-preserving homotopy equivalences of pointed topological spaces, then the induced map f \vee g: X \vee Y \to X' \vee Y' is also a homotopy equivalence.^[1] This follows from the fact that homotopy equivalences induce isomorphisms on all homotopy groups, and the wedge sum construction respects these isomorphisms, as detailed in classical treatments of pointed spaces. In the homotopy category of pointed topological spaces, denoted \mathbf{ho}(\mathbf{Top}_*), the wedge sum serves as the coproduct. Consequently, the homotopy groups of a wedge sum decompose as direct sums for dimensions greater than or equal to 2: if X and Y are path-connected and well-pointed (i.e., the inclusions of basepoint neighborhoods are cofibrations), then \pi_n(X \vee Y) \cong \pi_n(X) \oplus \pi_n(Y) for n \geq 2.^[1] For the fundamental group, under the hypotheses of the Seifert–van Kampen theorem (path-connected open neighborhoods of the basepoints deformation retracting to the basepoints with path-connected intersection), the decomposition is instead the free product: \pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y).^[1] This additivity extends to finite wedges of spaces and aligns with the coproduct structure in the homotopy category, where morphisms factor uniquely through the wedge. Regarding long exact sequences, the wedge sum induces splittings in certain contexts, such as the long exact sequence of a pair or excision sequences in homology. For instance, the reduced homology of a finite wedge \tilde{H}_n(\bigvee_\alpha X_\alpha) is the direct sum \bigoplus_\alpha \tilde{H}_n(X_\alpha), which implies that associated long exact sequences split additively.^[1] This splitting behavior reflects the disjoint cellular structure away from the basepoint, facilitating computations in homotopy theory. If X and Y are finite CW-complexes with basepoints chosen as 0-cells, then X \vee Y is also a finite CW-complex, obtained by attaching the cells of X and Y disjointly except at the shared basepoint.^[1] The cell structure is the union of the individual skeletons, preserving the finite dimensionality and enabling inductive arguments on homotopy and homology groups.^[1] In model category structures on pointed topological spaces, such as the Serre model structure on \mathbf{Top}_*, the wedge sum preserves weak homotopy equivalences. That is, if f: X \to X' and g: Y \to Y' are weak homotopy equivalences (inducing isomorphisms on all \pi_n), then f \vee g is a weak homotopy equivalence, as the wedge is a pushout along cofibrations (the inclusions of the basepoint), and model category axioms ensure that such pushouts preserve weak equivalences.^[6] This property holds for finite wedges and underpins the coproduct in the localized homotopy category.^[7]

Applications and Relations

In Homotopy Theory

In stable homotopy theory, the wedge sum of spectra plays a central role as the coproduct in the stable homotopy category. For spectra X and Y, their wedge sum X \vee Y is defined levelwise by taking the wedge of the underlying spaces at each dimension, preserving the structure maps, and this construction yields the categorical coproduct.^[8] This coproduct property facilitates decompositions and computations in the category, where homotopy groups of finite wedges add directly: \pi_k(X \vee Y) \cong \pi_k(X) \oplus \pi_k(Y).^[9] The wedge sum is particularly instrumental in the Adams spectral sequence, a key tool for computing stable homotopy groups of spheres and other spectra, as it allows resolution of spectra into wedges of Eilenberg-MacLane spectra or suspensions thereof, simplifying the E_2-term via Ext groups in the Steenrod algebra.^[10] The wedge sum also relates to suspensions through iterated constructions in desuspensions and dual decompositions. In the context of Moore-Postnikov towers, the dual Moore decomposition expresses simply connected spaces as homotopy colimits of wedges of suspended Moore spaces, where each Moore space M(\mathbb{Z}/m, n) captures torsion in homology, and the wedge sums assemble the filtration quotients under suspension. This splitting, established by Milnor for the suspension of infinite loop spaces, extends to general desuspensions in the stable range, enabling inductive computations of homotopy types via wedge decompositions.^[11] For Eilenberg-MacLane spaces K(G, n) and K(H, n) with n \geq 2, the wedge sum K(G, n) \vee K(H, n) has homotopy groups \pi_k(K(G, n) \vee K(H, n)) \cong G \oplus H for k = n and trivial otherwise in that dimension, as the spaces are simply connected and homotopy groups add under wedges for dimensions at or above the connectivity. This direct sum structure proves useful in analyzing cohomology rings, since the reduced cohomology of the wedge is the direct sum of the individual reduced cohomologies, \tilde{H}^*(K(G, n) \vee K(H, n); \mathbb{Z}) \cong \tilde{H}^*(K(G, n); \mathbb{Z}) \oplus \tilde{H}^*(K(H, n); \mathbb{Z}), facilitating computations of cup products and operations in generalized cohomology theories.^[1] Infinite wedges arise in profinite completions and shape theory to model wild compacta, such as the Hawaiian earring, whose shape is that of the infinite wedge of circles but with Čech homotopy groups incorporating profinite limits over finite subwedges. In shape theory, the pro-homotopy type of an infinite wedge captures the inverse system of finite approximations, relevant for profinite completions of fundamental groups in one-dimensional spaces.^[12]

Comparison to Other Constructions

The wedge sum of two pointed topological spaces X and Y, denoted X \vee Y, is constructed as the quotient of their disjoint union X \sqcup Y by identifying the basepoints x_0 \in X and y_0 \in Y to a single point.^[1] This identification distinguishes it from the disjoint union, which is the coproduct in the category of unpointed topological spaces and preserves the separate components without merging any points; in contrast, the natural map X \sqcup Y \to X \vee Y collapses the basepoints, potentially altering the homotopy type unless the basepoints are contractible.^[1] For instance, the reduced homology of the wedge sum satisfies \tilde{H}_n(X \vee Y) \cong \tilde{H}_n(X) \oplus \tilde{H}_n(Y) for all n, reflecting a direct sum structure, whereas the (unreduced) cohomology of the disjoint union satisfies H^n(X \sqcup Y; R) \cong H^n(X; R) \times H^n(Y; R) for all n.^[1] In comparison to the smash product X \wedge Y, defined as the quotient (X \times Y) / (X \times \{y_0\} \cup \{x_0\} \times Y), the wedge sum serves as the coproduct in the category of pointed spaces, while the smash product acts as a reduced tensor product, particularly in the stable homotopy category.^[1] The smash product collapses the entire wedge sum embedded in the product space, leading to different homotopy types; for example, S^1 \wedge S^1 \simeq S^2, but S^1 \vee S^1 is homotopy equivalent to a figure-eight, which is not homeomorphic to S^2.^[1] The wedge sum requires pointed spaces and preserves the basepoint structure through continuous maps that send basepoints to basepoints, whereas a one-point union in the unpointed category adjoins an external point to each space and identifies those added points without regard for existing basepoints.^[13] This makes the wedge sum the categorical coproduct in pointed topological spaces, ensuring compatibility with basepoint-preserving morphisms, unlike the unpointed variant which lacks this preservation.^[2] Algebraically, the wedge sum is analogous to the free product in the category of groups, where the fundamental group satisfies \pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y) under suitable conditions like path-connectedness and open neighborhoods deformation retracting to basepoints, as given by van Kampen's theorem.^[1] For abelian groups, it corresponds to the direct sum, mirroring the reduced homology isomorphism \tilde{H}_n(X \vee Y) \cong \tilde{H}_n(X) \oplus \tilde{H}_n(Y), which acts as a "one-point" version of these operations by amalgamating generators at the basepoint.^[1] Unlike the product topology, which preserves connectedness (the product of connected spaces is connected), the wedge sum does not inherently do so; if the basepoints lie in non-contractible components, it connects only those specific components while leaving others disconnected, potentially reducing the total number of path components compared to the original spaces.^[1]