Reduced HomologyEdit

Reduced homology is a variant of homology theory that refines the ordinary construction by removing the basepoint’s contribution. Defined for pointed spaces, it gives a cleaner account of how connected components and higher-dimensional holes interact, and it plays a central role in results about suspensions, wedge sums, and long exact sequences for pairs. In practical terms, it agrees with ordinary homology in positive degrees, but in degree zero it records the number of components minus one, which simplifies many formulas and proofs.

Definition Reduced homology is built from the reduced chain complex \tilde{C}*(X) associated to a pointed space (X, x0). Concretely, for n > 0 one takes \tilde{C}_n(X) = C_n(X), the usual singular chain group, while in degree zero one defines \tilde{C}_0(X) as the kernel of the augmentation ε: C_0(X) → Z that sends each 0-simplex to 1. The boundary maps are the same as in the ordinary chain complex, and the reduced homology groups are \tilde{H}_n(X) = H_n(\tilde{C}*(X)). This construction is standard in texts on singular homology and chain complex theory, and it uses the idea of focusing on the basepoint to kill the trivial contribution from components.

Basic properties - Functoriality: For a based map f: (X, x0) → (Y, y0), there is a induced homomorphism \tilde{f}*: \tilde{H}_n(X) → \tilde{H}_n(Y). See functor-style behavior in homology theory. - Relation to unreduced homology: For n > 0, there is a natural isomorphism \tilde{H}_n(X) ≅ H_n(X). In degree zero, one has 0 → \tilde{H}_0(X) → H_0(X) → Z → 0, where Z is the homology of the point, so \tilde{H}_0(X) ≅ H_0(X)/Z ≅ Z^{k-1} if X has k path components. - A point has trivial reduced homology: \tilde{H}_n(pt) = 0 for all n. This reflects the idea that a single component with no holes contributes nothing after the “basepoint removal.” - Wedge sums: For a family of pointed spaces {X_i}, one has \tilde{H}_n(\bigvee_i X_i) ≅ ⊕_i \tilde{H}_n(X_i) for n > 0. - Suspension: There is a suspension isomorphism \tilde{H}_n(ΣX) ≅ \tilde{H}{n-1}(X), which is particularly convenient for translating information about a space into information about its suspension suspension. - Pair spaces and long exact sequences: Reduced homology participates in a long exact sequence for pairs ... → \tilde{H}n(A) → \tilde{H}_n(X) → \tilde{H}_n(X,A) → \tilde{H}{n-1}(A) → ... mirroring the unreduced case but with the basepoint-normalization built in. See the discussion of relative homology and the long exact sequence in the literature.

Computations and examples - Spheres: For the n-sphere, one has \tilde{H}_k(S^n) ≅ Z if k = n, and 0 otherwise. This is the same as ordinary homology in positive degrees, but with the degree-zero nuance absorbed into the basepoint convention. See S^n for geometric intuition. - Intervals and contractible spaces: For a contractible space like the interval I, all reduced homology groups vanish: \tilde{H}_n(I) = 0 for all n. - Disjoint unions: If X is a disjoint union of k points, then H_0(X) ≅ Z^k while \tilde{H}_0(X) ≅ Z^{k-1}. In positive degrees, reduced and unreduced homology agree (both zero for such a space). - Wedge sums of spheres: By the wedge-sum formula, \tilde{H}_n(\bigvee_i S^{m_i}) ≅ ⊕_i \tilde{H}_n(S^{m_i}), which is particularly tidy in the case of spheres.

Relationships to other theories - Coherence with basepoints: The definition on pointed spaces makes reduced homology natural for statements that involve basepoints, such as the suspension isomorphism and wedge sums. When the basepoint is absent, one often rephrases results in terms of unreduced homology or uses reduced homology with a chosen basepoint. - Coefficients: Just as with ordinary homology, reduced homology can be defined with coefficients in a abelian group G, yielding \tilde{H}_n(X; G). The primary structural relationships carry over with the appropriate changes in coefficients. - Relation to cohomology: The dual theory, reduced cohomology, has analogous definitions and properties, including reduced versions of universal coefficient theorems and Kunneth-type formulas tailored to cohomology.

Historical notes and conventions Reduced homology arose as a convenient refinement to streamline statements about wedge sums, suspensions, and pair spaces. In some introductory treatments, the emphasis is on how reduced homology makes the basepoint behave like a neutral element, producing cleaner exact sequences and easier computations in a range of standard examples. Different texts emphasize different pedagogical routes: some introduce reduced homology early to motivate the suspension and wedge-sum results, while others prefer to delay the discussion until after ordinary homology is well established. The core mathematics is the same, and the choice of framework often comes down to which abstractions a reader finds most natural.

See also - homology - singular homology - simplicial homology - relative homology - pair (X,A) - chain complex - augmentation - S^n - suspension