Excision and Mayer–Vietoris

For a pair $(X,A)$, suppose $U\subseteq A$ and its closure in X lies inside the interior of A in X. Then inclusion $(X\setminus U,A\setminus U)\to(X,A)$ induces isomorphisms in singular homology with fixed coefficients. This is a standard sufficient excision hypothesis. The region removed is already contained well inside the part being treated relatively. The theorem does not assert that arbitrary points can be deleted from X without changing ordinary homology.

A proof subdivides singular simplices until each small piece lies in a suitable member of the relevant cover, then compares the small-chain complex with the full singular complex using chain homotopies. Compactness of each individual standard simplex supports the finite subdivision argument; the entire space need not be compact. Excision is therefore local in a carefully controlled algebraic sense. The boundary operation must remain compatible with subdivision, which is why simply discarding every simplex crossing a cut would not supply a valid proof.

Let $X=U\cup V$ with U and V open. The Mayer–Vietoris sequence relates the homology of the overlap, the direct sum of the two pieces, and the union. Use the map $(i_*,-j_*)$ from the overlap and the sum of the two inclusion maps into X. The composite is zero because the two ways to include the overlap in X agree. A connecting map then detects cycles in X assembled from chains in the two pieces whose boundaries cancel on the overlap.

The open-cover assumption guarantees that subdivision can reduce chains to pieces supported within the cover. CW subcomplex versions also exist under suitable excisive conditions, but no arbitrary closed cover is being asserted here. Exactness supplies kernels and cokernels of specific inclusion maps; the groups of the overlap alone are not enough. In low degree, the number and arrangement of overlap components can create a loop in the union even when both open pieces are individually contractible.

For connected contractible pieces, each overlap component maps to the single degree-zero generator of each piece. Under the sign convention above, its image is the same ordered pair with opposite signs. Differences between overlap-component generators then lie in the kernel, and exactness can identify these differences with first-homology classes of the union. This calculation uses ordinary H₀; reduced versions package the same information differently, but switching conventions halfway would remove the kernel one is trying to compute.

For a sphere covered by slightly enlarged hemispheres, the overlap deformation retracts to an equator while each piece is contractible. The reduced sequence shifts positive homology by one, allowing sphere groups to be calculated inductively. A cover by disjoint hemispheres would miss the equator and would not cover the connected sphere. The example illustrates the general habit: establish an actual cover and overlap homotopy types first, then determine the maps, and only then read exactness to infer the unknown groups.