Let \(X\) be a topological space and \(A,B\subset X\) open subspaces with \(X=A\cup B\). What does the Mayer–Vietoris theorem say about the homology groups of \(X\) and its subspaces?
Solution
For an open cover \(X=A\cup B\), Mayer–Vietoris gives a natural long exact sequence \[ \cdots\to H_n(A\cap B) \xrightarrow{(i_*,-j_*)} H_n(A)\oplus H_n(B) \xrightarrow{k_*+\ell_*} H_n(X) \xrightarrow{\partial} H_{n-1}(A\cap B)\to\cdots, \] where \(i,j\) are the inclusions of \(A\cap B\) into \(A,B\), and \(k,\ell\) are the inclusions of \(A,B\) into \(X\). The sequence continues through degree \(0\): \[ H_1(X)\to H_0(A\cap B)\to H_0(A)\oplus H_0(B)\to H_0(X)\to0. \] If \(A\cap B\ne\varnothing\), the same maps give an exact sequence in reduced homology, ending in \(\widetilde H_0(A)\oplus\widetilde H_0(B)\to\widetilde H_0(X)\to0\).