Cross product of fundamental classes

Let \(M^m\) and \(N^n\) be closed \(\mathbb R\)-orientable manifolds with chosen fundamental classes \[ [M]\in H_m(M;\mathbb R), \qquad [N]\in H_n(N;\mathbb R). \] We show that \[ [M]\times[N]\in H_{m+n}(M\times N;\mathbb R) \] is a fundamental class.

Fix \((x,y)\in M\times N\). The images of \([M]\) and \([N]\) in local homology are the chosen local orientation generators \[ [M]_x\in H_m(M,M-\{x\};\mathbb R), \qquad [N]_y\in H_n(N,N-\{y\};\mathbb R). \] By naturality of cross products, the image of \([M]\times[N]\) in local homology at \((x,y)\) is the image of \[ [M]_x\times[N]_y \] under \[ H_m(M,M-x)\otimes H_n(N,N-y) \longrightarrow H_{m+n}\bigl(M\times N, (M-x)\times N\cup M\times(N-y)\bigr). \] The subspace on the right is exactly \(M\times N-\{(x,y)\}\).

Choose orientation-preserving coordinate balls about \(x\) and \(y\). By excision the preceding local cross product identifies with \[ H_m(\mathbb R^m,\mathbb R^m-0;\mathbb R) \otimes H_n(\mathbb R^n,\mathbb R^n-0;\mathbb R) \longrightarrow H_{m+n}(\mathbb R^{m+n},\mathbb R^{m+n}-0;\mathbb R). \] The cross product of the two standard local generators is the standard generator in dimension \(m+n\); this follows directly from the relative Künneth theorem, or from the product orientation on Euclidean space. Hence the image of \([M]\times[N]\) is a generator of local homology at every \((x,y)\).

Therefore \([M]\times[N]\) is a fundamental class of \(M\times N\).