Table 1 The Euler characteristic of \(\:{B}_{N}\) and \(\:{B}_{N}-{E}_{N,m}\) and computed using our method, with calculated values shown in parentheses for precision. * and † denote the \(\:N\)-dimensional ball and spherical shell, respectively. For the 4-dimensional case, additional 2D manifolds embeddable in 4D space, the real projective plane (\(\:\text{R}{\text{P}}^{2})\) and the Klein bottle, are included.
\(N=2\) | |||||||
|---|---|---|---|---|---|---|---|
\(\:{B}_{2}\) | \(\:{B}_{2}-{E}_{\text{2,1}}\) | \(\:{B}_{2}-{E}_{\text{2,2}}\) | |||||
\(\:n\) | 1 (0.999)* | 2 (1.999) | 0 (0.000)† | ||||
\(N=3\) | |||||||
|---|---|---|---|---|---|---|---|
\(\:{B}_{3}\) | \(\:{B}_{3}-{E}_{\text{3,1}}\) | \(\:{B}_{3}-{E}_{\text{3,2}}\) | \(\:{B}_{3}-{E}_{\text{3,3}}\) | ||||
\(\:n\) | 1 (0.999)* | 2 (1.999) | 0 (0.000) | 2 (1.999)† | |||
\(N=4\) | |||||||
|---|---|---|---|---|---|---|---|
\(\:{B}_{4}\) | \(\:{B}_{4}-{E}_{\text{4,1}}\) | \(\:{B}_{4}-{E}_{\text{4,2}}\) | \(\:{B}_{4}-{E}_{\text{4,3}}\) | \(\:{B}_{4}-{E}_{\text{4,4}}\) | \(\:\text{R}{\text{P}}^{2}\) | Klein Bottle | |
\(\:n\) | 1 (0.999)* | 2 (1.999) | 0 (0.000) | 2 (1.997) | 0 (0.005)† | 1 (0.993) | 0 (0.000) |
