The group of concordances of the disc, the diffeomorphisms of \(D^n\times I\) that are the identity on the bottom edge \(D^n\times \{0\}\).
| \( D^0\) | \( D^1\) | \( D^2\) | \( D^3\) | \( D^4\) | \( D^5\) | \( D^6\) | \( D^7\) | \( D^8\) | \( D^9\) | \( D^{10}\) | \( D^{11}\) | \( D^{12}\) | \( D^{13}\) | \( D^{14}\) | \( D^{15}\) | \( D^{16}\) | \( D^{17}\) | \( D^{18}\) | \( D^{19}\) | \( D^{20}\) | \( D^{21}\) | \( D^{n}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\pi_0\) | \(\) | \(\) | \(\) | \(\) | \(0 \to\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | |
| \(\pi_1\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\mathbb{Z}_2 \to \) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
| \(\pi_2\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(0\to \) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
| \(\pi_3\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\mathbb{Z}\to \) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
| \(\pi_4\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(0\to \) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
| \(\pi_5\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\mathbb{Z}_2\to \) | \(\) | \(\) | \(\) |
| \(\pi_6\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
In the stable range for \(\pi_i\) we have the following further list. Since our technique requires knowning the K theory of the integers it fails in degrees that are congruent to \(2,6\) mod \(8\) as this is still an open problem. They may be known by other means, I just cant calculuate them. Rognes results are for regular primes, this can be stated as the prime does not divide the numerator of any Bernoilli number. The known even congruences for the K theory of the integers are given by just these numerators and so Rognes results are almost tight, that is sufficient in the case of computing the concordance groups. Havent thought about this enough though.
| \(i = \) | \( 7\) | \( 8\) | \( 9\) | \( 10\) | \( 11\) | \( 12\) | \(13\) | \(14\) | \(15\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|
| \(\pi_i\mathcal{C}(D^n)\) | \(\mathbb{Z}\oplus \mathbb{Z}_2\) | \(\mathbb{Z}_2^2\oplus \mathbb{Z}_8\) | \(\mathbb{Z}_2\oplus \mathbb{Z}_3\) | \(?\) | \(\mathbb{Z}\) | \(\mathbb{Z}_4\) | \(\mathbb{Z}_2^2\) | \(?\) | \(\mathbb{Z}_2^2\) | \(\mathbb{Z}_2^3\oplus \mathbb{Z}_{32} \oplus \mathbb{Z}_3\oplus \mathbb{Z}_5 \) |
Igusa says that the map \(\mathcal{C}(D^n) \to \mathcal{C}(D^{n+1})\) is \(k\) connected for \(n\geq \max(2k+7, 3k+4) \). The first option is bigger for \(k\leq 3\) and the second otherwise. Yellow represents this stable range.