Brunnian braids over the 2-sphere and Artin combed form

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

Bounding size of homotopy groups of Spheres

Let p be prime. We prove that, for n odd, the p-torsion part of πq(Sn) has cardinality at most $p^{2^{{1}/({p-1})(q-n+3-2p)}}$ and hence has rank at most 21/(p−1)(q−n+3−2p). for p = 2, these results also hold for n even. The best bounds proven in the existing literature are $p^{2^{q-n+1}}$ and 2q−n+1, respectively, both due to Hans–Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.

Stable homotopy groups of spheres

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

Classification of knotted tori

For a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

The group of sphere-trivial braids

We give a presentation for the group of Artin braids that become trivial when considered over the [Formula: see text]-sphere and explore the relation to Brunnian braids and homotopy groups of spheres.