The edge-regular complete maps

A map is called edge-regular if it is edge-transitive but not arc-transitive. In this paper, we show that a complete graph K n {K}_{n} has an orientable edge-regular embedding if and only if n = p d > 3 n={p}^{d}\gt 3 with p an odd prime such that p d ≡ 3 {p}^{d}\equiv 3 ( mod 4 ) (\mathrm{mod}\hspace{.25em}4) . Furthermore, K p d {K}_{{p}^{d}} has p d − 3 4 d ϕ ( p d − 1 2 ) \tfrac{{p}^{d}-3}{4d}\hspace{0.25em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) non-isomorphic orientable edge-regular embeddings.

The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

Generic immersions and totally real embeddings

We show that, for a closed orientable [Formula: see text]-manifold, with [Formula: see text] not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex [Formula: see text]-space ensures the existence of a totally real embedding into complex [Formula: see text]-space. This implies that a closed orientable [Formula: see text]-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex [Formula: see text]-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

CR regular embeddings and immersions of compact orientable 4-manifolds into ℂ3

We show that a compact orientable 4-manifold M has a CR regular immersion into ℂ3 if and only if both its first Pontryagin class p1(M) and its Euler characteristic χ(M) vanish, and has a CR regular embedding into ℂ3 if and only if in addition the second Stiefel–Whitney class w2(M) vanishes.