Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$.

Let $k$ be an infinite field. Let $R$ be the semi-local ring of a finite family of closed points on a $k$-smooth affine irreducible variety, let $K$ be the fraction field of $R$, and let $G$ be a reductive simple simply connected $R$-group scheme isotropic over $R$. Our Theorem 1.1 states that for any Noetherian $k$-algebra $A$ the kernel of the map

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$

$R$

$K$

$A=k$

$R$

In this paper we give lower bounds for the Mislin genus of the symplectic groups $\mathrm{Sp}(m)$. This result appears to be the exact analogue of Zabrodsky's theorem concerning the special unitary groups $\mathrm{SU}(n)$ . It is achieved by the determination of the stable genus of the quasi-projective quaternionic spaces $Q\mathbb{H}(m)$ , following the approach of McGibbon. It leads to a symplectic version of Zabrodsky's conjecture, saying that these lower bounds are in fact the exact cardinality of the genus sets. The genus of $\mathrm{Sp}(2)$ is well known to contain exactly two elements. We show that the genus of $\mathrm{Sp}(3)$ has exactly 32 elements and see that the conjecture is true in these two cases.

Independently, we also show that any homotopy type in the genus of $\mathrm{Sp}(m)$ fibers over the sphere $S^{4m-1}$ with fiber in the genus of $\mathrm{Sp}(m-1)$ , and that any homotopy type in the genus of $\mathrm{SU}(n)$ fibers over the sphere $S^{2n-1}$ with fiber in the genus of $\mathrm{SU}(n-1)$ . Moreover, these fibrations are principal with respect to some appropriate loop structures on the fibers. These constructions permit us to produce particular spaces realizing the lower bounds obtained.

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P → X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G$_2$ type (we consider both the coarse moduli space and the moduli stack).