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On Nori's Fundamental Group Scheme
We determine the quotient category which is the representation category of
the kernel of the homomorphism from Nori's fundamental group scheme to its
\'etale and local parts. Pierre Deligne pointed out an error in the first
version of this article. We profoundly thank him, in particular for sending us
his enlightning example reproduced in Remark 2.4 2).Comment: 29 page
Dressing preserving the fundamental group
In this note we consider the relationship between the dressing action and the
holonomy representation in the context of constant mean curvature surfaces. We
characterize dressing elements that preserve the topology of a surface and
discuss dressing by simple factors as a means of adding bubbles to a class of
non finite type cylinders.Comment: 36 pages, 1 figur
On non Fundamental Group Equivalent Surfaces
In this paper we present an example of two polarized K3 surfaces which are
not Fundamental Group Equivalent (their fundamental groups of the complement of
the branch curves are not isomorphic; denoted by FGE) but the fundamental
groups of their related Galois covers are isomorphic. For each surface, we
consider a generic projection to CP^2 and a degenerations of the surface into a
union of planes - the "pillow" degeneration for the non-prime surface and the
"magician" degeneration for the prime surface. We compute the Braid Monodromy
Factorization (BMF) of the branch curve of each projected surface, using the
related degenerations. By these factorizations, we compute the above
fundamental groups. It is known that the two surfaces are not in the same
component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove
that furthermore they are not FGE equivalent, and thus they are not of the same
Braid Monodromy Type (BMT) (which implies that they are not a projective
deformation of each othe
Parallizable manifolds and the fundamental group
ntroduction. Low-dimensional topology is dominated by the fundamental group. However, since every finitely presented group is the fundamental group of some closed 4-manifold, it is often stated that the effective influence of π1 ends in dimension three. This is not quite true, however, and there are some interesting border disputes. In this paper, we show that, by imposing the extra condition of parallelizability on the tangent bundle, the dominion of π1 is extended by an extra dimension
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