51 research outputs found
On the generic triangle group
We introduce the concept of a generic Euclidean triangle and study the
group generated by the reflection across the edges of . In
particular, we prove that the subgroup of all translations in
is free abelian of infinite rank, while the index 2 subgroup of all
orientation preserving transformations in is free metabelian of rank
2, with as the commutator subgroup. As a consequence, the group
cannot be finitely presented and we provide explicit minimal infinite
presentations of both and . This answers in the affirmative
the problem of the existence of a minimal presentation for the free metabelian
group of rank 2. Moreover, we discuss some examples of non-trivial relations in
holding for given non-generic triangles .Comment: 21 pages, 6 figure
The complex of pant decompositions of a surface
We exhibit a set of edges (moves) and 2-cells (relations) making the complex
of pant decompositions on a surface a simply connected complex. Our
construction, unlike the previous ones, keeps the arguments concerning the
structural transformations independent from those deriving from the action of
the mapping class group. The moves and the relations turn out to be supported
in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of
boundary components), illustrating in this way the so called Grothendieck
principle.Comment: Minor changes in the introductio
Branched coverings of and other basic 4-manifolds
We give necessary and sufficient conditions for a 4-manifold to be a branched
covering of , , and , which are expressed in terms of the Betti numbers and the
intersection form of the 4-manifold.Comment: 16 pages, 1 figure, 19 reference
Lifting Braids
In this paper we study the homeomorphisms of the disk that are liftable with
respect to a simple branched covering. Since any such homeomorphism maps the
branch set of the covering onto itself and liftability is invariant up to
isotopy fixing the branch set, we are dealing in fact with liftable braids. We
prove that the group of liftable braids is finitely generated by liftable
powers of half-twists around arcs joining branch points. A set of such
generators is explicitly determined for the special case of branched coverings
from the disk to the disk. As a preliminary result we also obtain the
classification of all the simple branched coverings of the disk.Comment: 20 page
On branched covering representation of 4-manifolds
We provide new branched covering representations for bounded and/or
non-compact 4-manifolds, which extend the known ones for closed 4-manifolds.
Assuming to be a connected oriented PL 4-manifold, our main results are the
following: (1) if is compact with (possibly empty) boundary, there exists a
simple branched cover , where the 's are disjoint PL 4-balls, is the
number of boundary components of ; (2) if is open, there exists a simple
branched cover , where
is the end space of tamely embedded in . In
both cases, the degree and the branching set of can be assumed
to satisfy one of these conditions: (1) and is a properly
self-transversally immersed locally flat PL surface; (2) and is
a properly embedded locally flat PL surface. In the compact (resp. open) case,
by relaxing the assumption on the degree we can have (resp. ) as the
base of the covering. We also define the notion of branched covering between
topological manifolds, which extends the usual one in the PL category. In this
setting, as an interesting consequence of the above results, we prove that any
closed oriented topological 4-manifold is a 4-fold branched covering of .
According to almost-smoothability of 4-manifolds, this branched cover could be
wild at a single point.Comment: 16 pages, 9 figure
A universal ribbon surface in B^4
We construct an orientable ribbon surface F in B^4, which is universal in the
following sense: any compact orientable pl 4-manifold having a handle
decomposition with 0-, 1- and 2-handles can be represented as a cover of B^4
branched over F.Comment: 19 pages, 28 figures, 28 references. LaTeX 2.09 file. Uses:
amstext.sty amscd.sty geom.sty epsf.st
Compact Stein surfaces with boundary as branched covers of \B^4
We prove that Stein surfaces with boundary coincide up to orientation
preserving diffeomorphisms with simple branched coverings of \B^4 whose
branch set is a positive braided surface. As a consequence, we have that a
smooth oriented 3-manifold is Stein fillable iff it has a positive open-book
decomposition.Comment: 25 pages, 20 postscript figures. LaTeX file. Uses: geom.sty epsf.st
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